# Chern Classes and Generalized Riemann-Roch Theorems

Chern classes are an ubiquitous concept in mathematics, being part of algebraic geometry, algebraic topology, and differential geometry. In this post we discuss Chern classes in the context of algebraic geometry, where they are part of intersection theory (see Algebraic Cycles and Intersection Theory). Among the applications of the theory of Chern classes is a higher-dimensional generalization of the Riemann-Roch theorem (see More on Sheaves) called the Hirzebruch-Riemann-Roch theorem. There is an even further generalization called the Grothendieck-Riemann-Roch theorem, which concerns a morphism of nonsingular projective varieties $f:X\rightarrow Y$, and for which the Hirzebruch-Riemann-Roch theorem is merely the case where $Y$ is a point.

Let $X$ be a nonsingular projective variety, and let $A(X)$ be the Chow ring of $X$ (see Algebraic Cycles and Intersection Theory). Let $\mathcal{E}$ be a locally free  sheaf of rank $r$ on $X$.

We recall that locally free  sheaves correspond to vector bundles (see Vector Fields, Vector Bundles, and Fiber Bundles and More on Sheaves). Therefore, their fibers are isomorphic to $\mathbb{A}^{r}$. The projective bundle $\mathbb{P}(\mathcal{E})$ associated to the locally free sheaf $\mathcal{E}$ is essentially obtained by replacing the fibers with projective spaces $\mathbb{A}\setminus\{0\}/k^{*}$ (see Projective Geometry).

Let $\xi\in A^{1}(\mathbb{P}(\mathcal{E}))$ be the class of the divisor corresponding to the twisting sheaf (see More on Sheaves) $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$. Let $\pi:\mathbb{P}(\mathcal{E})\rightarrow X$ be the projection of the fiber bundle $\mathbb{P}(\mathcal{E})$ to its “base space” $X$. Then the pullback $\pi^{*}:A^{i}(X)\rightarrow A^{i+r-1}(\mathbb{P}(\mathcal{E}))$ makes $A(\mathbb{P}(\mathcal{E}))$ into a free $A(X)$ module generated by $1$, $\xi$, $\xi^{2},...,\xi^{r-1}$.

We define the $i$-th Chern class $c_{i}(\mathcal{E})\in A^{i}(X)$ by the requirement that $c_{0}(\mathcal{E})=1$ and

$\displaystyle \sum_{i=0}^{r}(-1)^{i}\pi^{*}c_{i}(\mathcal{E}).\xi^{r-i}=0$

where the dot $.$ denotes the intersection product (see Algebraic Cycles and Intersection Theory).

Chern classes are associated to locally free sheaves, which, as we have already mentioned, correspond to vector bundles, and are elements of the Chow ring. We can therefore think of them as generalizing the correspondence between line bundles (vector bundles of dimension $1$) and elements of the Picard group, since, as mentioned in Algebraic Cycles and Intersection Theory, the Chow ring is kind of an analogue of the Picard group for higher dimensions.

We can also define the total Chern class

$\displaystyle c(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})...+c_{r}(\mathcal{E})$

and the Chern polynomial

$\displaystyle c_{t}(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})t+...+c_{r}(\mathcal{E})t^{r}$.

Chern classes satisfy the following important properties:

(i) If $\mathcal{E}$ is the line bundle $\mathcal{L}(D)$ associated to a divisor $D$, then $c_{t}=1+Dt$.

(ii) If $f:X'\rightarrow X$ is a morphism, and $\mathcal{E}$ is a locally free sheaf on $X$, then for each $i$,

$\displaystyle c_{i}(f^{*}\mathcal{E})=f^{*}c_{i}(\mathcal{E})$.

(iii) If $0\rightarrow\mathcal{E}'\rightarrow\mathcal{E}\rightarrow\mathcal{E}''\rightarrow 0$ is an exact sequence (see Exact Sequences) of locally free sheaves, then

$\displaystyle c_{t}(\mathcal{E})=c_{t}(\mathcal{E}')\cdot c_{t}(\mathcal{E}'')$

These three properties can also be considered as a set of axioms which define the Chern classes, instead of the definition that we gave earlier.

Another important property of Chern classes, which comes from the so-called splitting principle, allows us to factor the Chern polynomial into the Chern polynomials of line bundles, and so we have:

$c_{t}(\mathcal{E})=\prod_{i=1}^{r}(1+a_{i}t)$

The $a_{i}$ are called the Chern roots of $\mathcal{E}$.

We define the exponential Chern character (or simply Chern character) as

$\displaystyle \text{ch}(\mathcal{E})=\sum_{i=1}^{r}e^{a_{i}}$

and the Todd class as

$\displaystyle \text{td}(\mathcal{E})=\prod_{i=1}^{r}\frac{(a_{i})}{1-e^{-a_{i}}}$.

Now we can discuss the generalizations of the Riemann-Roch theorem. We first review the statement of the Riemann-Roch theorem for curves, but we restate it slightly in terms of the Euler characteristic.

The Euler characteristic of a coherent sheaf $\mathcal{E}$ on a projective scheme $X$ over a field $k$ is defined to be the alternating sum of the dimensions of the cohomology groups $H^{i}(X,\mathcal{F})$ (see Cohomology in Algebraic Geometry) as vector spaces over $k$.

$\displaystyle \chi(\mathcal{E})=\sum_{i}(-1)^{i}\text{dim}_{k}H^{i}(X,\mathcal{F})$.

Then we can state the Riemann-Roch theorem for curves as

$\chi(\mathcal{L}(D))=\text{deg}(D)+1-g$.

The connection of this formulation with the one we gave in More on Sheaves, where the left-hand side is given by $h^{0}(D)-h^{0}(K_{X}-D)$ is provided by the fact that $h^{0}(D)$ is the same as (and in fact defined as) $\text{dim}_{k}H^{0}(X, \mathcal{L}(D))$, together with the theorem known as Serre duality, which says that $H^{1}(X,\mathcal{L}(D))$ is dual to $H^{0}(X,\omega\otimes\mathcal{L}(D)^{\vee})$, where $\mathcal{L}(D)^{\vee}$ denotes the dual of the line bundle $\mathcal{L}(D)$.

The Hirzebruch-Riemann-Roch theorem says that

$\displaystyle \chi(\mathcal{E})=\text{deg}(\text{ch}(\mathcal{E}).\text{td}(\mathcal{T}_{X}))_{n}$

where $\mathcal{T}_{X}$ is the tangent bundle of $X$ (the dual of the cotangent bundle of $X$, as defined in More on Sheaves) and $(\quad)_{n}$ is the component of degree $n$ in $A(X)\otimes\mathbb{Q}$.

Finally we come to the even more general Grothendieck-Riemann-Roch theorem, but first we must introduce the Grothendieck group $K(X)$ of a scheme $X$, which eventually inspired the area of mathematics known as K-theory.

The Grothendieck group $K(X)$ of a scheme $X$ is defined to be the quotient of the free abelian group generated by the coherent sheaves on $X$ by the subgroup generated by expressions of the form $\mathcal{F}-\mathcal{F}'-\mathcal{F}''$ whenever there is an exact sequence

$\displaystyle 0\rightarrow\mathcal{F'}\rightarrow\mathcal{F}\rightarrow\mathcal{F''}\rightarrow 0$

of coherent sheaves on $X$. Intuitively, we may think of the Grothendieck group as follows. The isomorphism classes of vector bundles on $X$ form a commutative monoid under the operation of taking the direct sum of vector bundles (also called the Whitney sum). There is a way to obtain an abelian group from this monoid, called the group completion, and the abelian group we obtain is the Grothendieck group. The Chern classes and the Chern character are also defined on the Grothendieck group $K(X)$. In K-theory, the Grothendieck group $K(X)$ is also denoted $K_{0}(X)$.

If $f:X\rightarrow Y$ is a proper morphism (a morphism that is separable, of finite type, and universally closed, i.e. for every scheme $Z\rightarrow Y$ , the projection $X\times_{Y}Z\rightarrow Z$ maps closed sets to closed sets), we have a map $f_{!}:K(X)\rightarrow Y$ defined by

$\displaystyle f_{!}(\mathcal{F})=\sum_{i}(-1)^{i}R^{i}f_{*}(\mathcal{F})$

where the $R^{i}f_{*}$ are the higher direct image functors, which are defined as the right derived functors (The Hom and Tensor Functors) of the direct image functor $f_{*}$ (see Direct Images and Inverse Images of Sheaves).

The Grothendieck-Riemann-Roch theorem says that for any $x\in K(X)$, we have

$\displaystyle f_{*}(\text{ch}(x).\text{td}(\mathcal{T}_{X})=\text{ch}(f_{!}(x)).\text{td}(\mathcal{T}_{Y})$.

The Grothendieck-Riemann-Roch theorem is one of the most general versions of the Riemann-Roch theorem, a classic theorem whose origins date back to the 19th century. However, there are also other generalizations, such as the arithmetic Riemann-Roch theorem which is closely related to number theory, and the Atiyah-Singer index theorem which is closely related to physics. We leave these, and the many other details of the topics we have discussed in this post (along with the theory of Chern classes in the context of algebraic topology and differential geometry), to the references for now, until we can discuss them on this blog in the future.

The featured image on this post is a handwritten comment of Alexander Grothendieck, apparently from a lecture in 1971, featuring the Grothendieck-Riemann-Roch theorem.

References:

Chern Class on Wikipedia

Projective Bundle on Wikipedia

Hirzebruch-Riemann-Roch Theorem on Wikipedia

Grothendieck-Riemann-Roch Theorem on Wikipedia

Chern Classes: Part 1 on Rigorous Trivialities

Chern Classes: Part 2 on Rigorous Trivialities

The Chow Ring and Chern Classes on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, Stated on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, a Proof-Sketch on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

# The Field with One Element

##### Introduction: Analogies between Function Fields and Number Fields

As has been hinted at, and mentioned in passing, in several previous posts on this blog, there are important analogies between numbers and functions. The analogy can perhaps be made most explicit in the case of $\mathbb{Z}$ (the ring of ordinary integers) and $\mathbb{F}_{p}[t]$ (the ring of polynomials in one variable $t$ over the finite field $\mathbb{F}_{p}$). We also often say that the analogy is between $\mathbb{Q}$ (the field of rational numbers) and $\mathbb{F}_{p}(t)$ (the field of rational functions in one variable $t$ over the finite field $\mathbb{F}_{p}$), which are the respective fields of fractions of $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$. Recall also from Some Basics of Class Field Theory that $\mathbb{Q}$ and $\mathbb{F}_{p}(t)$ are examples of what we call global fields, together with their respective finite extensions.

Let us go back to $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$ and compare their similarities. They are both principal ideal domains, which means that all their ideals can be generated by a single element. They both have groups of units (elements which have multiplicative inverses) which are finite. They both have an infinite number of prime ideals (generated by prime numbers in the case of $\mathbb{Z}$, and by monic irreducible polynomials in the case of $\mathbb{F}_{p}[t]$), and finally, they share the property that their residue fields over these prime ideals are finite.

But of course, despite all these analogies, a rather obvious question still remains unanswered. Regarding this question we quote the words of the mathematician Christophe Soule:

“The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that $\text{Spec}(\mathbb{Z})$ (with a point at infinity added, as is familiar in Arakelov geometry) is like a (complete) curve, but over which field?”

This question led to the development of the idea of the “field with one element”, also written $\mathbb{F}_{1}$, or sometimes $\mathbb{F}_{\text{un}}$ (it’s a pun taken from “un”, the French word for “one”). Taken literally, there is no such thing  as a “field” with one element – the way we define a field, it must always have a “one” and a “zero”, and these two elements must be different. Instead, the idea of the “field with one element” is just a name for ideas that extend the analogy between function fields and number fields, as if this “field” really existed. The name itself has historical origins in the work of the mathematician Jacques Tits involving certain groups called Chevalley groups and Weil groups, where surprising results appear in the limit when the number of elements of the finite fields involved goes to one – but in most approaches now, the “field with one element” is not a field, and often has more than one element. The whole point is that these ideas may still work, even though the “field” itself may not even exist! As one might expect, in order to pursue these ideas one must think out of the box, and different mathematicians have approached this question in different ways.

