An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry

String theory is by far the most popular of the current proposals to unify the as of now still incompatible theories of quantum mechanics and general relativity. In this post we will give a short overview of the concepts involved in string theory, but not with the goal of discussing the theory itself in depth (hopefully there will be more posts in the future working towards this task). Instead, we will focus on introducing a very interesting and very beautiful branch of mathematics that arose out of string theory called mirror symmetry. In particular, we will focus on a version of it originally formulated by the mathematician Maxim Kontsevich in 1994 called homological mirror symmetry.

We will start with string theory. String theory started out as a theory of the nuclear forces that held together the protons and electrons in the nucleus of an atom. It was abandoned later on, due to a more successful theory called quantum chromodynamics taking its place. However, it was soon found out that string theory could model the elusive graviton, a particle “carrier” of gravity in the same way that a photon is a particle “carrier” of electromagnetism (the photon is more popularly referred to as a particle of light, but because light itself is an electromagnetic wave, it is also a manifestation of an electromagnetic field), and since then physicists have started developing string theory, no longer in the sole context of nuclear forces, but as a possible candidate for a working theory of quantum gravity.

The incompatibility of quantum mechanics and general relativity (which is currently our accepted theory of gravity) arises from the nonrenormalizability of gravity. In calculations in quantum field theory (see Some Basics of Relativistic Quantum Field Theory and Some Basics of (Quantum) Electrodynamics), there appear certain “nonsensical” quantities which are made sense of via a “corrective” procedure called renormalization (not to be confused with some other procedures called “normalization”). While the way that renormalization works is not really completely understood at the moment, it is known that this procedure at least “works” – this means that it produces the correct values of quantities, as can be checked via experiment.

Renormalization, while it works for the other forces, however fails for gravity. Roughly this is sometimes described as gravity “wildly fluctuating” at the smallest scales. What we know is that this signals, for us, a lack of knowledge of  what physics is like at these extremely small scales (much smaller than the current scale of quantum mechanics).

String theory attempts to solve this conundrum by proposing that particles, at the very smallest scales, are not “particles” at all, but “strings”. This takes care of the problem of fluctuations at the smallest scales, since there is a limit to how small the scale can be, set by the length of the strings. It is perhaps worth noting at this point that the next most popular contender to string theory, loop quantum gravity, tackles this problem by postulating that space itself is not continuous, but “discretized” into units of a certain length. For both theories, this length is predicted to be around 10^{-35} meters, a constant quantity which is known as the Planck length.

Over time, as string theory was developed, it became more ambitious, aiming to provide not only the unification of quantum mechanics and general relativity, but also the unification of the four fundamental forces – electromagnetism, the weak nuclear force, the strong nuclear force, and gravity, under one “theory of everything“. At the same time, it needed more ingredients – to be able to account for bosons, the particles carrying “forces”, such as photons and gravitons, and the fermions, particles that make up matter, such as electrons, protons, and neutrons, a new ingredient had to be added, called supersymmetry. In addition, it worked not in the four dimensions of spacetime that we are used to, but instead required ten dimensions (for the “bosonic” string theory, before supersymmetry, the number of dimensions required was a staggering twenty-six)!

How do we explain spacetime having ten dimensions, when we experience only four? It turns out, even before string theory, the idea of extra dimensions was already explored by the physicists Theodor Kaluza and Oskar Klein. They proposed a theory unifying electromagnetism and gravity by postulating an “extra” dimension which was “curled up” into a loop so small we could never notice it. The usual analogy is that of an ant crossing a wire – when the radius of the wire is big, the ant realizes that it can go sideways along the wire, but when the radius of the wire is small, it is as if there is only one dimension that the ant can move along.

So we now have this idea of six curled up dimensions of spacetime, in addition to the usual four. It turns out that there are so many ways that these dimensions can be curled up. This phenomenon is called the string theory landscape, and it is one of the biggest problems facing string theory today. What could be the specific “shape” in which these dimensions are curled up, and why are they not curled up in some other way? Some string theorists answer this by resorting to the controversial idea of a multiverse, so that there are actually several existing universes, each with its own way of how the extra six dimensions are curled up, and we just happen to be in this one because, perhaps, this is the only one where the laws of physics (determined by the way the dimensions are curled up) are able to support life. This kind of reasoning is called the anthropic principle.

