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

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