Homology, Direct Limit

Direct Limit

A generalized definition of direct limit can be found in category theory, but that probably confuses more than it helps. Let me explain the direct limit as it relates to homology.

Let B be an R module and let B₁ B₂ B₃ etc be a collection of submodules of B. Often the submodules are the finitely generated submodules of B. Every two submodules can be combined to make a third submodule on the list. For instance, the generators of B₃ and the generators of B₅ combine to generate B₇. If all finitely generated submodules are included, there is no trouble here. Finally, all the submodules, taken together, span B. We say B is the direct limit of the underlying system of submodules.

A directed system, with a direct limit, often includes a function into another module, such as C. Each B_i maps into C_i, and when taken together, B maps into C. Of course the maps must be consistent. Whenever they intersect, B_i and B_j must both map x to the same element in C_i and C_j.

The Direct Limit is Exact

Assume A B and C are direct limits. Let the following sequence be exact for each submodule in the system.

0 → A_i → B_i → C_i → 0

If x in B-A maps to 0 in C, find B_i containing x, and x maps to 0 in C_i. Since only A maps to 0, this is a contradiction. Similarly, x in A lies in some A_i, and maps to 0 in C_i, and in C. We have exactness at B.

The same reasoning holds at A and at C, and the sequence 0 → A → B → C → 0 is exact.

The Homology of the Direct Limit

A generalization of the above does not require exactness. Let f(A) map into B, and let g(B) map into C, such that fg = 0. There is indeed a homology at B, which I will call H.

Let all three modules be direct limits, such that f, restricted to A_i, maps A_i into B_i, and g maps B_i into C_i. Furthermore, A_i must be the preimage of B_i under f. This is assured if A is a subset of B, and f is simple inclusion. Now there is a homology H_i associated with B_i.

Let x be an element of B_i representing a member of H_i. The same x can be used to represent a member of H_j (where B_j contains B_i), or H itself. If x was in the image of A_i it would be in the image of A_j, and A, hence the map on H_i is well defined. Also, if x is in the image of A_j, or A, it is in the image of A_i, which necessarily includes the preimage of x. Therefore the map is injective. Each H_i embeds in H_j (where B_j contains B_i), and in H itself.

If x represents a member of H, find B_i containing x, and H_i covers that part of H. Thus H is covered, and H is the direct limit of H_i.

Direct Limit and Tensor Product

Let B be the direct limit of various submodules B_i. To be general, let B map into C, with B_i mapping into C_i. Tensor each submodule, and the associated homomorphisms, with the module M. If j is a stepup from i, so that B_j contains B_i, and C_j contains C_i, tensor with M and B_jM contains B_iM, and C_jM contains C_iM, and the function on B_jM is backward compatible with the function on C_iM. For any i and j, we can find an upper bound B_kM containing B_iM and B_jM, and the function on B_kM is compatible with the functions on B_iM and B_jM. Also, everything in B×M and C×M is covered. Therefore, tensoring with M builds a directed system, whose direct limit is B → C.

This is sometimes used in concert with the earlier result on exact sequences, so that a directed system of exact sequences can be tensored with M, at every level, to give a new directed system of exact sequences, leading to an exact sequence of tensored direct limits. This is a way to step up from finitely generated modules to infinitely generated modules.

A Directed System of Homomorphisms

Don't be frightened, but I'm going to take one more step towards category theory. A directed system does not have to consist of submodules. Let B be a module and let B_i be a collection of modules, partially ordered by module homomorphisms. Thus B_i < B_j if there is a designated homomorphism that maps B_i into B_j. This homomorphism is part of the directed system. If B_j then maps into B_k, the homomorphism from B_i into B_k is the composition of the two underlying homomorphisms. Every two modules map into a third, and homomorphisms are consistent wherever modules intersect. Finally, each B_i maps into B.

Everything we did above is a special case of this definition, where each homomorphism is inclusion. But this is more general. As we move from B_i to B_j, modules could get smaller, rather than bigger. Here is an example that comes in handy.

Let H be an ideal of R and let H_i be the system of finitely generated ideals within H. Let B = R/H, and let B_i = R/H_i. If H_j contains H_i, B_i maps onto the smaller quotient ring B_j. And they all map onto B. This may remind you of galois theory, where one system seems to stand the other on its head.

Once again, a set of parallel functions can map one directed system into another, but we need a commutative diagram. If f_i maps B_i into C_i, we can go from B_i to C_i and on to C_j, or from B_i to B_j and over to C_j. The result is the same. (When the internal homomorphisms were simple inclusions, we simply said the functions were consistent.)

There is now a function f from B into C. Given x in B, findd B_i(y) = x, apply f_i into C_i, and map the result to C. Consistency within the two directed systems, and commutativity, combine to prove f is well defined. Suppose y in B_i and z in B_j lead to different values in C. Map y and z down to w in B_k, which is also a preimage of x in B. Map y and z into C_i and C_j, then take those images down to f_k(w) in C_k, and on to a common element of C, thus f is well defined. Furthermore, f is commutative with each f_i.

Let's continue the example we started earlier. For each finitely generated ideal H_i in a larger ideal H, write this exact sequence.

0 → H_i → R → R/H_i → 0

The system denoted by H_i is one of inclusion, with direct limit H. The middle module is the trivial directed system in which each module is R, and each homomorphism is the identity. The last system consists of decreasing quotient rings, moving towards R/H. Functions carry the first system into the second, and the second onto the third, and they commute, as they should.

Return to A → B → C, not necessarily exact, with a directed system A_i → B_i → C_i. As before, A_i is the preimage of B_i, and C_i contains the image of B_i. As you recall, each B_i has its homology H_i. Map an x in B_i, representing a member of H_i, to a member of H, which is the homology of B. The proof is essentially the same as that shown above. Thus H is the directed limit of H_i.

If each H_i is 0 then H = 0. A directed system of exact sequences becomes exact in the limit.

Finally, tensor each B_i → C_i with M. The function f_i, represented by the arrow, is tensored with the identity map on M. This is sometimes written f_i×M, or f_iM. Sharpen your pencil and perform a lot of algebra to show that B_iM forms a directed system with limit BM, C_iM forms a directed system with limit CM, and f_iM becomes a valid map between the two directed systems. Furthermore, f_iM implies a function from BM into CM, which is equal to fM. If we know what happens to each function f_i when tensored with M, we may be able to infer the properties of fM, as it maps BM into CM.