In this post, we will look at four approaches to the field with one element, developed by the mathematicians Anton Deitmar, Christophe Soule, Bertrand Toen and Michel Vaquie, and James Borger. There are many more approaches besides these, but we will perhaps discuss them in future posts.

Note: Throughout this post it will be helpful to remind ourselves that since there exists a map from the integers $\mathbb{Z}$ to any ring, we can think of rings as $\mathbb{Z}$-algebras. One of the ways the idea of the field with one element is approached is by exploring what $\mathbb{F}_{1}$-algebras mean, if ordinary rings are $\mathbb{Z}$-algebras.

##### The Approach of Deitmar

Deitmar defines the “category of rings over $\mathbb{F}_{1}$” (this is the term Deitmar uses, but we can also think of this as the category of $\mathbb{F}_{1}$-algebras) as simply the category of monoids. A monoid $A$ is also written as $\mathbb{F}_{A}$ to emphasize its nature as a “ring over $\mathbb{F}_{1}$“. The “field with one element” $\mathbb{F}_{1}$ is simply defined to be the trivial monoid.

For an $\mathbb{F}_{1}$-ring $\mathbb{F}_{A}$ we define the base extension (see Grothendieck’s Relative Point of View) to $\mathbb{Z}$ by taking the “monoid ring” $\mathbb{Z}[A]$:

$\displaystyle \mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[A]$

We may think of this monoid ring as a ring whose elements are formal sums of elements of the monoid $A$ with integer coefficients, and with a multiplication provided by the multiplication on $A$, commuting with the scalar multiplication.

Meanwhile we also have the forgetful functor $F$ which simply “forgets” the additive structure of a ring, leaving us with a monoid under its multiplication operation. The base extension functor $-\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ is left adjoint to the forgetful functor $F$, i.e. for every ring $R$ and every $\mathbb{F}_{A}/\mathbb{F}_{1}$ we have $\text{Hom}_{\text{Rings}}(\mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}, R)\cong\text{Hom}_{\mathbb{F}_{1}}(\mathbb{F}_{A},F(R))$ (see also Adjoint Functors and Monads).

The important concepts of “localization” and “ideals” in the theory of rings, important to construct the structure sheaf of a variety or a scheme, have analogues in the theory of monoids. The idea is that they only make use of the multiplicative structure of rings, so we can forget the additive structure and consider monoids instead. Hence, we can define varieties or schemes over $\mathbb{F}_{1}$. Many other constructions of algebraic geometry can be replicated with only monoids instead of rings, such as sheaves of modules over the structure sheaf. Deitmar then defines the zeta function of a scheme over $\mathbb{F}_{1}$, and hopes to connect this with known ideas about zeta functions (see for example our discussion in The Riemann Hypothesis for Curves over Finite Fields).

Deitmar’s idea of using monoids is one of the earlier approaches to the idea of the field with one element, and has become somewhat of a template for other approaches. One may be able to notice the influence of Deitmar’s work in the other approaches that we will discuss in this post.

##### The Approach of Soule

Soule’s question, as phrased in his paper On the Field with One Element, is as follows:

“Which varieties over $\mathbb{Z}$ are obtained by base change from $\mathbb{F}_{1}$ to $\mathbb{Z}$?”

Soule’s approach to answering this question then makes use of three concepts. The first one is a suggestion from the early days of the development of the idea of the field with one element, apparently due to the mathematicians Andre Weil and Kenkichi Iwasawa, that the finite field extensions of the field with one element should consist of the roots of unity, together with zero.

The second concept is an important point that we only touched on briefly from Algebraic Spaces and Stacks, namely, that we may identify the functor of points of a scheme with the scheme itself. Now the functor of points of a scheme is uniquely determined by its values on affine schemes, and the category of affine schemes is the opposite category to the category of rings; therefore, we now redefine a scheme simply as a covariant functor from the category of rings to the category of sets, which is representable.

The third concept is the idea of an evaluation of a function at a point. Soule implements this concept by including a $\mathbb{C}$-algebra as part of his definition of a variety over $\mathbb{F}_{1}$, together with a natural transformation that expresses this evaluation.

We now give the details of Soule’s construction, proceeding in four steps. Taking into account the first concept mentioned earlier,  we consider the following expression, the base extension of $\mathbb{F}_{1^{n}}$ to $\mathbb{Z}$ over $\mathbb{F}_{1}$:

$\displaystyle \mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[T]/(T^{n}-1)=\mathbb{Z}[\mu_{n}]$

We shall also denote this ring by $R_{n}$. We can form a category whose objects are the finite tensor products of $R_{n}$, for $n\geq 1$, and we denote this category by $\mathcal{R}$.

An affine gadget over $\mathbb{F}_{1}$ is a triple $(\underline{X},\mathcal{A}_{X},e_{X})$ where $\underline{X}$ is a covariant functor from the category $\mathcal{R}$ to the category of sets, $\mathcal{A}_{X}$ is a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{X}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of affine gadgets consists of a natural transformation $\underline{\phi}:\underline{X}\rightarrow\underline{Y}$ and a morphism of algebras $\phi^{*}:\mathcal{A}_{X}\rightarrow\mathcal{A}_{Y}$ such that $f(\underline{\phi}(P))=(\phi^{*}(f))(P)$. A morphism $(\underline{\phi}, \phi^{*})$ is also called an immersion if $\underline{\phi}$ and $\phi^{*}$ are both injective.

An affine variety over $\mathbb{F}_{1}$ is an affine gadget $X=(\underline{X},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{X}(R)$ is finite, and

(ii) there exists an affine scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{G}(X_{\mathbb{Z}})$ with the universal property that for any other affine scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{G}(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{G}(V)$ such that $\varphi=\mathcal{G}(\varphi_{\mathbb{Z}})\circ i$.

An object over $\mathbb{F}_{1}$ is a triple $(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ where $\underline{\underline{X}}$ is a contravariant functor from the category of affine gadgets over $\mathbb{F}_{1}$$\mathcal{A}_{X}$ is once again a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{\underline{X}}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of objects is defined in the same way as a morphism of affine gadgets.

A variety over $\mathbb{F}_{1}$ is an object $X=(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{\underline{X}}(\text{Spec}(R))$ is finite, and

(ii) there exists a scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{O}b(X_{\mathbb{Z}})$ with the universal property that for any other scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{O}b(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{O}b(V)$ such that $\varphi=\mathcal{O}b(\varphi_{\mathbb{Z}})\circ i$.

Like Deitmar, Soule constructs the zeta function of a variety over $\mathbb{F}_{1}$, and furthermore explores connections with certain kinds of varieties called “toric varieties”, which are also of interest in other approaches to the field with one element, and the theory of motives (see The Theory of Motives).

##### The Approach of Toen and Vaquie

We recall from Grothendieck’s Relative Point of View that we call a scheme $X$ a scheme “over” $S$, or an $S$-scheme, if there is a morphism of schemes from $X$ to $S$, and if $S$ is an affine scheme defined as $\text{Spec}(R)$ for some ring $R$, we also refer to it as a scheme over $R$, or an $R$-scheme. We recall also every scheme is a scheme over $\text{Spec}(\mathbb{Z})$, or a $\mathbb{Z}$-scheme. The approach of Toen and Vaquie is to construct categories of schemes “under” $\text{Spec}(\mathbb{Z})$.

From Monoidal Categories and Monoids we know that rings are the monoid objects in the monoidal category of abelian groups, and abelian groups are $\mathbb{Z}$-modules.

More generally, for a symmetric monoidal category $(\textbf{C}, \otimes, \mathbf{1})$ that is complete, cocomplete, and closed (i.e. possesses internal Homs related to the monoidal structure $\otimes$, see again Monoidal Categories and Monoids), we have in $\textbf{C}$ a notion of monoid, for such a monoid $A$ a notion of an $A$-module, and for a morphism of monoids $A\rightarrow B$ a notion of a base change functor $-\otimes_{A}B$ from $A$-modules to $B$-modules.

Therefore, if we have a category $\textbf{C}$ with a symmetric monoidal functor $\textbf{C}\rightarrow \mathbb{Z}\text{-Mod}$, we obtain a notion of a “scheme relative to $\textbf{C}$” and a base change functor to $\mathbb{Z}$-schemes. This gives us our sought-for notion of schemes under $\text{Spec}(\mathbb{Z})$.

In particular, there exists a notion of commutative monoids (associative and with unit) in $\textbf{C}$, and they form a category which we denote by $\textbf{Comm}(\textbf{C})$. We define the category of affine schemes related to $\textbf{C}$ as $\textbf{Aff}_{\textbf{C}}:= \textbf{Comm}(\textbf{C})^{\text{op}}$.

These constructions satisfy certain properties needed to define a category of schemes relative to $(\textbf{C},\otimes,\mathbf{1})$, such as a notion of Zariski topology. A relative scheme is defined as a sheaf on the site $\textbf{Aff}_{\textbf{C}}$ provided with the Zariski topology, and which has a covering by affine schemes. The category of schemes obtained is denoted $\textbf{Sch}(\textbf{C})$. It is a subcategory of the category of sheaves on $\textbf{Aff}_{\textbf{C}}$ which is closed under the formation of fiber products and disjoint unions. It contains a full subcategory of affine schemes, given by the representable sheaves, and which is equivalent to the category $\textbf{Comm}(\textbf{C})^{\text{op}}$. The purely categorical nature of the construction makes the category $\textbf{Sch}(\textbf{C})$ functorial in $\textbf{C}$.

In their paper, Toen and Vacquie give six examples of their construction, one of which is just the ordinary category of schemes, while the other five are schemes “under $\text{Spec}(\mathbb{Z})$“.

First we let $(C,\otimes,\mathbf{1})=(\mathbb{Z}\text{-Mod},\otimes,\mathbb{Z})$, the symmetric monoidal category of abelian groups (for the tensor product). The category of schemes obtained $\mathbb{Z}\text{-Sch}$ is equivalent to the category of schemes in the usual sense.

The second example will be $(C,\otimes,\mathbf{1})=(\mathbb{N}\text{-Mod},\otimes,\mathbb{N})$ the category of commutative monoids, or abelian semigroups, with the tensor product, which could also be called $\mathbb{N}$-modules. The category of schemes in this case will be denoted $\mathbb{N}\text{-Sch}$, and the subcategory of affine schemes is equivalent to the opposite category of commutative semirings.

The third example is $(C,\otimes,\mathbf{1})=(\text{Ens},\times, *)$, the symmetric monoidal category of sets with the direct product. The category of relative schemes will be denoted $\mathbb{F}_{1}\text{-Sch}$, and we can think of them as schemes or varieties defined on the field with one element. By definition, the subcategory of affine $\mathbb{F}_{1}$-schemes is equivalent to the opposite category of commutative monoids.

We have the base change functors

$-\otimes_{\mathbb{F}_{1}}\mathbb{N}:\mathbb{F}_{1}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{N}}\mathbb{Z}:\mathbb{N}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

We can compose these base change functors and represent it with the following diagram:

$\text{Spec}(\mathbb{Z})\rightarrow\text{Spec}(\mathbb{N})\rightarrow\text{Spec}(\mathbb{F}_{1})$.

The final three examples of “schemes under $\text{Spec}(\mathbb{Z})$” given by Toen and Vaquie make use of ideas from “homotopical algebraic geometry“. Homotopical algebraic geometry is a very interesting subject that unfortunately we have not discussed much on this blog. Roughly, in homotopical algebraic geometry the role of rings in ordinary algebraic geometry is taken over by ring spectra – spectra (in the sense of Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) with a “smash product” operation. This allows us to make use of concepts from abstract homotopy theory. In this post we will only introduce some very basic concepts that we will need to discuss Toen and Vaquie’s examples, and leave the rest to the references.

