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 and are two modules whose ring of scalars is the ring (we refer to and as -modules), then we denote by the set of linear transformations (see Vector Spaces, Modules, and Linear Algebra) from to . It is worth noting that this set has an abelian group structure (see Groups).
We define the functor as the functor that assigns to an -module the abelian group of linear transformations from to . Similarly, the functor assigns to the -module the abelian group of linear transformations from to .
These functors and , 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 , we say that it is exact if for an exact sequence of modules
is also exact. The concept of an exact sequence of sets of linear transformations of -modules makes sense because of the abelian group structure on these sets. In this case we also say that the -module is projective.
Similarly, an -module is injective if the functor is exact, i.e. if for an exact sequence of modules
is also exact.
We introduce another functor, which we write . This functor assigns to an -module the tensor product (see More on Vector Spaces and Modules) . Similarly, we also have the functor , which assigns to an -module the tensor product . If our ring is commutative, then there will be no distinction between the functors and . We will continue assuming that our rings are commutative (an example of a noncommutative ring is the ring of matrices).
We say that a module is flat if the functor is exact, i.e. if for an exact sequence of modules
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 , an -algebra is a ring and a ring homomorphism , which makes into an -module via the following definition of the scalar multiplication:
The notion of an algebra will be useful in defining the notion of a flat morphism. A ring homomorphism is a flat morphism if the functor is exact. Since is an -algebra, and an -algebra is also an -module, this means that is a flat morphism if is flat as an -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 , , and , 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 for a fixed -module , is calculated by taking an injective resolution of ,
then applying the functor :
we “remove” to obtain the chain complex
Then is the -th homology group (see Homology and Cohomology) of this chain complex.
Alternatively, we can also define the Ext functor for a fixed -module by taking a projective resolution of ,
then then applying the functor , which “dualizes” the chain complex:
we again “remove” to obtain the chain complex
and is once again given by the -th homology group of this chain complex.
The Tor functor, meanwhile, written for a fixed -module , is calculated by taking a projective resolution of and applying the functor , followed by “removing” :
is then given by the -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.
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