In addition to the string theory landscape, there was also the problem of having several different versions of string theory. These problems were perhaps alleviated by the discovery of mysterious dualities. For example, there is the so-called T-duality, where a compactification (a “curling up”) with a bigger radius gives the same laws of physics as a compactification with a smaller, “reciprocal” radius. Not only do the concept of dualities connect the different ways in which the extra dimensions are curled up, they also connect the several different versions of string theory! In 1995, the physicist Edward Witten conjectured that this is perhaps because all these different versions of string theory come from a single “mother theory”, which he called “M-theory“.

In 1991, physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes used these dualities to solve a mathematical problem that had occupied mathematicians for decades, that of counting curves on a certain manifold (a manifold is a shape without sharp corners or edges) known as a Calabi-Yau manifold. In the context of Calabi-Yau manifolds, which are some of the shapes in which the extra dimensions of spacetime are postulated to be curled up, these dualities are known as mirror symmetry. With the success of Candelas, de la Ossa, Green, and Parkes, mathematicians would take notice of mirror symmetry and begin to study it as a subject of its own.

Calabi-Yau manifolds are but special cases of Kahler manifolds, which themselves are very interesting mathematical objects because they can be studied using three aspects of differential geometry – Riemannian geometry, symplectic geometry, and complex geometry.

We have already encountered examples of Kahler manifolds on this blog – they are the elliptic curves (see Elliptic Curves and The Moduli Space of Elliptic Curves). In fact elliptic curves are not only Kahler manifolds but also Calabi-Yau manifolds, and they are the only two-dimensional Calabi-Yau manifolds (we sometimes refer to them as “one-dimensional” when we are considering “complex dimensions”, as is common practice in algebraic geometry – this apparent “discrepancy” in counting dimensions arises because we need two real numbers to specify a complex number). In string theory of course we consider six-dimensional (three-dimensional when considering complex dimensions) Calabi-Yau manifolds, since there are six extra curled up dimensions of spacetime, but often it is also fruitful to study also the other cases, especially the simpler ones, since they can serve as our guide for the study of the more complicated cases.

Riemannian geometry studies Riemannian manifolds, which are manifolds equipped with a metric tensor, which intuitively corresponds to an “infinitesimal distance formula” dependent on where we are on the manifold. We have already encountered Riemannian geometry before in Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry. There we have seen that Riemannian geometry is very important in the mathematical formulation of general relativity, since in this theory gravity is just the curvature of spacetime, and the metric tensor expresses this curvature by showing how the formula for the infinitesimal distance between two points (actually the infinitesimal spacetime interval between two events) changes as we move around the manifold.

Symplectic geometry, meanwhile, studies symplectic manifolds. If Riemannian manifolds are equipped with a metric tensor that measures “distances”, symplectic manifolds are equipped with a symplectic form that measures “areas”. The origins of symplectic geometry are actually related to William Rowan Hamilton’s formulation of classical mechanics (see Lagrangians and Hamiltonians), as developed later on by Henri Poincare. There the object of study is phase space, which gives the state of a system based on the position and momentum of the objects that comprise it. It is this phase space that is expressed as a symplectic manifold.

Complex geometry, following our pattern, studies complex manifolds. These are manifolds which locally look like \mathbb{C}^{n}, in the same way that ordinary differentiable manifolds locally look like \mathbb{R}^{n}. Just as Riemannian geometry has metric tensors and symplectic geometry has symplectic forms, complex geometry has complex structures, mappings of tangent spaces with the property that applying them twice is the same as multiplication by -1, mimicking the usual multiplication by the imaginary unit i on the complex plane.