We will need the concepts of $\Gamma$-spaces and simplicial sets. We define the category $\Gamma^{0}$ to be the category whose objects are “pointed” finite sets (a finite set where one element is defined to be the “basepoint”) and whose morphisms are maps of finite sets that preserve the basepoint. We also define the category $\Delta$ to be the category whose objects are finite ordered sets $[n]=\{0<1<2... and whose morphisms are monotone (non-decreasing) maps of finite ordered sets. A $\Gamma$-space is then simply a covariant functor from the category $\Gamma^{0}$ to the category of pointed sets, while a simplicial set is a covariant functor from the category $\Delta$ to the category of sets. Simplicial sets are rather abstract constructions, but they are inspired by simplices and simplicial complexes in algebraic topology (see Simplices).

Let $M$ be a $\Gamma$-space. If there is a monoid structure on $\pi_{0}M(1_{+})$ (see Homotopy Theory), then we say that $M$ is a special $\Gamma$-space. If, in addition, this structure is also an abelian group structure, then we say that $M$ is a very special $\Gamma$-space.

The category of $\Gamma$-spaces and the category of simplicial sets are both symmetric monoidal categories, which we need to define relative schemes. For the category of $\Gamma$-spaces, we have the smash product, defined by the requirement that any morphism $F_{1}\wedge F_{2}\rightarrow G$ to any functor $G$ from $\Gamma^{0}\times \Gamma^{0}$ to the category of pointed sets be a natural transformation, i.e. there are maps of pointed sets from $F_{1}\wedge F_{2}(X\wedge Y)$ to $G(X\wedge Y)$, natural in $X$ and $Y$ (here $X\wedge Y$ refers to the smash product of pointed sets obtained by taking the direct product and collapsing the wedge sum, see Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories),  and the unit is the sphere spectrum $\mathbb{S}$, which is just the inclusion functor from the category of pointed finite sets to the category of pointed sets.

For the category of simplicial sets, we have the direct product, defined as the functor $X\times Y$ which sends the finite ordered set $[n]$ to the set $X([n])\times Y([n])$, for two simplicial sets $X$ and $Y$, and the unit is the functor $*$, which sends any finite ordered set to the set with a single element.

We now go back to Toen and Vaquie’s final three examples of relative schemes. The first of these examples is when one has $(C,\otimes,\mathbf{1}) = (\mathcal{GS},\wedge,\mathbb{S})$, the category of very special $\Gamma$-spaces. We thus have a category of schemes relative to $\mathcal{GS}$, which we will denote $\mathbb{S}\text{-Sch}$, where the notation $\mathbb{S}$ recalls the sphere spectrum.

The second example is $(C,\otimes,\mathbf{1})=(\mathcal{MS},\wedge,\mathbb{S}_{+})$, the category of special $\Gamma$-spaces. The category of relative schemes will be noted $\mathbb{S}_{+}\text{-Sch}$, and its affine objects are homotopical analogs of commutative semirings. The notation $\mathbb{S}_{+}$ intuitively means the semiring in spectra of positive integers, and is a homotopical version of the semiring $\mathbb{N}$.

The third example is $(C,\otimes,\mathbf{1})=(\text{SEns},\times,*)$, the category of simplicial sets with its direct product. The schemes that we obtain are homotopical versions of the $\mathbb{F}_{1}$-schemes, and will be called $\mathbb{S}_{1}$-schemes, where $\mathbb{S}_{1}$ may be thought of as the “ring spectrum with one element”, in analogy with $\mathbb{F}_{1}$, the “field with one element”.

Similar to the earlier cases, we also have the base change functors

$-\otimes_{\mathbb{S}_{1}}\mathbb{S}_{+}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{S}_{+}\text{-Sch}$

and

$-\otimes_{\mathbb{S}_{+}}\mathbb{S}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{S}\text{-Sch}$

which we can also compose and represent it with the following diagram:

$\text{Spec}(\mathbb{S})\rightarrow\text{Spec}(\mathbb{S}_{+})\rightarrow\text{Spec}(\mathbb{S}_{1})$.

Moreover, we also have the following functors:

$-\otimes_{\mathbb{S}_{1}}\mathbb{F}_{1}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{F}_{1}\text{-Sch}$

$-\otimes_{\mathbb{S}_{+}}\mathbb{N}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{S}}\mathbb{Z}:\mathbb{S}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

which relate the “homotopical” relative schemes to the ordinary relative schemes. And so, all these schemes, both the new schemes “under $\text{Spec}(\mathbb{Z})$” as well as the ordinary schemes over $\text{Spec}(\mathbb{Z})$, are related to each other.

##### The Approach of Borger

Borger’s approach makes use of the idea of adjoint triples (see Adjoint Functors and Monads). Before we discuss the field with one element in this approach, let us first discuss something more elementary. Consider a field $K$ and and a field extension $L$ of $K$, and let $G=\text{Gal}(L/K)$. We have the following adjoint triple:

$\displaystyle \text{Weil restrict}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

$\displaystyle -\otimes_{K}L: K\textbf{-Alg}\rightarrow L\textbf{-Alg}$

$\displaystyle \text{forget base}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

Grothendieck’s abstract reformulation of Galois theory says that there is an equivalence of categories between the category of $K$-algebras and the category of $L$-algebras with an action of $G$. This means that we can also consider the above adjoint triple in the following sense:

$\displaystyle A\rightarrow\otimes_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

$\displaystyle \text{fgt}: L\textbf{-Alg}\text{ (with }G\text{-action)}\rightarrow L\textbf{-Alg}$

$\displaystyle A\rightarrow\prod_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

Let us now go back to the field with one element. We want to construct the following adjoint triple:

$\displaystyle \text{Weil restrict}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

$\displaystyle -\otimes_{\mathbb{F}_{1}}\mathbb{Z}:\mathbb{F}_{1}\textbf{-Alg}\rightarrow\mathbb{Z}\textbf{-Alg}$

$\displaystyle \text{forget base}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

Following the above example of the field $K$ and the field extension $L$ of $K$, we will approach the construction of this adjoint triple by considering instead the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

We must now discuss the meaning of the concepts involved in the last adjoint triple. In particular, the must define the category of $\Lambda$-rings, as well as the adjoint functors $\Lambda\odot-$, $\text{fgt}$, and $W(-)$ that form the adjoint triple.

Let $R$ be a ring and let $p$ be a prime number. A Frobenius lift is a ring homomorphism $\psi_{p}:R\rightarrow R$ such that $F\circ q=q\circ\psi_{p}$ where $q:R\rightarrow R/pR$ is the quotient map and $F:R/pR\rightarrow R/pR$ is the Frobenius map which sends an element $x$ to the element $x^{p}$.

Closely related to the idea of Frobenius lifts is the idea of $p$-derivations. If the terminology is reminiscent of differential calculus, this is because Borger’s approach is closely related to the mathematician Alexandru Buium’s theory of “arithmetic differential equations“. If numbers are like functions, then what Buium wants to figure out is what the analogue of a derivative of a function should be for numbers.

Let

$\displaystyle \psi_{p}(x)=x^{p}+p\delta_{p}(x)$.

Being a ring homomorphism means that $\psi$ satisfies the following properties:

(1) $\psi_{p}(x+y)=\psi_{p}(x)+\psi_{p}(y)$

(2) $\psi_{p}(xy)=\psi_{p}(x)\psi_{p}(y)$

(3) $\psi_{p}(1)=1$

(4) $\psi_{p}(0)=0$

Recalling that $\psi_{p}(x)=x^{p}+p\delta_{p}(x)$, this means that $\delta_{p}(x)$ must satisfy the following properties corresponding to the above properties for $\psi_{p}(x)$:

(1) $\delta_{p}(x+y)=\delta_{p}(x)+\delta_{p}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

(2) $\delta_{p}(xy)=x^{p}\delta_{p}(y)+y^{p}\delta_{p}(x)+p\delta_{p}(x)\delta_{p}(y)$

(3) $\delta_{p}(1)=0$

(4) $\delta_{p}(0)=0$.

Let

$\displaystyle \Lambda_{p}\odot A=\mathbb{Z}[\delta_{p}^{\circ n}(a)|n\geqslant 0,a\in A]/\sim$

where $\sim$ is the equivalence relation given by the “Liebniz rule”, i.e.

$\displaystyle \delta_{p}^{\circ 0}(x+y)=\delta_{p}^{\circ 0}(x)+\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 0}(xy)=\delta_{p}^{\circ 0}(x)\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 1}(x+y)=\delta_{p}^{\circ 1}(x)+\delta_{p}^{\circ 1}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

$\displaystyle \delta_{p}^{\circ 1}(xy)=\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)+p\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)$

and so on.

We now discuss the closely related (and an also important part of modern mathematical research)  notion of Witt vectors. We define the ring of Witt vectors of the ring $A$ by

$\displaystyle W_{p}(A)=A\times A\times...$

with ring operations given by

$\displaystyle (a_{0},a_{1},...)+(b_{0},b_{1},...)=(a_{0}+b_{0},a_{1}+b_{1}-\sum_{i=1}^{p-1}\frac{1}{p}\binom{p}{i}a_{0}^{i}b_{0}^{p-i},...)$

$\displaystyle (a_{0},a_{1},...)(b_{0},b_{1},...)=(a_{0}b_{0},a_{0}^{p}b_{1}+a_{1}b_{0}^{p}+pa_{1}b_{1},...)$

$\displaystyle 0=(0,0,...)$

$\displaystyle 1=(1,0,...)$

The functors

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

form an adjoint triple.

A $\Lambda_{p}$-ring is defined to be the smallest $\Lambda_{p}^{'}$-ring that contains $e$, where a $\Lambda_{p}^{'}$-ring is in turn defined to be a $p$-torsion free ring together with a Frobenius lift. But it so happens that a $\Lambda_{p}$-ring is also the same thing as a $\delta_{p}$-ring, so we also have the following adjoint triple:

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda_{p}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

Now that we know the basics of a “$p$-typical” $\Lambda$-ring, which is a ring together with a Frobenius morphism $\psi_{p}$ for one fixed $p$, we can also consider a ring together with a Frobenius morphism $\psi_{p}$ for every prime $p$, to form a “big” $\Lambda$-ring. We will then obtain the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

This is just the adjoint triple that we were looking for in the beginning of this section. In other words, we now have what we need to construct rings over $\mathbb{F}_{1}$, or $\mathbb{F}_{1}$-algebras and moreover, we have an adjoint triple that relates them to ordinary rings (or $\mathbb{Z}$-algebras).

We can then generalize these constructions from rings to schemes. The definition of a $\Lambda$-structure on general schemes is complicated and left to the references, but when the scheme $X$ is flat over $\mathbb{Z}$ (see The Hom and Tensor Functors), a $\Lambda$-structure on $X$ is simply defined to be a commuting family of endomorphisms $\psi_{p}$, one for each prime $p$, such that they agree with the $p$-th power Frobenius map on the fibers $X\times_{\text{Spec}(\mathbb{Z})}\mathbb{F}_{p}$.

One may notice that in Borger’s approach an $\mathbb{F}_{1}$-scheme has more structure than a $\mathbb{Z}$-scheme, whereas in Deitmar’s approach $\mathbb{F}_{1}$-schemes, being commutative monoids, have less structure than $\mathbb{Z}$-schemes. One may actually think of the $\Lambda$-structure as “descent data” to $\mathbb{F}_{1}$. In other words, the $\Lambda$-structure tells us how a scheme defined over $\mathbb{Z}$ is defined over $\mathbb{F}_{1}$. There is actually a way to use a monoid $M$ to construct a $\Lambda$-ring $\mathbb{Z}[M]$, where $\mathbb{Z}[M]$ is just the monoid ring as described earlier in the approach of Deitmar, and the Frobenius lifts are defined by $\psi_{p}=m^{p}$ for $m\in M$. We therefore have some sort of connection between Deitmar’s approach (which is also easily seen to be closely related to Soule’s and Toen and Vaquie’s approach) with Borger’s approach.

