More on Sheaves

In Sheaves we introduced the concept of sheaves as mathematical objects that “live on” a space and can be “patched together ” in a certain way. As an example we introduced the concept of the sheaf of regular functions on the complex plane $\mathbb{C}$ . The complex plane is one example of a variety, which we defined in Basics of Algebraic Geometry as a “shape” that is described by the zero set of polynomial equations. The concept of the sheaf of regular functions can also be generalized to varieties other than $\mathbb{C}$ . The sheaf of regular functions on a variety $X$ is also called the structure sheaf on $X$ and is written $\mathcal{O}_{X}$ .

In this post we will give some more important kinds of sheaves.

Twisting Sheaves

We will start with the twisting sheaf. For this we need the notion of projective space, which we introduced in Projective Geometry. We know that projective space provides us with many advantages, in particular “points at infinity”, but they come at the cost of some new language – for example, we require that our polynomials be homogeneous, which means that every term of such a polynomial must be of the same degree. The zero set of such a polynomial then defines a projective variety.

The definition of the sheaf of regular functions on a projective variety also has some differences compared to that on an affine space. To protect our definition of projective space, we need the numerator and the denominator to always have the same degree. This has the effect that the only regular functions defined everywhere on a projective variety are the constant functions.

The twisting sheaves, written $\mathcal{O}_{X}(n)$ for an integer $n$ , are made up of expressions $\frac{f}{g}$ where $f$ and $g$ are homogeneous polynomials and the degree of $f$ is equal to $d+n$ , where $d$ is the degree of $g$ . We also require for each open set $U$ that $g$ never be zero on any point of $U$ , as in the definition of the regular functions on $U$ . The sheaf of regular functions on $X$ is then just the twisting sheaf when $n=0$ . Twisting sheaves are isomorphic to the sheaf of regular functions “locally“, i.e. on open sets of the space, but not “globally“.

Sheaves of Modules and Quasi-Coherent Sheaves

Twisting sheaves can be thought of as sheaves of modules, with the sheaf of regular functions serving as their “scalars”. More generally, sheaves of modules play an important part in algebraic geometry. In the same way that a ring $R$ determines the sheaf of regular functions $\mathcal{O}_{X}$ on the affine scheme $X=\text{Spec}(R)$ , $R$ -modules can always give rise to sheaves of $\mathcal{O}_{X}$ -modules on $X$ . However, not all sheaves of $\mathcal{O}_{X}$ -modules come from $R$ -modules. In the special case that they do, they are referred to as quasi-coherent sheaves. Quasi-coherent sheaves are interesting because we have ways of constructing new modules from old ones, for instance using the tensor product or direct product, hence, we can also construct new sheaves of modules from old ones.

Locally Free Sheaves, Vector Bundles, and Line Bundles

A quasi-coherent sheaf $\mathcal{F}$ such that $\mathcal{F}|_{U_{i}}$ is isomorphic to the quasi-coherent sheaf $\mathcal{O}_{U_{i}}^{\oplus^{r}}$ ,i.e. a direct sum of $r$ copies of the sheaf of regular functions, is called a locally free sheaf of rank $r$ . Locally free sheaves correspond to vector bundles, which we have already discussed in the context of differential geometry and algebraic topology (see Vector Fields, Vector Bundles, and Fiber Bundles). A locally free module of rank $1$ is also known as a line bundle. As we have mentioned earlier, a twisting sheaf is locally isomorphic to the sheaf of regular functions, therefore, it is an example of a line bundle.

Sheaves of Differentials and the Cotangent Bundle

We now discuss the concept of differentials. As may be inferred from the name, this concept is somewhat related to concepts in calculus, such as tangents. However, in algebraic geometry we want to be able to define things algebraically, as this contributes to the strength of algebraic geometry in relating algebra and geometry. In addition, in algebraic geometry we may consider not only real and complex numbers but also rational numbers or even finite fields, and some of the methods we have developed in calculus may not always be immediately applicable to the case at hand. Therefore we must “redefine” these objects algebraically, even if they are going to be conceptually inspired by the objects we are already familiar with from calculus.

We now give the definition of differentials (which in the context of algebraic geometry are also called Kahler differentials). Given a homomorphism of rings $S\rightarrow R$ , the module of relative differentials, denoted $\Omega_{R/S}$ , to be the free $R$ -module generated by the formal symbols $\{dr|r\in R\}$ , modulo the following relations:

$\displaystyle d(r_{1}+r_{2})=dr_{1}+dr_{2}$ for $r_{1},r_{2}\in R$

$\displaystyle d(r_{1}r_{2})=r_{1}dr_{2}+r_{2}dr_{1}$ for $r_{1},r_{2}\in R$

$\displaystyle ds=0$ for $s\in S$

If we have schemes $X$ and $Y$ whose open subsets are given by the spectrum $\text{Spec}(R)$ of some ring $R$ , and a morphism $X\rightarrow Y$ , we have for each of these open subsets a module of relative differentials which we can “glue together” to form a quasi-coherent sheaf called the sheaf of relative differentials, written using the symbol $\Omega_{X/Y}$ . If $Y$ is a point, i.e. it is the spectrum $\text{Spec }k$ of some field $k$ , we simply write $\Omega_{X}$ .