Complex manifolds are not only part of differential geometry, they are also often studied using the methods of algebraic geometry! We recall (see Basics of Algebraic Geometry) that algebraic geometry studies varieties and schemes, which are shapes such as lines, conic sections (parabolas, hyperbolas, ellipses, and circles), and elliptic curves, that can be described by polynomials (their modern definitions are generalizations of this concept). In fact, all Calabi-Yau manifolds can be described by polynomials, such as the following example, due to user Andrew J. Hanson of Wikipedia:

CalabiYau5

This is a visualization (actually a sort of “cross section”, since we can only display two dimensions and this object is actually six-dimensional) of the Calabi-Yau manifold described by the following polynomial equation:

\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0

This polynomial equation (known as the Fermat quintic) actually describes the Calabi-Yau manifold  in projective space using homogeneous coordinates. This means that we are using the concepts of projective geometry (see Projective Geometry) to include “points at infinity“.

We note at this point that Kahler manifolds and Calabi-Yau manifolds are interesting in their own right, even outside of the context of string theory. For instance, we have briefly mentioned in Algebraic Cycles and Intersection Theory the Hodge conjecture, one of seven “Millenium Problems” for which the Clay Mathematics Institute is currently offering a million-dollar prize, and it concerns Kahler manifolds. Perhaps most importantly, it “unifies” several different branches of mathematics; as we have already seen, the study of Kahler manifolds and Calabi-Yau manifolds involves Riemannian geometry, symplectic geometry, complex geometry, and algebraic geometry. The more recent version of mirror symmetry called homological mirror symmetry further adds category theory and homological algebra to the mix.

Now what mirror symmetry more specifically states is that a version of string theory called Type IIA string theory, on a spacetime with extra dimensions compactified onto a certain Calabi-Yau manifold V, is the same as another version of string theory, called Type IIB string theory, on a spacetime with extra dimensions compactified onto another Calabi-Yau manifold W, which is “mirror” to the Calabi-Yau manifold V.

The statement of homological mirror symmetry (which is still conjectural, but mathematically proven in certain special cases) expresses the idea of the previous paragraph as follows (quoted verbatim from the paper Homological Algebra of Mirror Symmetry by Maxim Kontsevich):

Let (V,\omega) be a 2n-dimensional symplectic manifold with c_{1}(V)=0 and W be a dual n-dimensional complex algebraic manifold.

The derived category constructed from the Fukaya category F(V) (or a suitably enlarged one) is equivalent to the derived category of coherent sheaves on a complex algebraic variety W.

The statement makes use of the language of category theory and homological algebra (see Category TheoryMore Category Theory: The Grothendieck ToposEven More Category Theory: The Elementary ToposExact SequencesMore on Chain Complexes, and The Hom and Tensor Functors), but the idea that it basically expresses is that there exists a relation between the symplectic aspects of the Calabi-Yau manifold V, as encoded in its Fukaya category, and the complex aspects of the Calabi-Yau manifold W, as encoded in its category of coherent sheaves (see Sheaves and More on Sheaves). As we have said earlier, the subjects of algebraic geometry and complex geometry are closely related, and hence the language of sheaves show up in (and is an important part of) both subjects. The concept of derived categories, which generalize derived functors like the Ext and Tor functors, allow us to relate the two categories, which otherwise would be expressing different concepts. Inspired by string theory, therefore, we have now a deep and beautiful idea in geometry, relating its different aspects.

Is string theory the correct way towards a complete theory of quantum gravity, or the so-called “theory of everything”? As of the moment, we don’t know. Quantum gravity is a very difficult problem, and the scales involved are still far out of our reach – in order to probe smaller and smaller scales we need particle accelerators with higher and higher energies, and right now the technologies that we have are still very, very far from the scales which are relevant to quantum gravity. Still, it is hoped for that whatever we find in experiments in the near future, not only in the particle accelerators but also in the radio telescopes that look out into space, will at least guide us towards the correct path.