##### Conclusion

We have mentioned only four approaches to the idea of the field with one element in this rather lengthy post. There are many others, and these approaches are often related to each other. In addition, there are other approaches to uncovering even more analogies between function fields and number fields that are not commonly classified as being part of this circle of ideas. To end this post, we just mention that many open problems in mathematics, such as the abc conjecture and the Riemann hypothesis, have function field analogues that have already been solved (we have already discussed the function field analogue of the Riemann hypothesis in The Riemann Hypothesis for Curves over Finite Fields) – perhaps an investigation of these analogies would lead to the solution of their number field analogues – or, in the other direction, perhaps work on these problems would help uncover more aspects of these mysterious and beautiful analogies.

References:

Field with One Element on Wikipedia

Field with One Element on the nLab

Function Field Analogy on the nLab

Schemes over F1 by Anton Deitmar

Lectures on Algebraic Varieties over F1 by Christophe Soule

Les Varieties sur le Corps a un Element by Christophe Soule

On the Field with One Element by Christophe Soule

Under Spec Z by Bertrand Toen and Michel Vaquie

Lambda-Rings and the Field with One Element by James Borger

Witt Vectors, Lambda-Rings, and Arithmetic Jet Spaces by James Borger

Mapping F1-Land: An Overview of Geometries over the Field with One Element by Javier Lopez-Pena and Oliver Lorscheid

Geometry and the Absolute Point by Lieven Le Bruyn

This Week’s Finds in Mathematical Physics (Week 259) by John Baez

Algebraic Number Theory by Jurgen Neukirch

The Local Structure of Algebraic K-Theory by Bjorn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy

# Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

# Algebraic Spaces and Stacks

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category $\mathcal{C}$, we say that some other category $\mathcal{S}$ is a category over $\mathcal{C}$ if there is a functor $p$ from $\mathcal{S}$ to $\mathcal{C}$ (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If $\mathcal{S}$ is a category over some other category $\mathcal{C}$, we say that it is a fibered category (over $\mathcal{C}$) if for every object $U=p(x)$ and morphism $f: V\rightarrow U$ in $\mathcal{C}$, there is a strongly cartesian morphism $\phi: f^{*}x\rightarrow x$ in $\mathcal{S}$ with $f=p(\phi)$.

This means that any other morphism $\psi: z\rightarrow x$ whose image $p(\psi)$ under the functor $p$ factors as $p(\psi)=p(\phi)\circ h$ must also factor as $\psi=\phi\circ \theta$ under some unique morphism $\theta: z\rightarrow f^{*}x$ whose image under the functor $p$ is $h$. We refer to $f^{*}x$ as the pullback of $x$ along $f$.

Under the functor $p$, the objects of $\mathcal{S}$ which get sent to $U$ in $\mathcal{C}$ and the morphisms of $\mathcal{S}$ which get sent to the identity morphism $i_{U}$ in $\mathcal{C}$ form a subcategory of $\mathcal{S}$ called the fiber over $U$. We will also write it as $\mathcal{S}_{U}$.

An important example of a fibered category is given by an ordinary presheaf on a category $\mathcal{C}$, i.e. a functor $F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set})$; we can consider it as a category fibered in sets $\mathcal{S}_{F}\rightarrow\mathcal{C}$.

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let $\mathcal{U}=\{f_{i}:U_{i}\rightarrow U\}$ be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object $U$ of $\mathcal{C}$. An object with descent data is a collection of objects $X_{i}$ in $\mathcal{S}_{U}$ together with transition isomorphisms $\varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j}$ in $\mathcal{S}_{U_{i}\times_{U}U_{j}}$, satisfying the cocycle condition

$\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}$

The morphisms $\text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i}$ and the $\text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j}$ are the projection morphisms. The notations $\text{pr}_{0}^{*}X_{i}$ and $\text{pr}_{1}^{*}X_{j}$ means that we are “pulling back” $X_{i}$ and $X_{j}$ from $\mathcal{S}_{U_{i}}$ and $\mathcal{S}_{U_{j}}$, respectively, to $\mathcal{S}_{U_{i}\times_{U}U_{j}}$.

A morphism between two objects with descent data is a a collection of morphisms $\psi_{i}:X_{i}\rightarrow X'_{i}$ in $\mathcal{S}_{U_{i}}$ such that $\varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}$. Therefore we obtain a category, the category of objects with descent data, denoted $\mathcal{DD}(\mathcal{U})$.

We can define a functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ by assigning to each object $X$ of $\mathcal{S}_{U}$ the object with descent data given by the pullback $f_{i}^{*}X$ and the canonical isomorphism $\text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X$. An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is faithful if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is injective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is full if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is surjective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is essentially surjective if any object $y$ of $\mathcal{B}$ is isomorphic to the image $F(x)$ of some object $x$ in $\mathcal{A}$ under $F$.

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ we have defined earlier.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is fully faithful, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a prestack.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is an equivalence of categories, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) $F$ is a prestack if and only if it is a separated functor,

(ii) $F$ is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(X,U)$, for some fixed object $U$ of $\mathcal{C}$.

A representable presheaf is a contravariant functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(U,X)$, for some fixed object $U$ of $\mathcal{C}$. If $\mathcal{C}$ is the category of schemes, the latter functor is also called the functor of points of the object $U$.

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points $h_{U}$ of a scheme $U$ may be identified with $U$ itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let $F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors, and let $a:F\rightarrow G$ and $b:H\rightarrow G$ be natural transformations between these functors. Then the fiber product $F\times_{a,G,b}H$ is a functor from $\mathcal{C}^{\text{op}}$ to $\textbf{Sets}$, and is given by the formula

$\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)$

for any object $X$ of $\mathcal{C}$.

Let $F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors. We say that a natural transformation $a:F\rightarrow G$ is representable, or that $F$ is relatively representable over $G$ if for every $U\in\text{Ob}(\mathcal{C})$ and any $\xi\in G(U)$ the functor $h_{U}\times_{G}F$ is representable.

We now let $(\text{Sch}/S)_{\text{fppf}}$ be the site (a category with a Grothendieck topology –  see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of $S$-schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme $S$ is a presheaf

$\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}$

with the following properties

(1) The presheaf $F$ is a sheaf.

(2) The diagonal morphism $F\rightarrow F\times F$ is representable.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and a map $h_{U}\rightarrow F$ which is surjective, and etale (This is often written simply as $U\rightarrow F$). The scheme $U$ is also called an atlas.

The diagonal morphism being representable implies that the natural transformation $h_{U}\rightarrow F$ is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have $U$ is the disjoint union of affine schemes $U_{i}$ and where the map $h_{U}\rightarrow F$ is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let $\mathcal{C}$ be a category. A category fibered in groupoids $p:\mathcal{S}\rightarrow\mathcal{C}$ is called representable if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j:\mathcal{S}\rightarrow \mathcal{C}/X$ (The notation $\mathcal{C}/X$ signifies a slice category, whose objects are morphisms $f:U\rightarrow X$ in $\mathcal{C}$, and whose morphisms are morphisms $h:U\rightarrow V$ in $\mathcal{C}$ such that $f=g\circ h$, where $g:U\rightarrow X$).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let $\mathcal{X}$ be a category fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$. Then $\mathcal{X}$ is representable by a scheme if there exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and an equivalence $j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

A category fibered in groupoids $p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$ is representable by an algebraic space over $S$ if there exists an algebraic space $F$ over $S$ and an equivalence $j:\mathcal{X}\rightarrow \mathcal{S}_{F}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let $F:\mathcal{X}\rightarrow\mathcal{S}$ and $G:\mathcal{Y}\rightarrow\mathcal{S}$ be morphisms of categories over $\mathcal{C}$. The fiber product $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is given by the following description:

(1) an object of $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is a quadruple $(U,x,y,f)$, where $U\in\text{Ob}(\mathcal{C})$, $x\in\text{Ob}(\mathcal{X}_{U})$, $y\in\text{Ob}(\mathcal{Y}_{U})$, and $f : F(x)\rightarrow G(y)$ is an isomorphism in $\mathcal{S}_{U}$,

(2) a morphism $(U,x,y,f) \rightarrow (U',x',y',f')$ is given by a pair $(a,b)$, where $a:x\rightarrow x'$ is a morphism in $X$, and $b:y\rightarrow y'$ is a morphism in $Y$ such that $a$ and $b$ induce the same morphism $U\rightarrow U'$, and $f'\circ F(a)=G(b)\circ f$.

Let $S$ be a scheme. A morphism $f:\mathcal{X}\rightarrow \mathcal{Y}$ of categories fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ is called representable by algebraic spaces if for any $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and any $y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y}$ the category fibered in groupoids

$\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}$

over $(\text{Sch}/U)_{\text{fppf}}$ is representable by an algebraic space over $U$.

An algebraic stack (or Artin stack) over a scheme $S$ is a category

$\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$

with the following properties:

(1) The category $\mathcal{X}$ is a stack in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ .

(2) The diagonal $\Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X}$ is representable by algebraic spaces.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch/S})_{\text{fppf}})$ and a morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ which is surjective and smooth (This is often written simply as $U\rightarrow\mathcal{X}$). Again, the scheme $U$ is called an atlas.

If the morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by $\mathcal{M}_{1,1}$ – the reason for this notation will become clear later). A family of elliptic curves over some “base space” $B$ is a fibration $\pi:X\rightarrow B$ with a section $O:B\rightarrow X$ such that the fiber $\pi^{-1}(b)$ over any point $b$ of $B$ is an elliptic curve with origin $O(b)$.

Ideally what we want is to be able to obtain every family $X\rightarrow B$ by pulling back a “universal object” $E\rightarrow\mathcal{M}_{1,1}$ via the map $B\rightarrow\mathcal{M}_{1,1}$. This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by $M_{1,1}$). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus $g$ with $\nu$ marked points, denoted $\mathcal{M}_{g,\nu}$. The moduli stack of elliptic curves is simply the special case $\mathcal{M}_{1,1}$. Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.

References:

Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

# The Theory of Motives

The theory of motives originated from the observation, sometime in the 1960’s, that in algebraic geometry there were several different cohomology theories (see Homology and Cohomology and Cohomology in Algebraic Geometry), such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology. The search for a “universal cohomology theory”, such that all these other cohomology theories could be obtained from such a universal cohomology theory is what led to the theory of motives.

The four cohomology theories enumerated above are examples of what is called a Weil cohomology theory. A Weil cohomology theory, denoted $H^{*}$, is a functor (see Category Theory) from the category $\mathcal{V}(k)$ of smooth projective varieties over some field $k$ to the category $\textbf{GrAlg}(K)$ of graded $K$-algebras, for some other field $K$ which must be of characteristic zero, satisfying the following axioms:

(1) (Finite-dimensionality) The homogeneous components $H^{i}(X)$ of $H^{*}(X)$ are finite dimensional for all $i$, and $H^{i}(X)=0$ whenever $i<0$ or $i>2n$, where $n$ is the dimension of the smooth projective variety $X$.

(2) (Poincare duality) There is an orientation isomorphism $H^{2n}\cong K$, and a nondegenerate bilinear pairing $H^{i}(X)\times H^{2n-i}(X)\rightarrow H^{2n}\cong K$.

(3) (Kunneth formula) There is an isomorphism

$\displaystyle H^{*}(X\times Y)\cong H^{*}(X)\otimes H^{*}(Y)$.

(4) (Cycle map) There is a mapping $\gamma_{X}^{i}$ from $C^{i}(X)$, the abelian group of algebraic cycles of codimension $i$ on $X$ (see Algebraic Cycles and Intersection Theory), to $H^{i}(X)$, which is functorial with respect to pullbacks and pushforwards, has the multiplicative property $\gamma_{X\times Y}^{i+j}(Z\times W)=\gamma_{X}^{i}(Z)\otimes \gamma_{Y}^{j}(W)$, and such that $\gamma_{\text{pt}}^{i}$ is the inclusion $\mathbb{Z}\hookrightarrow K$.

(5) (Weak Lefschetz axiom) If $W$ is a smooth hyperplane section of $X$, and $j:W\rightarrow X$ is the inclusion, the induced map $j^{*}:H^{i}(X)\rightarrow H^{i}(W)$ is an isomorphism for $i\leq n-2$, and a monomorphism for $i\leq n-1$.