The sheaf of relative differentials $\Omega_{X}$ is also known as the cotangent bundle, since it is dual to the tangent bundle. From the cotangent bundle we can form the canonical bundle by taking exterior products. The exterior product $x\wedge y$ is the tensor product of $x$ and $y$ modulo the relation $x\wedge y=-y\wedge x$ . The canonical bundle is then the top exterior power of the cotangent bundle, i.e. $\wedge^{\text{dim}(X)}\Omega_{X}$ . It is yet another example of a line bundle.

Line Bundles and Divisor Classes

Line bundles (including the canonical bundle) on curves are closely related to divisors (see Divisors and the Picard Group). In fact, the set of all line bundles on a curve $X$ is the same as the Picard group (the group of divisor classes) of $X$ . We will not prove this, but we will elaborate a little on the construction that gives the correspondence between line bundles and divisor classes. Since a line bundle is locally isomorphic to the sheaf of regular functions, a section $s$ of the line bundle corresponds, at least on some open set $U$ to some regular function on $U$ that we denote by $\psi(s)$ . Let $P$ be a point in $U$ . We define the order of vanishing $\text{ord}_{P}s$ of the section $s$ as the order of vanishing of the regular function $\psi(s)$ at $P$ .

A rational section of a line bundle is a section of the bundle possibly multiplied by a rational function (which may not necessarily be a function in the set-theoretic sense but merely an expression which is a “fraction” of polynomials). Similar to the case of ordinary sections of the line bundle and regular functions, there is also a correspondence between rational sections and rational functions. We then define the divisor $(s)$ associated to a rational section $s$ by

$\displaystyle (s)=\sum_{P\in X}\text{ord}_{P}s\cdot P$

On the other hand, given a divisor $D$ , we may obtain a line bundle by associating to the divisor $D$ the set of all rational functions with divisor $(\psi)$ , such that

$\displaystyle (\psi)+D\geq 0$ .

The notation means that when we formally add the divisors $(\psi)$ and $D$ , the resulting sum has coefficients which are all greater than or equal to $0$ . We refer to such a divisor as an effective divisor. Thus we have a means of associating divisors to line bundles and vice-versa, and it is a theorem, which we will not prove, that this gives a correspondence between line bundles and divisor classes.

Preview of the Riemann-Roch Theorem

The correspondence between line bundles and divisor classes will allow us to state the Riemann-Roch theorem (once again, without proof, for now) for the case of complex smooth projective curves. Let $h^{0}(D)$ denote the dimension of the vector space of global sections of the line bundle $\mathcal{L}$ , with corresponding divisor $D$ . We recall that the degree $\text{deg}(D)$ of a divisor $D$ is the sum of its coefficients. The genus $g$ of a curve roughly gives the “number of holes” of the curve as a space whose points have coordinates that are complex numbers (recall that the complex points of a curve actually form a surface – for example, an elliptic curve is actually a torus, which has genus equal to $1$ ). The Riemann-Roch theorem relates all these concepts. Let $K_{X}$ denote the divisor corresponding to the canonical bundle of the curve $X$ , and let $D$ be any divisor on $X$ . Then the Riemann-Roch theorem is the following statement:

$\displaystyle h^{0}(D)+h^{0}(K_{X}+D)=\text{deg}(D)+1-g$

More on the Riemann-Roch theorem, including its proof, examples of its applications, and generalization to varieties other than curves, will be left to the references for now. It is intended and hoped for, however, that these subjects will be tackled at some later time on this blog.

References:

Sheaf of Modules on Wikipedia

Coherent Sheaf on Wikipedia

Kahler Differential on Wikipedia

Cotangent Sheaf on Wikipedia

Cotangent Bundle on Wikipedia

Canonical Bundle on Wikipedia

Sheaves of Modules by Charles Siegel on Rigorous Trivialities

Locally Free Sheaves and Vector Bundles by Charles Siegel on Rigorous Trivialities

Line Bundles and the Picard Group by Charles Siegel on Rigorous Trivialities

Differential Forms and the Canonical Bundle by Charles Siegel on Rigorous Trivialities

Riemann-Roch Theorem for Curves by Charles Siegel on Rigorous Trivialities