There are some who believe that, in the absence of definitive experimental evidence, mathematical beauty is our next best guide. And, without a doubt, string theory is related to, and has inspired, some very beautiful and very interesting mathematics, including that which we have discussed in this post. Still, physics, like all natural science, is empirical (based on evidence and observation), and hence it is ultimately physical evidence that will be the judge of correctness. It may yet turn out that string theory is wrong, and that it is a different theory which describes the fundamental physical laws of nature, or that it needs drastic modifications to its ideas. This will not invalidate the mathematics that we have described here, anymore than the discoveries of Copernicus invalidated the mathematics behind the astronomical model of Ptolemy – in fact this mathematics not only outlived the astronomy of Ptolemy, but served the theories of Copernicus, and his successors, just as well. Hence we cannot really say that the efforts of Ptolemy were wasted, since even though his scientific ideas were shown to be wrong, still his mathematical methods were found very useful by those who succeeded him. Thus, while our current technological limitations prohibit us from confirming or ruling out proposals for a theory of quantum gravity such as string theory, there is still much to be gained from such continued efforts on the part of theory, while experiment is still in the process of catching up.

Our search for truth continues. Meanwhile, we have beauty to cultivate.

References:

String Theory on Wikipedia

Mirror Symmetry on Wikipedia

Homological Mirror Symmetry on Wikipedia

Calabi-Yau Manifold on Wikipedia

Kahler Manifold on Wikipedia

Riemannian Geometry on Wikipedia

Symplectic Geometry on Wikipedia

Complex Geometry on Wikipedia

Fukaya Category on Wikipedia

Coherent Sheaf on Wikipedia

Derived Category on Wikipedia

Image by User Andrew J. Hanson of Wikipedia

Homological Algebra of Mirror Symmetry by Maxim Kontsevich

The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory by Brian Greene

String Theory by Joseph Polchinski

String Theory and M-Theory: A Modern Introduction by Katrin Becker, Melanie Becker, and John Schwarz

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The Hom and Tensor Functors

We discussed functors in Category Theory, and in this post we discuss certain functors important to the study of rings and modules. Moreover, we look at these functors and how they affect exact sequences, whose importance was discussed in Exact Sequences. Our discussion in this post will also be related to some things that we discussed in More on Chain Complexes.

If M and N are two modules whose ring of scalars is the ring R (we refer to M and N as R-modules), then we denote by \text{Hom}_{R}(M,N) the set of linear transformations (see Vector Spaces, Modules, and Linear Algebra) from M to N. It is worth noting that this set has an abelian group structure (see Groups).

We define the functor \text{Hom}_{R}(M,-) as the functor that assigns to an R-module N the abelian group \text{Hom}_{R}(M,N) of linear transformations from M to N. Similarly, the functor \text{Hom}_{R}(-,N) assigns to the R-module M the abelian group \text{Hom}_{R}(M,N) of linear transformations from M to N.

These functors \text{Hom}_{R}(M,-) and \text{Hom}_{R}(-,N), combined with the idea of exact sequences, give us new definitions of projective and injective modules, which are equivalent to the old ones we gave in More on Chain Complexes.

We say that a functor is an exact functor if preserves exact sequences. In the case of \text{Hom}_{R}(M,-), we say that it is exact if for an exact sequence of modules

0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0

the sequence

0\rightarrow \text{Hom}_{R}(M,A)\rightarrow \text{Hom}_{R}(M,B)\rightarrow \text{Hom}_{R}(M,C)\rightarrow 0

is also exact. The concept of an exact sequence of sets of linear transformations of R-modules makes sense because of the abelian group structure on these sets. In this case we also say that the R-module M is projective.

Similarly, an R-module N is injective if the functor \text{Hom}_{R}(-,N) is exact, i.e. if for an exact sequence of modules

0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0

the sequence

0\rightarrow \text{Hom}_{R}(A,N)\rightarrow \text{Hom}_{R}(B,N)\rightarrow \text{Hom}_{R}(C,N)\rightarrow 0

is also exact.

We introduce another functor, which we write M\otimes_{R}-. This functor assigns to an R-module N the tensor product (see More on Vector Spaces and Modules) M\otimes_{R}N. Similarly, we also have the functor -\otimes_{R}N, which assigns to an R-module M the tensor product M\otimes_{R}N. If our ring R is commutative, then there will be no distinction between the functors M\otimes_{R}- and -\otimes_{R}M. We will continue assuming that our rings are commutative (an example of a noncommutative ring is the ring of n\times n matrices).