(6) (Hard Lefschetz axiom) The Lefschetz operator

$\displaystyle \mathcal{L}:H^{i}(X)\rightarrow H^{i+2}(X)$

given by

$\displaystyle \mathcal{L}(x)=x\cdot\gamma_{X}^{1}(W)$

for some smooth hyperplane section $W$ of $X$, with the product $\cdot$ provided by the graded $K$-algebra structure of $H^{*}(X)$, induces an isomorphism

$\displaystyle \mathcal{L}^{i}:H^{n-i}(X)\rightarrow H^{n+i}(X)$.

The idea behind the theory of motives is that all Weil cohomology theories should factor through a “category of motives”, i.e. any Weil cohomology theory

$\displaystyle H^{*}: \mathcal{V}(k)\rightarrow \textbf{GrAlg}(K)$

can be expressed as the following composition of functors:

$\displaystyle H^{*}: \mathcal{V}(k)\xrightarrow{h} \mathcal{M}(k)\rightarrow\textbf{GrAlg}(K)$

where $\mathcal{M}(k)$ is the category of motives. We can get different Weil cohomology theories, such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology, via different functors (called realization functors) from the category of motives to a category of graded algebras over some field $K$. This explains the term “motive”, which actually comes from the French word “motif”, which itself is already used in music and visual arts, among other things, as some kind of common underlying “theme” with different possible manifestations.

Let us now try to construct this category of motives. This category is often referred to in the literature as a “linearization” of the category of smooth projective varieties. This means that we obtain it from some sense starting with the category of smooth projective varieties, but we also want to modify it so that it we can do linear algebra, or more properly homological algebra, in some sense. In other words, we want it to behave like the category of modules over some ring. With this in mind, we want the category to be an abelian category, so that we can make sense of notions such as kernels, cokernels, and exact sequences.

An abelian category is a category that satisfies the following properties:

(1) The morphisms form an abelian group.

(2) There is a zero object.

(3) There are finite products and coproducts.

(4) Every morphism $f:X\rightarrow Y$ has a kernel and cokernel, and satisfies a decomposition

$\displaystyle K\xrightarrow{k} X\xrightarrow{i} I\xrightarrow{j} Y\xrightarrow{c} K'$

where $K$ is the kernel of $f$, $K'$ is the cokernel of $f$, and $I$ is the kernel of $c$ and the cokernel of $k$ (not to be confused with our notation for fields).

In order to proceed with our construction of the category of motives, which we now know we want to be an abelian category, we discuss the notion of correspondences.

The group of correspondences of degree $r$ from a smooth projective variety $X$ to another smooth projective variety $Y$, written $\text{Corr}^{r}(X,Y)$, is defined to be the group of algebraic cycles of $X\times Y$ of codimension $n+r$, where $n$ is the dimension of $X$, i.e.

$\text{Corr}^{r}(X,Y)=C^{n+r}(X\times Y)$

A morphism (of varieties, in the usual sense) $f:Y\rightarrow X$ determines a correspondence from $X$ to $Y$ of degree $0$ given by the transpose of the graph of $f$ in $X\times Y$. Therefore we may think of correspondences as generalizations of the usual concept of morphisms of varieties.

As we have learned in Algebraic Cycles and Intersection Theory, whenever we are dealing with algebraic cycles, it is often useful to consider them only up to some equivalence relation. In the aforementioned post we introduced the notion of rational equivalence. This time we consider also homological equivalence and numerical equivalence between algebraic cycles.

We say that two algebraic cycles $Z_{1}$ and $Z_{2}$ are homologically equivalent if they have the same image under the cycle map, and we say that they are numerically equivalent if the intersection numbers $Z_{1}\cdot Z$ and $Z_{2}\cdot Z$ are equal for all $Z$ of complementary dimension. There are other such equivalence relations on algebraic cycles, but in this post we will only mostly be using rational equivalence, homological equivalence, and numerical equivalence.

Since correspondences are algebraic cycles, we often consider them only up to these equivalence relations, and denote the quotient group we obtain by $\text{Corr}_{\sim}^{r}(X,Y)$, where $\sim$ is the equivalence relation imposed, for example, for numerical equivalence we write $\text{Corr}_{\text{num}}^{r}(X,Y)$.

Taking the tensor product of the abelian group $\text{Corr}_{\sim}^{r}(X,Y)$ with the rational numbers $\mathbb{Q}$, we obtain the vector space

$\displaystyle \text{Corr}_{\sim}^{r}(X,Y)_{\mathbb{Q}}=\text{Corr}_{\sim}^{r}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q}$

To obtain something closer to an abelian category (more precisely, we will obtain what is known as a pseudo-abelian category, but in the case where the equivalence relation is numerical equivalence, we will actually obtain an abelian category), we need to consider “projectors”, correspondences $p$ of degree $0$ from a variety $X$ to itself such that $p^{2}=p$. So now we form a category, whose objects are $h(X,p)$ for a variety $X$ and projector $p$, and whose morphisms are given by

$\displaystyle \text{Hom}(h(X,p),h(Y,q))=q\circ\text{Corr}_{\sim}^{0}(X,Y)_{\mathbb{Q}}\circ p$.

We call this category the category of pure effective motives, and denote it by $\mathcal{M}_{\sim}^{\text{eff}}(k)$. The process described above is also known as passing to the pseudo-abelian (or Karoubian) envelope.

We write $h^{i}(X,p)$ for the objects of $\mathcal{M}_{\sim}^{\text{eff}}(k)$ that map to $H^{i}(X)$. In the case that $X$ is the projective line $\mathbb{P}^{1}$, and $p$ is the diagonal $\Delta_{\mathbb{P}^{1}}$, we find that

$h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=h^{0}\mathbb{P}^{1}\oplus h^{2}\mathbb{P}^{1}$

which can be rewritten also as

$\displaystyle h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=\mathbb{I}\oplus\mathbb{L}$

where $\mathbb{I}$ is the image of a point in the category of pure effective motives, and $\mathbb{L}$ is known as the Lefschetz motive. It is also denoted by $\mathbb{Q}(-1)$. The above decomposition corresponds to the projective line $\mathbb{P}^{1}$ being a union of the affine line $\mathbb{A}^{1}$ and a “point at infinity”, which we may denote by $\mathbb{A}^{0}$:

$\displaystyle \mathbb{P}^{1}=\mathbb{A}^{0}\cup\mathbb{A}^{1}$

More generally, we have

$\displaystyle h(\mathbb{P}^{n},\Delta_{\mathbb{P}^{n}})=\mathbb{I}\oplus\mathbb{L}\oplus...\oplus\mathbb{L}^{n}$

corresponding to

$\displaystyle \mathbb{P}^{n}=\mathbb{A}^{0}\cup\mathbb{A}^{1}\cup...\cup\mathbb{A}^{n}$.

The category of effective pure motives is an example of a tensor category. This means it has a bifunctor $\otimes: \mathcal{M}_{\sim}^{\text{eff}}\times\mathcal{M}_{\sim}^{\text{eff}}\rightarrow\mathcal{M}_{\sim}^{\text{eff}}$ which generalizes the usual notion of a tensor product, and in this particular case it is given by taking the product of two varieties. We can ask for more, however, and construct a category of motives which is not just a tensor category but a rigid tensor category, which provides us with a notion of duals.

By formally inverting the Lefschetz motive (the formal inverse of the Lefschetz motive is then known as the Tate motive, and is denoted by $\mathbb{Q}(1)$), we can obtain this rigid tensor category, whose objects are triples $h(X,p,m)$, where $X$ is a variety, $e$ is a projector, and $m$ is an integer. The morphisms of this category are given by

$\displaystyle \text{Hom}(h(X,p,n),h(Y,q,m))=q\circ\text{Corr}_{\sim}^{n-m}(X,Y)_{\mathbb{Q}}\circ p$.

This category is called the category of pure motives, and is denoted by $\mathcal{M}_{\sim}(k)$. The category $\mathcal{M}_{\text{rat}}(k)$ is called the category of Chow motives, while the category $\mathcal{M}_{\text{num}}(k)$ is called the category of Grothendieck (or numerical) motives.

The category of Chow motives has the advantage that it is known to be “universal”, in the sense that every Weil cohomology theory factors through it, as discussed earlier; however, in general it is not even abelian, which is a desirable property we would like our category of motives to have. Meanwhile, the category of Grothendieck motives is known to be abelian, but it is not yet known if it is universal. If the so-called “standard conjectures on algebraic cycles“, which we will enumerate below, are proved, then the category of Grothendieck motives will be known to be universal.

We have seen that the category of pure motives forms a rigid tensor category. Closely related to this concept, and of interest to us, is the notion of a Tannakian category. More precisely, a Tannakian category is a $k$-linear rigid tensor category with an exact faithful functor (called a fiber functor) to the category of finite-dimensional vector spaces over some field extension $K$ of $k$.

One of the things that makes Tannakian categories interesting is that there is an equivalence of categories between a Tannakian category $\mathcal{C}$ and the category $\text{Rep}_{G}$ of finite-dimensional linear representations of the group of automorphisms of its fiber functor, which is also known as the Tannakian Galois group, or, if the Tannakian category is a “category of motives” of some sort, the motivic Galois group. This aspect of Tannakian categories may be thought of as a higher-dimensional analogue of the classical theory of Galois groups, which can be stated as an equivalence of categories between the category of finite separable field extensions of a field $k$ and the category of finite sets equipped with an action of the Galois group $\text{Gal}(\bar{k}/k)$, where $\bar{k}$ is the algebraic closure of $k$.

So we see that being a Tannakian category is yet another desirable property that we would like our category of motives to have. For this not only do we have to tweak the tensor product structure of our category, we also need certain conjectural properties to hold. These are the same conjectures we have hinted at earlier, called the standard conjectures on algebraic cycles, formulated by Alexander Grothendieck at around the same time he initially developed the theory of motives.

These conjectures have some very important consequences in algebraic geometry, and while they remain unproved to this day, the search for their proof (or disproof) is an important part of modern mathematical research on the theory of motives. They are the following:

(A) (Standard conjecture of Lefschetz type) For $i\leq n$, the operator $\Lambda$ defined by

$\displaystyle \Lambda=(\mathcal{L}^{n-i+2})^{-1}\circ\mathcal{L}\circ (\mathcal{L}^{n-i}):H^{i}\rightarrow H^{i-2}$

$\displaystyle \Lambda=(\mathcal{L}^{n-i})\circ\mathcal{L}\circ (\mathcal{L}^{n-i+2})^{-1}:H^{2n-i+2}\rightarrow H^{2n-i}$

is induced by algebraic cycles.

(B) (Standard conjecture of Hodge type) For all $i\leq n/2$, the pairing

$\displaystyle x,y\mapsto (-1)^{i}(\mathcal{L}x\cdot y)$

is positive definite.

(C) (Standard conjecture of Kunneth type) The projectors $H^{*}(X)\rightarrow H^{i}(X)$ are induced by algebraic cycles in $X\times X$ with rational coefficients. This implies the following decomposition of the diagonal:

$\displaystyle \Delta_{X}=\pi_{0}+...+\pi_{2n}$

which in turn implies the decomposition

$\displaystyle h(X,\Delta_{X},0)=h(X,\pi_{0},0)\oplus...\oplus h(X,\pi_{2n},0)$

which, writing $h(X,\Delta_{X},0)$ as $hX$ and $h(X,\pi_{i},0)$ as $h^{i}(X)$, we can also compactly and suggestively write as

$\displaystyle hX=h^{0}X\oplus...\oplus h^{2n}X$.

In other words, every object $hX=h(X,\Delta_{X},0)$ of our “category of motives” decomposes into graded “pieces” $h^{i}(X)=h(X,\pi_{i},0)$ of pure “weight$i$. We have already seen earlier that this is indeed the case when $X=\mathbb{P}^{n}$. We will need this conjecture to hold if we want our category to be a Tannakian category.

