Assume the left square is a pushout. Then the right square is a pushout if and only if the big rectangle is.

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1 COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY VIVEK SHENDE Last time we learned about Yoneda s lemma, and various universal constructions initial and final objects, products and coproducts (which turned out to be the same for R-modules); pullbacks and pushouts. We connected this back to the usual discussion of exact sequences, by noting that if you turn into 0 A C 0 A 0 C then the sequence is left-exact if this diagram is a pullback; right exact if this diagram is a pushout, and exact if it s both. We left off by noting that the first isomorphism theorem says that whenever A, there s a pushout-pullback diagram A 0 /A The real content is in the assertion that it s a pushout (i.e. right exact); once you know this, the fact that it s a pullback (i.e. left exact) is already contained in the hypothesis that A. Here s a really important fact about pushouts. (There s a similar one about pullbacks.) Lemma 1. Suppose given a diagram like this: Assume the left square is a pushout. Then the right square is a pushout if and only if the big rectangle is. Proof. Exercise. 1

3 COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY 3 That is, both sides define functors C op D set, and we demand the data of natural transformations between these functors, which compose both ways to the identity. Exercise: using Yoneda s lemma or otherwise, show that an adjoint, if it exists, is unique up to unique natural transformation. The prototypical example of an adjunction is (free construction, forgetful functor). E.g, we have a functor Rings Sets, which forgets the ring structure and just remembers the elements of the ring. It s a forgetful functor. On the other hand, given a set S, we can form the polynomial ring Z[S], where the variables are elements of S. There s a natural isomorphism Hom Rings (Z[S], R) = Hom Sets (S, F orget(r)) Actually, being the left adjoint of the forgetful functor is what it means to be a free construction. You should regard the above isomorphism as the assertion that there s a such thing as a free ring on a set S, and it s Z[S]. Similarly, you can forget that a module is a module, and remember only that it s a set. Given a set S, you can form the R-module with basis S, I ll denote it R S. This is the free R-module on S, i.e. there s an adjunction Hom R mod (R S, M) = Hom Sets (S, F orget(m)) The most useful consequence of the existence of an adjunction is: Lemma 2. Left adjoints preserve colimits and right adjoints preserve limits. We ll prove this shortly, but first let s review (or learn) what limits and colimits are. You ve seen many universal constructions, described in the following way. There s a diagram of objects and morphisms, and you ask for an object with morphisms (from) to all objects in the diagram, commuting with all morphisms in the diagram, which is (co) universal in the sense that any such object and collection of maps will factor through the a map (from) to the (co) universal one. E.g. given X W Y, the pullback is some X W Y, such that there s a commutative diagram X W Y Y X such that any other commutative diagram W Z Y X W is obtained from it by factoring through some unique map Z X W Y.

5 COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY 5 It follows from the existence of these adjoints that taking limits or colimits commutes with forgetting the S-module structure. Note in particular that, taking R = Z, the underlying abelian group of a limit or colimit is the limit or colimit of underlying abelian groups. Example. Consider an R-module, M. Then for any other R-modules, L, N, we have Hom R (L R M, N) = Hom R (L, Hom R (M, N)) Thus we learn that R M preserves colimits, and in particular right exact sequences, and that Hom R (M, ) preserves limits, and in particular, left exact sequences. Example. Using the above adjunction both ways, we deduce Hom R (M, Hom R (L, N)) = Hom R (L, Hom R (M, N)) This doesn t look like an adjunction. However, let s think more carefully about Hom R (, N). It is contravariant, which means there s two ways of thinking of it: Thus we can rewrite the above equality as Hom R (, N) : R mod op R mod Hom R mod op(n, ) : R mod R mod op Hom R mod op(hom R mod op(n, L), M) = Hom R (L, Hom R (M, N)) Now you see that again Hom R (, N) is a left adjoint. Thus it takes left-exact sequences in R mod op aka right exact sequences in R mod to left-exact sequences in R mod. 3 You might call this an op-fuscation of the previous example. Finally, let s prove lemma 2. We ll do the right adjoints. Say we have some diagram of objects d α in D; assume lim d α exists. We compute: Hom C (c, g(lim d α )) = Hom D (f(c), lim d α ) = lim Hom D (f(c), d α ) = lim Hom C (c, g(d α )) The first and third equality use the adjunction; the second holds by definition of limit. Comparing the left and right most terms, we learn (by the definition of limit) g(lim d α ) = lim g(d α ). 3 We haven t really explained why exact sequences in R mod op is a thing that makes sense. Here s why: taking opposite category interchanges limits and colimits, so it s clear that R mod op has a zero (= initial and final) object, and pushouts are interchanged with pullbacks. So we can ask for pushout-pullback squares with a zero in a corner.

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