We say that a module N is flat if the functor -\otimes_{R}N is exact, i.e. if for an exact sequence of modules

0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0

the sequence

0\rightarrow A\otimes_{R}N\rightarrow B\otimes_{R}N\rightarrow C\otimes_{R}N\rightarrow 0

is also exact.

We make a little digression to introduce the concept of an algebra. The word “algebra” has a lot of meanings in mathematics, but in our context, as a mathematical object in the subject of abstract algebra and linear algebra, it means a set with both a ring and a module structure. More technically, for a ring A, an A-algebra is a ring B and a ring homomorphism f:A\rightarrow B, which makes B into an A-module via the following definition of the scalar multiplication:

ab=f(a)b for a\in A, b\in B.

The notion of an algebra will be useful in defining the notion of a flat morphism. A ring homomorphism f: A\rightarrow B is a flat morphism if the functor -\otimes_{A}B is exact. Since B is an A-algebra, and an A-algebra is also an A-module, this means that f: A\rightarrow B is a flat morphism if B is flat as an A-module. The notion of a flat morphism is important in algebraic geometry, where the “points” of schemes are given by the prime ideals of a ring, since it corresponds to a “continuous” family of schemes parametrized by the “points” of another scheme.

Finally, the functors \text{Hom}_{R}(M,-), \text{Hom}_{R}(-,N), and -\otimes_{R}N, which we will also refer to as the “Hom” and “Tensor” functors, can be used to define the derived functors “Ext” and “Tor”, to which we have given a passing mention in More on Chain Complexes. We now elaborate on these constructions.

The Ext functor, written \text{Ext}_{R}^{n}(M,N) for a fixed R-module M, is calculated by taking an injective resolution of B,

0\rightarrow N\rightarrow E^{0}\rightarrow E^{1}\rightarrow ...

then applying the functor \text{Hom}_{R}(M,-):

0 \rightarrow \text{Hom}_{R}(M,N)\rightarrow \text{Hom}_{R}(M,E^{0})\rightarrow \text{Hom}_{R}(M,E^{1})\rightarrow ...

we “remove” \text{Hom}_{R}(M,N) to obtain the chain complex

0 \rightarrow \text{Hom}_{R}(M,E^{0})\rightarrow \text{Hom}_{R}(M,E^{1})\rightarrow ...

Then \text{Ext}_{R}^{n}(M,N) is the n-th homology group (see Homology and Cohomology) of this chain complex.

Alternatively, we can also define the Ext functor \text{Ext}_{R}^{n}(M,N) for a fixed R-module N by taking a projective resolution of M,

...\rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0

then then applying the functor \text{Hom}_{R}(-,N), which “dualizes” the chain complex:

0 \rightarrow \text{Hom}_{R}(M,N)\rightarrow \text{Hom}_{R}(P_{0},N)\rightarrow \text{Hom}_{R}(P_{1},N)\rightarrow ...

we again “remove” \text{Hom}_{R}(M,N) to obtain the chain complex

0 \rightarrow \text{Hom}_{R}(P_{0},N)\rightarrow \text{Hom}_{R}(P_{1},N)\rightarrow ...

and \text{Ext}_{R}^{n}(M,N) is once again given by the n-th homology group of this chain complex.

The Tor functor, meanwhile, written \text{Tor}_{n}^{R}(M,N) for a fixed R-module N, is calculated by taking a projective resolution of M and applying the functor -\otimes_{R}N, followed by “removing” M\otimes_{R}N:

0\rightarrow M\otimes_{R}P_{0}\rightarrow M\otimes_{R}P_{1}\rightarrow ...

\text{Tor}_{n}^{R}(M,N) is then given by the n-th homology group of this chain complex.

The Ext and Tor functors were originally developed to study the concepts of “extension” and “torsion” of groups in abstract algebra, hence the names, but they have since then found utility in many other subjects, in particular algebraic topology, algebraic geometry, and algebraic number theory. Our exposition here has been quite abstract; to find more motivation, aside from checking out the references listed below, the reader may also compare with the ordinary homology and cohomology theories in algebraic topology. Hopefully we will be able to flesh out more aspects of what we have discussed here in future posts.