(D) (Standard conjecture on numerical equivalence and homological equivalence) If an algebraic cycle is numerically equivalent to zero, then its cohomology class is zero. If the category of Grothendieck motives is to be “universal”, so that every Weil cohomology theory factors through it, this conjecture must be satisfied.

In Algebraic Cycles and Intersection Theory and Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry, we have made mention of the two famous conjectures in algebraic geometry known as the Hodge conjecture and the Tate conjecture. In fact, these two closely related conjectures can be phrased in the language of motives as the conjectures stating that the realization functors from the category of motives to the category of pure Hodge structures and continuous $l$-adic representations of $\text{Gal}(\bar{k}/k)$, respectively, be fully faithful. These conjectures are closely related to the standard conjectures on algebraic cycles as well.

We have now constructed the category of pure motives, for smooth projective varieties. For more general varieties and schemes, there is an analogous idea of “mixed motives“, which at the moment remain conjectural, although there exist several related constructions which are the closest thing we currently have to such a theory of mixed motives.

If we want to construct a theory of mixed motives, instead of Weil cohomology theories we must instead consider what are known as “mixed Weil cohomology theories“, which are expected to have the following properties:

(1) (Homotopy invariance) The projection $\pi:X\rightarrow\mathbb{A}^{1}$ induces an isomorphism

$\displaystyle \pi^{*}:H^{*}(X)\xrightarrow{\cong}H^{*}(X\times\mathbb{A}^{1})$

(2) (Mayer-Vietoris sequence) If $U$ and $V$ are open coverings of $X$, then there is a long exact sequence

$\displaystyle ...\rightarrow H^{i}(U\cap V)\rightarrow H^{i}(X)\rightarrow H^{i}(U)\oplus H^{i}(V)\rightarrow H^{i}(U\cap V)\rightarrow...$

(3) (Duality) There is a duality between cohomology $H^{*}$ and cohomology with compact support $H_{c}^{*}$.

(4) (Kunneth formula) This is the same axiom as the one in the case of pure motives.

We would like a category of mixed motives, which serves as an analogue to the category of pure motives in that all mixed Weil cohomology theories factor through it, but as mentioned earlier, no such category exists at the moment. However, the mathematicians Annette Huber-Klawitter, Masaki Hanamura, Marc Levine, and Vladimir Voevodsky have constructed different versions of a triangulated category of mixed motives, denoted $\mathcal{DM}(k)$.

A triangulated category $\mathcal{T}$ is an additive category with an automorphism $T: \mathcal{T}\rightarrow\mathcal{T}$ called the “shift functor” (we will also denote $T(X)$ by $X[1]$, and $T^{n}(X)$ by $X[n]$, for $n\in\mathbb{Z}$) and a family of “distinguished triangles

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

which satisfies the following axioms:

(1) For any object $X$ of $\mathcal{T}$, the triangle $X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow X[1]$ is a distinguished triangle.

(2) For any morphism $u:X\rightarrow Y$ of $\mathcal{T}$, there is an object $Z$ of $\mathcal{T}$ such that $X\xrightarrow{u}Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle.

(3) Any triangle isomorphic to a distinguished triangle is a distinguished triangle.

(4) If $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle, then the two “rotations” $Y\rightarrow Z\rightarrow Z[1]\rightarrow Y[1]$ and $Z[-1]\rightarrow X\rightarrow Y\rightarrow Z$ are also distinguished triangles.

(5) Given two distinguished triangles $X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]$ and $X'\xrightarrow{u'}Y'\xrightarrow{v'}Z'\xrightarrow{w'}X'[1]$ and morphisms $f:X\rightarrow X'$ an $g:Y\rightarrow Y'$ such that the square “commutes”, i.e. $u'\circ f=g\circ u$, there exists a morphisms $h:Z\rightarrow Z$ such that all other squares commute.

(6) Given three distinguished triangles $X\xrightarrow{u}Y\xrightarrow{j}Z'\xrightarrow{k}X[1]$$Y\xrightarrow{v}Z\xrightarrow{l}X'\xrightarrow{i}Y[1]$, and $X\xrightarrow{v\circ u}Z\xrightarrow{m}Y'\xrightarrow{n}X[1]$, there exists a distinguished triangle $Z'\xrightarrow{f}Y'\xrightarrow{g}X'\xrightarrow{h}Z'[1]$ such that “everything commutes”.

A $t$-structure on a triangulated category $\mathcal{T}$ is made up of two full subcategories $\mathcal{T}^{\geq 0}$ and $\mathcal{T}^{\leq 0}$ satisfying the following properties (writing $\mathcal{T}^{\leq n}$ and $\mathcal{T}^{\leq n}$ to denote $\mathcal{T}^{\leq 0}[-n]$ and $\mathcal{T}^{\geq 0}[-n]$ respectively):

(1) $\mathcal{T}^{\leq -1}\subset \mathcal{T}^{\leq 0}$ and $\mathcal{T}^{\geq 1}\subset \mathcal{T}^{\geq 0}$

(2) $\displaystyle \text{Hom}(X,Y)=0$ for any object $X$ of $\mathcal{T}^{\leq 0}$ and any object $Y$ of $\mathcal{T}^{\geq 1}$

(3) for any object $Y$ of $\mathcal{T}$ we have a distinguished triangle

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

where $X$ is an object of $\mathcal{T}^{\leq 0}$ and $Z$ is an object of $\mathcal{T}^{\geq 1}$.

The full subcategory $\mathcal{T}^{0}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0}$ is called the heart of the $t$-structure, and it is an abelian category.

It is conjectured that the category of mixed motives $\mathcal{MM}(k)$ is the heart of the $t$-structure of the triangulated category of mixed motives $\mathcal{DM}(k)$.

Voevodsky’s construction proceeds in a manner somewhat analogous to the construction of the category of pure motives as above, starting with schemes (say, over a field $k$, although a more general scheme may be used) as objects and correspondences as morphisms, but then makes use of concepts from abstract homotopy theory, such as taking the bounded homotopy category of bounded complexes, and localization with respect to a certain subcategory, before passing to the pseudo-abelian envelope and then formally inverting the Tate object $\mathbb{Z}(1)$. The triangulated category obtained is called the category of geometric motives, and is denoted by $\mathcal{DM}_{\text{gm}}(k)$. The schemes and correspondences involved in the construction of $\mathcal{DM}_{\text{gm}}(k)$ are required to satisfy certain properties which eliminates the need to consider the equivalence relations which form a large part of the study of the category of pure motives.

Closely related to the triangulated category of mixed motives is motivic cohomology, which is defined in terms of the former as

$\displaystyle H^{i}(X,\mathbb{Z}(m))=\text{Hom}_{\mathcal{DM}(k)}(X,\mathbb{Z}(m)[i])$

where $\mathbb{Z}(m)$ is the tensor product of $m$ copies of the Tate object $\mathbb{Z}(1)$, and the notation $\mathbb{Z}(m)[i]$ tells us that the shift functor of the triangulated category is applied to the object $\mathbb{Z}(m)$ $i$ times.

Motivic cohomology is related to the Chow group, which we have introduced in Algebraic Cycles and Intersection Theory, and also to algebraic K-theory, which is another way by which the ideas of homotopy theory are applied to more general areas of abstract algebra and linear algebra. These ideas were used by Voevodsky to prove several related theorems, from the Milnor conjecture to its generalization, the Bloch-Kato conjecture (also known as the norm residue isomorphism theorem).

Historically, one of the motivations for Grothendieck’s attempt to obtain a universal cohomology theory was to prove the Weil conjectures, which is a higher-dimensional analogue of the Riemann hypothesis for curves over finite fields first proved by Andre Weil himself (see The Riemann Hypothesis for Curves over Finite Fields). In fact, if the standard conjectures on algebraic cycles are proved, then a proof of the Weil conjectures would follow via an approach that closely mirrors Weil’s original proof (since cohomology provides a Lefschetz fixed-point formula –  we have mentioned in The Riemann Hypothesis for Curves over Finite Fields that the study of fixed points is an important part of Weil’s proof). The last of the Weil conjectures were eventually proved by Grothendieck’s student Pierre Deligne, but via a different approach that bypassed the standard conjectures. A proof of the standard conjectures, which would lead to a perhaps more elegant proof of the Weil conjectures, is still being pursued to this day.

The theory of motives is not only related to analogues of the Riemann hypothesis, which concerns the location of zeroes of L-functions, but to L-functions in general. For instance, it is also related to the Langlands program, which concerns another aspect of L-functions, namely their analytic continuation and functional equation, and to the Birch and Swinnerton-Dyer conjecture, which concerns their values at special points.

We recall in The Riemann Hypothesis for Curves over Finite Fields that the Frobenius morphism played an important part in counting the points of a curve over a finite field, which in turn we needed to define the zeta function (of which the L-function can be thought of as a generalization) of the curve. The Frobenius morphism is an element of the Galois group, and we recall that a category of motives which is a Tannakian category is equivalent to the category of representations of its motivic Galois group. Therefore we can see how we can define “motivic L-functions” using the theory of motives.

As the L-functions occupy a central place in many areas of modern mathematics, the theory of motives promises much to be gained from its study, if only we could make progress in deciphering the many mysteries that surround it, of which we have only scratched the surface in this post. The applications of motives are not limited to L-functions either – the study of periods, which relate Betti cohomology and de Rham cohomology, and lead to transcendental numbers which can be defined using only algebraic concepts, is also strongly connected to the theory of motives. Recent work by the mathematicians Alain Connes and Matilde Marcolli has also suggested applications to physics, particularly in relation to Feynman diagrams in quantum field theory. There is also another generalization of the theory of motives, developed by Maxim Kontsevich, in the context of noncommutative geometry.

References:

Weil Cohomology Theory on Wikipedia

Motive on Wikipedia

Standard Conjectures on Algebraic Cycles on Wikipedia

Motive on nLab

Pure Motive on nLab

Mixed Motive on nLab

The Tate Conjecture over Finite Fields on Hard Arithmetic

What is…a Motive? by Barry Mazur

Motives – Grothendieck’s Dream by James S. Milne

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Algebraic Cycles and the Weil Conjectures by Steven L. Kleiman

The Standard Conjectures by Steven L. Kleiman

Feynman Motives by Matilde Marcolli

Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Periodes) by Yves Andre

# Some Useful Links on the History of Algebraic Geometry

It’s been a while since I’ve posted on this blog, but there are some posts I’m currently working on about some subjects I’m also currently studying (that’s why it’s taking so long, as I’m trying to digest the ideas as much as I can before I can post about it). But anyway, for the moment, in this short post I’ll be putting up some links to articles on the history of algebraic geometry. Aside from telling an interesting story on its own, there is also much to be learned about a subject from studying its historical development.

We know that the origins of algebraic geometry can be traced back to Rene Descartes and Pierre de Fermat in the 17th century. This is the high school subject also known as “analytic geometry” (which, as we have mentioned in Basics of Algebraic Geometry, can be some rather confusing terminology, because in modern times the word “analytic” is usually used to refer to concepts in complex calculus).

The so-called “analytic geometry” seems to be a rather straightforward subject compared to modern-day algebraic geometry, which, as may be seen on many of the previous posts on this blog, is very abstract (but it is also this abstraction that gives it its power). How did this transformation come to be?

The mathematician Jean Dieudonne, while perhaps more known for his work in the branch of mathematics we call analysis (the more high-powered version of calculus), also served as adviser to Alexander Grothendieck, one of the most important names in the development of modern algebraic geometry. Together they wrote the influential work known as Elements de Geometrie Algebrique, often simply referred to as EGA. Dieudonne was also among the founding members of the “Bourbaki group”, a group of mathematicians who greatly influenced the development of modern mathematics. Himself a part of its development, Dieudonne wrote many works on the history of mathematics, among them the following article on the history of algebraic geometry which can be read for free on the website of the Mathematical Association of America:

The Historical Development of Algebraic Geometry by Jean Dieudonne

But before the sweeping developments instituted by Alexander Grothendieck, the modern revolution in algebraic geometry was first started by the mathematicians Oscar Zariski and Andre Weil (we discussed some of Weil’s work in The Riemann Hypothesis for Curves over Finite Fields). Zariski himself learned from the so-called “Italian school of algebraic geometry”, particularly the mathematicians Guido Castelnuovo, Federigo Enriques, and Francesco Severi.