References:

Hom Functor on Wikipedia

Tensor Product of Modules on Wikipedia

Flat Module on Wikipedia

Associative Algebra on Wikipedia

Derived Functor on Wikipedia

Ext Functor on Wikipedia

Tor Functor on Wikipedia

Abstract Algebra by David S. Dummit and Richard B. Foote

Commutative Algebra by M. F. Atiyah and I. G. MacDonald

An Introduction to Homological Algebra by Joseph J. Rotman

More on Chain Complexes

In Homology and Cohomology we used the concept of chain complexes to investigate topological spaces. In Exact Sequences we saw examples of chain complexes generalized to abelian groups other than that made out of topological spaces. In this post we study chain complexes in the context of linear algebra (see Vector Spaces, Modules, and Linear Algebra).

We start with some definitions regarding modules. In More on Vector Spaces and Modules we gave the definition of a basis of a vector space. It is known that any vector space can always have a basis. However, the same is not true for modules. It is only a certain special kind of module called a free module which has the property that one can always find a basis for it.

Alternatively, a free module over a ring R may be thought of as being a module that is isomorphic to a direct sum of several copies of the ring R.

An example of a module that is not free is the module \mathbb{Z}/2\mathbb{Z} over the ring \mathbb{Z}. It is a module over \mathbb{Z} since it is closed under addition and under multiplication by any element of \mathbb{Z}, however a basis that will allow it to be written as a unique linear combination of elements of the basis cannot be found, nor is it a direct sum of copies of \mathbb{Z}.

Although not all modules are free, it is actually a theorem that any module is a quotient of a free module. Let A be a module over a ring R. The theorem says that this module is the quotient of some free module, which we denote by F_{0}, by some other module which we denote by K_{1}. In other words,

A=F_{0}/K_{1}

We can write this as the following chain complex, which also happens to be an exact sequence (see Exact Sequences):

0\rightarrow K_{1}\xrightarrow{i_{1}} F_{0}\xrightarrow{\epsilon} A\rightarrow 0

We know that the module F is free. However, we do not know if the same holds true for K_{1}. Regardless, the theorem says that any module is a quotient of a free module. Therefore we can write

0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{\epsilon_{1}} K_{1}\rightarrow 0

We can therefore put these chain complexes together to get

0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{\epsilon_{1}} K_{1}\xrightarrow{i_{1}} F_{0}\xrightarrow{\epsilon} A\rightarrow 0

However, this sequence of modules and morphisms is not a chain complex since the image of \epsilon_{1} is not contained in the kernel of i_{1}. But if we compose these two maps together, we obtain

0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{d_{1} }F_{0}\xrightarrow{\epsilon} A\rightarrow 0

where d_{1}=i_{1}\circ \epsilon_{1}. This is a chain complex as one may check. We can keep repeating the process indefinitely to obtain

...\xrightarrow{d_{3}} F_{2}\xrightarrow{d_{2} } F_{1}\xrightarrow{d_{1} } F_{0}\xrightarrow{\epsilon} A\rightarrow 0

This chain complex is called a free resolution of A. A free resolution is another example of an exact sequence.

We now introduce two more special kinds of modules.

A projective module is a module P such for any surjective morphism p: A\rightarrow A'' between two modules A and A'' and morphism h: P\rightarrow A'', there exists a morphism g: P\rightarrow A such that p\circ g=h.

It is a theorem that a module is projective if and only if it is a direct summand of a free module. This also means that a free module is automatically also projective.

An injective module is a module E such for any injective morphism i: A\rightarrow B between two modules A and B and morphism f: A\rightarrow E, there exists a morphism g: B\rightarrow E such that g\circ i=f.

Similar to our discussion regarding free resolutions earlier, we can also have projective resolutions and injective resolutions. A projective resolution is a chain complex

...\xrightarrow{d_{3}} P_{2}\xrightarrow{d_{2} } P_{1}\xrightarrow{d_{1} } P_{0}\xrightarrow{\epsilon} A\rightarrow 0

such that the P_{n} are projective modules.

Meanwhile, an injective resolution is a chain complex

...0\rightarrow A\xrightarrow{\eta} E^{0}\xrightarrow{d^{0} } E^{1}\xrightarrow{d^{1}} E^{2}\xrightarrow{d^{2}} ...

such that the E^{n} are injective modules.

Since projective and injective resolutions are chain complexes, we can use the methods of homology and cohomology to study them (Homology and Cohomology) even though they may not be made up of topological spaces. However, the usual procedure is to consider these chain complexes as forming an “abelian category” and then applying certain functors (see Category Theory) such as what are called the “Tensor” and “Hom” functors before applying the methods of homology and cohomology, resulting in what are known as “derived functors“. This is all part of the subject known as homological algebra.

References:

Free Module on Wikipedia

Projective Module on Wikipedia

Injective Module on Wikipedia

Resolution on Wikipedia

An Introduction to Homological Algebra by Joseph J. Rotman

Abstract Algebra by David S. Dummit and Richard M. Foote

Exact Sequences

In Homology and Cohomology we introduced the idea of chain complexes to help us obtain information about topological spaces. We recall that a chain complex is made up of abelian groups of spaces C_{n} and boundary homomorphisms \partial_{n}: C_{n}\rightarrow C_{n-1} such that for all n the composition of successive boundary homomorphisms \partial_{n-1}\circ \partial_{n}: C_{n}\rightarrow C_{n-2} sends every element in C_{n} to the zero element in C_{n-2}.

Chain complexes can be expressed using the following diagram:

...\xrightarrow{\partial_{n+3}}C_{n+2}\xrightarrow{\partial_{n+2}}C_{n+1}\xrightarrow{\partial_{n+1}}C_{n}\xrightarrow{\partial_{n}}C_{n-1}\xrightarrow{\partial_{n-1}}C_{n-2}\xrightarrow{\partial_{n-2}}...

We now abstract this idea, generalizing it so that the groups C_{n} do not necessarily have to be topological spaces, and show an example of a chain complex that is ubiquitous in mathematics.

First we recall some ideas from Homology and Cohomology. Our “important principle” was summarized in the following statement:

All boundaries are cycles.

Boundaries in C_{n} are elements of the image of the boundary homomorphism \partial_{n+1}. Cycles in C_{n} are elements of the kernel of the boundary homomorphism \partial_{n}. Therefore, we can also state our “important principle” as follows:

\text{Im }\partial_{n+1}\subseteq \text{Ker }\partial_{n} for all n

This is of course just another restatement of the defining property of all chain complexes that two successive boundary functions when composed send every element of its domain to the zero element of its range.

There is an important kind of chain complex with the following property:

\text{Im }\partial_{n+1}=\text{Ker }\partial_{n} for all n

Such a chain complex is called an exact sequence. Sometimes we just say that the chain complex is exact. We will show some simple examples of exact sequences, but for these examples we will drop the notation of the boundary homomorphism \partial_{n} to show that many properties of ordinary functions can be expressed in terms of exact sequences.

Consider, for example, abelian groups A, B, and C. The identity elements of A, B, and C will be denoted by 0, writing 0\in A, 0\in B, and 0\in C if necessary. We will also write 0 to denote the trivial abelian group consisting only of the single element 0. Let us now look at the exact sequence

0\rightarrow A\xrightarrow{f} B

where 0\rightarrow A is the inclusion function sending 0\in 0 to 0\in A. The image of this inclusion function is therefore 0\in A. By the defining property of exact sequences, this is also the kernel of the function f:A\rightarrow B. In other words, f sends 0\in A to 0\in B. It is a property of group homomorphisms that whenever the kernel consists of only one element, the homomorphism is an injective, or one-to-one, function. This means that no more than one element of the domain gets sent to the same element in the range. Since this function is also a homomorphism, it is also called a monomorphism.

Meanwhile, let us also consider the exact sequence

B\xrightarrow{g} C\rightarrow 0

where C\rightarrow 0 is the “constant” function that sends any element in C to 0. The kernel of this constant function is therefore the entirety of C. By the defining property of exact sequences, this is also the image of the function B\rightarrow C. In other words, the image of the function g is the entirety of C, or we can also say that every element of C is assigned by g to some element of B. Such a function is called surjective, or onto. Since this function is also a homomorphism, it is also called an epimorphism.