At the International Congress of Mathematicians in 1950, both Zariski and Weil presented, separately, a survey of the developments in algebraic geometry at the time, explaining how the new “abstract algebraic geometry” was different from the old “classical algebraic geometry”, and the new advantages it presented. The proceedings of this conference are available for free online:

Proceedings of the 1950 International Congress of Mathematicians, Volume I

Proceedings of the 1950 International Congress of Mathematicians, Volume II

The articles by Weil and Zariski can be found in the second volume, but I included also the first volume for “completeness”.

All proceedings of the International Congress of Mathematicians, which is held every four years, are actually available for free online:

Proceedings of the International Congress of Mathematicians, 1983-2010

The proceedings of the 2014 International Congress of Mathematicians in Seoul, Korea, can be found here:

Proceedings of the 2014 International Congress of Mathematicians

Going back to algebraic geometry, a relatively easy to understand (for those with some basic mathematical background, anyway) summary of the work of Alexander Grothendieck’s work in algebraic geometry can be found in the following article by Colin McLarty, published in April 2016 issue of the Notices of the American Mathematical Society:

How Grothendieck Simplified Algebraic Geometry by Colin McLarty

# Tangent Spaces in Algebraic Geometry

We have discussed the notion of a tangent space in Differentiable Manifolds Revisited in the context of differential geometry. In this post we take on the same topic, but this time in the context of algebraic geometry, where it is also known as the Zariski tangent space (when no confusion arises, however, it is often simply referred to as the tangent space).

This will present us with challenges, since the concept of the tangent space is perhaps best tackled using the methods of calculus, but in algebraic geometry, we want to have a notion of tangent spaces in cases where we would not usually think of calculus as being applicable, for instance in the case of varieties over finite fields. In other words, we want our treatment to be algebraic. Nevertheless, we will use the methods of calculus as an inspiration.

We don’t want to be too dependent on the parts of calculus that make use of properties of the real and complex numbers that will not carry over to the more general cases. Fortunately, if we are dealing with polynomials, we can just “borrow” the “power rule” of calculus, since that “rule” only makes use of algebraic procedures, and we need not make use of sequences, limits, and so on. Namely, if we have a polynomial given by

$\displaystyle f=\sum_{j=1}^{n}ax^{j}$

We set

$\displaystyle \frac{\partial f}{\partial x}=\sum_{j=1}^{n}jax^{j-1}$

We recall the rules for partial derivatives – in the case that we are differentiating over some variable $x$, we simply treat all the other variables as constants, and follow the usual rules of differential calculus. With these rules, we can now make the definition of the tangent space at the point $P$ with coordinates $(a_{1},a_{2},...,a_{n})$ as the algebraic set which satisfies the equation

$\displaystyle \sum_{j}\frac{\partial f}{\partial x_{j}}(P)(x_{j}-a_{j})=0$

For example, consider the parabola given by the equation $y-x^{2}=0$. Let us take the tangent space at the point $P$ with coordinates $x=1$, $y=1$. The procedure above gives us

$\displaystyle \frac{\partial f}{\partial x}(P)(x-1)+\frac{\partial f}{\partial y}(P)(y-1)=0$

Since

$\displaystyle \frac{\partial f}{\partial x}=-2x$

$\displaystyle \frac{\partial f}{\partial y}=1$

We then have

$\displaystyle -2x|_{x=1,y=1}(x-1)+1|_{x=1,y=1}(y-1)=0$

$\displaystyle -2(1)(x-1)+1(y-1)=0$

$\displaystyle -2x+2+y-1=0$

$\displaystyle y-2x+1=0$

The parabola is graphed (its real part, at least, using the Desmos graphing calculator) in the diagram below in red, with its tangent space, a line, in blue:

In case the reader is not convinced by our “borrowing” of concepts from calculus and claiming that they are “algebraic” in the specific case we are dealing with, another way to look at things without making reference to calculus is the following procedure, which comes from basic high school-level “analytic geometry”. First we translate the coordinate system so that the origin is at the point $P$ where we want to take the tangent space. Then we simply take the “linear part” of the polynomial equation, then translate again so that the origin is where it used to be originally. This gives the same results as the earlier procedure (the technical justification is given by the theory of Taylor series). More explicitly we have:

$\displaystyle y-x^{2}=0$

Translating the origin of coordinates to the point $x=1$, $y=1$, we have

$\displaystyle (y+1)-(x+1)^{2}=0$

$\displaystyle y+1-(x^{2}+2x+1)=0$

$\displaystyle y+1-x^{2}-2x-1=0$

$\displaystyle y-x^{2}-2x=0$

We take only the linear part, which is

$\displaystyle y-2x=0$

And then we translate the origin of coordinates back to the original one:

$\displaystyle (y-1)-2(x-1)=0$

$\displaystyle y-1-2x+2=0$

$\displaystyle y-2x+1=0$

which is the same result we had earlier.

But it may happen that the polynomial has no “linear part”. In this case the tangent space is the entirety of the ambient space. However, there is another related concept which may be useful in these cases, called the tangent cone. The tangent cone is the algebraic set which satisfies the equations we get by extracting the lowest degree part of the polynomial, which may or may not be the linear part. In the case that the lowest degree part is the linear part, the tangent space and the tangent cone coincide, and if this holds for all points of a variety, we say that the variety is nonsingular.

To give an explicit example, consider the curve $y^{2}=x^{3}+x^{2}$, as seen in the diagram below in red (its real part graphed once again using the Desmos graphing calculator):

The equation that defines this curve has no linear part. Therefore the tangent space at the origin consists of all $x$ and $y$ which satisfy the trivial equation $0=0$; but then, all values of $x$ and $y$ satisfy this equation, and therefore the tangent space is the “affine plane” $\mathbb{A}^{2}$. However, the lowest order part is $y^{2}=x^{2}$, which is satisfied by all points which also satisfy either of the two equations $y=x$ or $y=-x$. These points form the blue and orange diagonal lines in the diagram. Since the tangent space and the tangent cone do not agree, the curve is singular at the origin.

We can also define the tangent space in a more abstract manner, using the concepts we have discussed in Localization. Let $\mathfrak{m}$ be the unique maximal ideal of the local ring $O_{X,P}$, and let $\mathfrak{m}^{2}$ be the product ideal whose elements are the sums of products of elements of $\mathfrak{m}$. The quotient $\mathfrak{m}/\mathfrak{m}^{2}$ is then a vector space over the residue field $k$. The tangent space of $X$ at $P$ is then defined as the dual of this vector space (the vector space of linear transformations from $\mathfrak{m}/\mathfrak{m}^{2}$ to $k$). The vector space $\mathfrak{m}/\mathfrak{m}^{2}$ itself is called the cotangent space of $X$ at $P$. We can think of its elements as linear polynomial functions on the tangent space. There is an analogous abstract definition of the tangent cone, namely as the spectrum of the graded ring $\oplus_{i\geq 0}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$.

References:

Zariski Tangent Space on Wikipedia

Tangent Cone on Wikipedia

Desmos Graphing Calculator

Algebraic Geometry by J.S. Milne

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

# Grothendieck’s Relative Point of View

In Varieties and Schemes Revisited we defined the notion of schemes, which is a far-reaching generalization inspired by the concept of varieties, which is essentially a kind of “shape” defined by polynomials in some way. However, the definition of schemes were but one of many innovations in algebraic geometry developed by the mathematician Alexander Grothendieck. In this post, we discuss another of these innovations, the so-called “relative point of view“, in which the focus is not just on schemes in isolation, but schemes relative to (with a morphism to) some “base scheme”.

Let $S$ be a scheme. A scheme over $S$, or an $S$-scheme, is a scheme $X$ with a morphism $f:X\rightarrow S$ called the structural morphism. If $Y$ is another $S$-scheme with structural morphism $g:Y\rightarrow S$, a morphism of $S$-schemes is a morphism $u:X\rightarrow Y$ such that $f=g\circ u$.

If the scheme $S$ is the spectrum of some ring $R$, we may also refer to $X$ above as a scheme over $R$. Every ring has a morphism from the ring of ordinary integers $\mathbb{Z}$, and every scheme therefore has a morphism to the scheme $\text{Spec}(\mathbb{Z})$, so we may think of all schemes as schemes over $\mathbb{Z}$.

Given two schemes $X$ and $Y$ over a third scheme $S$, we define the fiber product $X\times_{S}Y$ to be a scheme together with projection morphisms $\pi_{X}:X\times_{S}Y\rightarrow X$ and $\pi_{Y}:X\times_{S}Y\rightarrow Y$ such that $f\circ\pi_{X}=g\circ\pi_{Y}$, and such that for any other scheme $Z$ and morphisms $p:Z\rightarrow X$ and $q:Z\rightarrow Y$, there is a unique morphism $Z\rightarrow X\times_{S}Y$ up to isomorphism (the concept of fiber product is part of category theory – see also More Category Theory: The Grothendieck Topos).

We can use the fiber product to introduce the concept of base change. Given a scheme $X$ over a scheme $S$, and a morphism $S'\rightarrow S$, the fiber product $X\times_{S}S'$ is a scheme over $S'$. We may think of it as being “induced” by the morphism $S'\rightarrow S$. One of the things that can be done with this idea of base change is to look at the properties of $X\times_{S}S'$ and see if we can use these to learn about the properties of $X$, which may be useful if the properties of $X$ are difficult to determine directly compared to the properties of $X\times_{S}S'$ (in essence we want to be able to attack a difficult problem indirectly by first attacking an easier problem related to it, which is a common strategy in mathematics).

A special case of base change is when $S'$ is given by the spectrum of the residue field (see Localization) $k$ corresponding to a point $P$ of $S$. There is a morphism of schemes $\text{Spec}(k)\rightarrow S$ which we may think of as the inclusion of the point $P$ into the scheme $X$. Then the fiber product $X\times_{S}\text{Spec}(k)$ is called the fiber of $X$ at the point $P$. The terminology is perhaps reminiscent of fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles), and is also rather similar to the concept of covering spaces (see Covering Spaces) in that we have some kind of space “over” every point of our “base” scheme. However, unlike those two earlier concepts, the spaces which make up our fibers may now vary as the points vary.

Actually, the concept that this special case of fiber product and base change should bring to mind is that of a moduli space (see The Moduli Space of Elliptic Curves), where every point represents a space, and the spaces vary as the points vary. Or, as we worded it in The Moduli Space of Elliptic Curves, every point of the moduli space (given by the base scheme) corresponds to a space (given by the fiber), and the moduli space tells us how these spaces vary, so that spaces which are similar to each other in some way correspond to points in the moduli space that are close together.

The lecture notes of Andreas Gathmann listed among the references below contain some nice diagrams to help visualize the idea of the fiber product and base change (these can be found in chapter 5 of the 2002 version). To see these ideas in action, one can look at the article Arithmetic on Curves by Barry Mazur (also among the references) which discusses, among other things, the approach taken by Gerd Faltings in proving the famous conjecture of Louis J. Mordell which says that there is a finite number of rational points on a curve of genus greater than $1$.

References:

Grothendieck’s Relative Point of View on Wikipedia

Arithmetic on Curves by Barry Mazur

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

# Varieties and Schemes Revisited

In Basics of Algebraic Geometry we introduced the idea of varieties and schemes as being kinds of “shapes” defined by polynomials (or rings, more generally) in some way. In this post we discuss the definitions of these concepts in more technical detail, and introduce other important concepts related to algebraic geometry as well.

##### I. Preliminaries: Affine Space, Algebraic Sets and Ringed Spaces

Affine $n$-space, written $\mathbb{A}^{n}$, is the set of all $n$-tuples of elements of a field $k$, i.e.

$\displaystyle \mathbb{A}^{n}=\{(a_{1},...,a_{n})|a_{i}\in k \text{ for }1\leq i\leq n\}$.

An algebraic set is a subset of $\mathbb{A}^{n}$ that is the zero set $Z(T)$ of some set $T$ of polynomials, i.e. $Y=Z(T)$, where

$\displaystyle Z(T)=\{P\in \mathbb{A}^{n}|f(P)=0 \text{ for all } f\in T\}$.

Intuitively, we want to define a “variety” as some kind of space which “locally” looks like an irreducible algebraic set. “Irreducible” means it cannot be expressed as the union of other algebraic sets. However, we want to think of a variety as more than just a space; we want to think of it as a space with things (namely functions) “living on it”. This leads us to the notion of a ringed space.

A ringed space is simply a pair $(X,\mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf (see  Sheaves) of rings on $X$. A morphism of ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$.

Recall that a morphism of sheaves of rings $\varphi:\mathcal{F}\rightarrow \mathcal{G}$ for sheaves of rings $\mathcal{F}$ and $\mathcal{G}$ on X is given by a morphism of rings $\varphi(U): \mathcal{F}(U)\rightarrow \mathcal{G}(U)$ for every open set $U$ of $X$ such that for $V\subseteq{U}$ we have $\rho_{U,V}\circ\varphi(U)=\varphi(V)\circ\rho'_{U,V}$, where $\rho_{U,V}$ and $\rho'_{U,V}$ are the restriction maps of $\mathcal{F}$ and $\mathcal{G}$.

We might as well mention locally ringed spaces here, since they will be used to define the concept of schemes later on:

A locally ringed space is a ringed space $(X,\mathcal{O}_{X})$ such that for each point $P$ of $X$, the stalk $\mathcal{O}_{X,P}$ is a local ring (see Localization). A morphism of locally ringed spaces from $(X,O_{X})$ to $(Y,O_{Y})$ is given by a continuous map $f: X\rightarrow Y$ and a morphism of sheaves of rings $f^{\#}: \mathcal{O}_{Y}\rightarrow f_{*}\mathcal{O}_{X}$ such that $(f_{P}^{\#})^{-1}(\mathfrak{m}_{X,P})=\mathfrak{m}_{Y,f(P)}$ for all $P$ where $f_{P}^{\#}: \mathcal{O}_{Y,f(P)}\rightarrow \mathcal{O}_{X,P}$ is the map induced on the stalk at $P$.

##### II. Varieties in Three Steps:  Affine Varieties, Prevarieties, and Varieties

We now set out to accomplish our goal of defining “varieties” as spaces that locally look like irreducible algebraic sets. We first start with a ringed space that just “looks like” an irreducible algebraic set:

An affine variety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and $X$ is isomorphic to an irreducible algebraic set in $\mathbb{A}^{n}$.

Next, we define a more general kind of ringed space, that is required to look like an irreducible algebraic set only “locally”:

A prevariety is a ringed space $(X,\mathcal{O}_{X})$ such that $X$ is irreducible, $O_{X}$ is a sheaf of $k$-valued functions, and there is a finite open cover $U_{i}$ such that $(U_{i},\mathcal{O}_{X}|_{U_{i}})$ is an affine variety for all $i$.

We are almost done. However, there is one more nice property that we would like our varieties to have. A topological space $X$ is said to have the Hausdorff property if two distinct points always have two disjoint neighborhoods. With the Zariski topology this is almost always impossible; however there is an analogous notion which is satisfied if the image of the “diagonal morphism” which sends the point $P$ in $X$ to the point $(P,P)$ in $X\times X$ is closed in $X\times X$. There is an analogous notion of “product” in algebraic geometry; therefore, we can define the concept of variety as follows:

A variety is a prevariety $X$ such that the diagonal morphism is closed in $X\times X$. In the rest of this post, we will refer to this property as the “algebro-geometric” analogue of the Hausdorff property.

##### III. Schemes

We now define the concept of schemes, which, as we shall show in the next section, generalize the concept of varieties, i.e. varieties are just a special case of schemes. Inspired by the correspondence between the maximal ideals of the “ring of polynomial functions” (with coefficients in an “algebraically closed field” like the complex numbers) of an algebraic set and the points of the algebraic set mentioned in Basics of Algebraic Geometry, we go further and consider a ringed space whose underlying topological space has points corresponding to the prime ideals of a ring (which is not necessarily a ring of polynomials – we might even consider, for example, the ring of ordinary integers $\mathbb{Z}$, or the ring of integers of an algebraic number field –  see Algebraic Numbers).

The spectrum (note that the word “spectrum” has many different meanings in mathematics, and this particular usage is different, say, from that in Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) of a ring is a  locally ringed space $(\text{Spec}(A)),\mathcal{O}$, where $\text{Spec}(A)$ is the set of prime ideals of $A$ equipped with the Zariski topology, and $\mathcal O$ is a sheaf on $\text{Spec}(A)$ given by defining $\mathcal{O}(U)$ to be the set of functions $s:U\rightarrow \coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$, such that $s(\mathfrak{p})\in A_\mathfrak{p}$ for each $\mathfrak{p}\in U$, and such that for each $\mathfrak{p}\in U$, there is an open set $V\subseteq U$ containing $\mathfrak{p}$ and elements $a,f\in A$ such that for each $\mathfrak{q}\in V$, $f\notin \mathfrak{q}$, and $s(\mathfrak{q})=a/f$ in $A_{\mathfrak{q}}$.

We now proceed to define schemes, closely mirroring how we defined varieties earlier:

An affine scheme is a locally ringed space $(X,\mathcal{O}_{X})$ that is isomorphic as a locally ringed space to the spectrum of some ring.

A scheme is a locally ringed space $(X,\mathcal{O}_{X})$ where every point is contained in some open set $U$ such that $U$ considered as a topological space, together with the restricted sheaf $\mathcal{O}_{X}|_{U}$, is an affine scheme. A morphism of schemes is a morphism as locally ringed spaces.

Finally, to complete the analogy with varieties, we refer to schemes which have the (analogue of the) Hausdorff property as separated schemes.

Note: In some of the (mostly older) literature, what we refer to as schemes in this post are instead referred to as preschemes, in analogy with prevarieties. What they call a scheme is what we refer to as a separated scheme, i.e. a scheme possessing the Hausdorff property. I have no idea at the moment as to why this rather nice terminology was changed, but in this post we stick with the modern convention.

##### IV. Prevarieties and Varieties as Special Kinds of Schemes

We now discuss varieties as special cases of schemes. First we need to define what properties we would like our schemes to have, in order to fit with how we described varieties earlier (as ringed spaces which locally look like irreducible spaces defined by polynomials). Therefore, we have to mimic certain properties of polynomial rings.

We first note that polynomials over a field are finitely generated algebras over some field $k$. A scheme is said to be of finite type over the field $k$ if the affine open sets are each isomorphic to the spectrum of some ring which is a finitely generated algebra over $k$. More generally, given a morphism of schemes $X\rightarrow Y$, there is a concept of $X$ being a scheme of finite type over $Y$, but we will leave this to the references for now.

Next we note that polynomials over a field are integral domains. This means that whenever there are two polynomials $f$ and $g$ with the property that $fg=0$, then either $f=0$ or $g=0$. A scheme is integral if each the affine open sets are each isomorphic to the spectrum of some ring which is an integral domain. An equivalent condition is for the scheme to be irreducible and reduced (this means that the ring specified above has no nilpotent elements, i.e. elements where some power is equal to zero).

We therefore redefine a prevariety as an integral scheme of finite type over the field $k$. As with the earlier definition, a variety is a prevariety with the (analogue of the) Hausdorff property (i.e. an integral separated scheme of finite type over $k$).

##### V. Conclusion

In conclusion, we have started with essentially the same ideas as the “analytic geometry” of Pierre de Fermat and Rene Descartes, familiar to high school students everywhere, used to describe shapes such as lines, circles, conics (parabolas, hyperbolas, circles, and ellipses), and so on. From there we generalized to get more shapes, which resemble only these old shapes “locally” (we may also think of these new shapes as being “glued” from the old ones). To maintain certain familiar properties expected of shapes, we impose the analogue of the Hausdorff property. We then obtain the concept of a variety.

But we can generalize much, much farther to more than just polynomial rings. We can define “spaces” which come from rings which need not be polynomial rings, such as the ring of ordinary integers $\mathbb{Z}$ (or more generally algebraic integers – we have actually hinted at these applications of algebraic geometry in Divisors and the Picard Group). We can then have a kind of “geometry” of these rings, which gives us methods analogous to the powerful methods of geometry, which can be applied to branches of mathematics we would not usually think of as being “geometric”, such as number theory, as we have mentioned above. We end this post with quotes from two of the pioneers of modern mathematics (these quotes are also found in the book Algebra by Michael Artin):

“To me algebraic geometry is algebra with a kick.”

-Solomon Lefschetz

“In helping geometry, modern algebra is helping itself above all.”

-Oscar Zariski

References:

Algebraic Variety on Wikipedia

Scheme on Wikipedia

Ringed Space on Wikipedia

Abstract Varieties on Rigorous Trivialities

Schemes on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

Algebra by Michael Artin

# Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry

I’m going to be fairly busy in the coming days, so instead of the usual long post, I’m going to post here some links to interesting stuff I’ve found online (related to the subjects stated on the title of this post).

In the previous post, An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, we discussed Calabi-Yau manifolds (which are special cases of Kahler manifolds) and how their interesting properties, namely their Riemannian, symplectic, and complex aspects figure into the branch of mathematics called mirror symmetry, which is inspired by the famous, and sometimes controversial, proposal for a theory of quantum gravity (and more ambitiously a candidate for the so-called “Theory of Everything”), string theory.

We also mentioned briefly a famous open problem concerning Kahler manifolds called the Hodge conjecture (which was also mentioned in Algebraic Cycles and Intersection Theory). The links I’m going to provide in this post will be related to this conjecture.

As with the post An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, aside from introducing the subject itself, another of the primary intentions will be to motivate and explore aspects of algebraic geometry such as complex algebraic geometry, and their relation to other branches of mathematics.

Here is the page on the Hodge conjecture, found on the website of the Clay Mathematics Institute:

Hodge Conjecture on Clay Mathematics Institute

We have mentioned before that the Hodge conjecture is one of seven “Millenium Problems” for which the Clay Mathematics Institute is offering a million dollar prize. The page linked to above contains the official problem statement as stated by Pierre Deligne, and a link to a lecture by Dan Freed, which is intended for a general audience and quite understandable. The lecture by Freed is also available on Youtube:

Dan Freed on the Hodge Conjecture at the Clay Mathematics Institute on Youtube

Unfortunately the video of that lecture has messed up audio (although the lecture remains understandable – it’s just that the audio comes out of only one side of the speakers or headphones). Here is another set of videos by David Metzler on Youtube, which explains the Hodge conjecture (along with the other Millennium Problems) to a general audience:

Millennium Problem Talks on Youtube

The Hodge conjecture is also related to certain aspects of number theory. In particular, we have the Tate conjecture, which is another conjecture similar to the Hodge conjecture, but more related to Galois groups (see Galois Groups). Alex Youcis discusses it on the following post on his blog Hard Arithmetic:

The Tate Conjecture over Finite Fields on Hard Arithmetic

On the same blog there is also a discussion of a version of the Hodge conjecture called the $p$-adic Hodge conjecture on the following post:

An Invitation to p-adic Hodge Theory; or How I Learned to Stop Worrying and Love Fontaine on Hard Arithmetic

The first part of the post linked to above discusses the Hodge conjecture in its classical form, while the second part introduces $p$-adic numbers and related concepts, some aspects of which were discussed on this blog in Valuations and Completions.

A more technical discussion of the Hodge conjecture, Kahler manifolds, and complex algebraic geometry can be found in the following lecture of Claire Voisin, which is part of the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad, India:

On the Cohomology of Algebraic Varieties by Claire Voisin

More about these subjects will hopefully be discussed on this blog at sometime in the future.