The exact sequence

0\rightarrow A\xrightarrow{f} B\xrightarrow{g} C\rightarrow 0

is important in many branches of mathematics, and is called a short exact sequence. This means that f is a monomorphism, g is an epimorphism, and that \text{im }f=\text{ker g} in B. As an example of a short exact sequence of abelian groups, we have

0\rightarrow 2\mathbb{Z}\xrightarrow{f} \mathbb{Z}\xrightarrow{g} \mathbb{Z}/2\mathbb{Z}\rightarrow 0

(see also Modular Arithmetic and Quotient Sets). The monomorphism f takes the abelian group of even integers 2\mathbb{Z} and “embeds” them into the abelian group of the integers \mathbb{Z}. The epimorphism g then sends the integers in \mathbb{Z} to the element 0 in \mathbb{Z}/2\mathbb{Z} if they are even, and to the element 1 in  \mathbb{Z}/2\mathbb{Z} if they are odd. We see that every element in \mathbb{Z} that comes from 2\mathbb{Z}, i.e. the even integers, gets sent to the identity element or zero element 0 of the abelian group \mathbb{Z}/2\mathbb{Z}.

In the exact sequence

0\rightarrow A\xrightarrow{f} B\xrightarrow{g} C\rightarrow 0

The abelian group B is sometimes referred to as the extension of the abelian group C by the abelian group A.

We recall the definition of the homology groups H_{n}:

H_{n}=\text{Ker }\partial_{n}/\text{Im }\partial_{n+1}.

We can see from this definition that a chain complex is an exact sequence (we can also say that the chain complex is acyclic) if all of its homology groups are zero. So in a way, the homology groups “measure” how much a chain complex “deviates” from being an exact sequence.

We also have the idea of a long exact complex, which usually comes from the homology groups of chain complexes which themselves form a short exact sequence. In order to discuss this we first need the notion of a chain map between chain complexes. If we have a chain complex

...\xrightarrow{\partial_{A, n+3}}A_{n+2}\xrightarrow{\partial_{A, n+2}}A_{n+1}\xrightarrow{\partial_{A, n+1}}A_{n}\xrightarrow{\partial_{A, n}}A_{n-1}\xrightarrow{\partial_{A, n-1}}A_{n-2}\xrightarrow{\partial_{A, n-2}}...

and another chain complex

...\xrightarrow{\partial_{B, n+3}}B_{n+2}\xrightarrow{\partial_{B, n+2}}B_{n+1}\xrightarrow{\partial_{B, n+1}}B_{n}\xrightarrow{\partial_{B, n}}B_{n-1}\xrightarrow{\partial_{B, n-1}}B_{n-2}\xrightarrow{\partial_{B, n-2}}...

a chain map is given by homomorphisms

f_{n}: A_{n}\rightarrow B_{n} for all n

such that the homomorphisms f_{n} commute with the boundary homomorphisms \partial_{A, n} and \partial_{B, n}, i.e.

\partial_{B, n}\circ f_{n}=f_{n-1}\circ \partial_{A, n} for all n.

A short exact sequence of chain complexes is then a short exact sequence

0\rightarrow A_{n}\xrightarrow{f_{n}} B_{n}\xrightarrow{g_{n}} C_{n}\rightarrow 0 for all n

where the homomorphisms f_{n} and g_{n} satisfy the conditions for them to form a chain map, i.e. they commute with the boundary homomorphisms in the sense shown above.

In the case that we have a short exact sequence of chain complexes, their homology groups will then form a long exact sequence:

...\rightarrow H_{n}(A)\xrightarrow{f_{*}}H_{n}(B)\xrightarrow{g_{*}}H_{n}(C)\xrightarrow{\partial}H_{n-1}(A)\xrightarrow{f_{*}}...

Long exact sequences are often used for calculating the homology groups of complicated topological spaces related in some way to simpler topological spaces whose homology groups are already known.

References:

Chain Complex on Wikipedia

Exact Sequence on Wikipedia

Algebraic Topology by Allen Hatcher

A Concise Course in Algebraic Topology by J. P. May

Abstract Algebra by David S. Dummit and Richard M. Foote