Derived Functors II

First off, we need to get this pesky result out of the way that derived functors are independent of the choice of resolution. So we do this by proving a related result. Suppose \cdots\rightarrow F_i \stackrel{\phi_i}{\rightarrow}F_{i-1}\rightarrow \cdots \rightarrow F_1\stackrel{\phi_1}{\rightarrow} F_0 is a a complex of projective A-modules, and \cdots G_1\stackrel{\psi_1}{\rightarrow} G_0 is a complex of A-modules. Let M=coker \ \phi_1 and N=coker \ \psi_1. Suppose the homology of G vanishes except H_0(G)=N. Then every map \beta\in Hom_A(M, N) is a map induced on H_0 by a map of complexes \alpha : F\to G and is determined up to homotopy by \beta.

Before proving this, note that as a corollary we get that any two projective resolutions are homotopy equivalent, and hence the derived functors have constructed on different resolutions have a natural isomorphism between them.

Proof: I knew I should never have tried to do homological algebra without a good way to do diagrams on wordpress. This is clearer if you draw it out…but the idea for existence is to inductively lift your maps. Lift F_0\to M\to N to \alpha_0: F_0\to G_0, then \alpha_0\phi_1: F_1\to ker(G_0\to N)=im(G_1\to G_0). Thus we lift this to \alpha_1: F_1\to G_1 and continue this process. This gives the map of the complexes.

We now want uniqueness up to homotopy. Suppose we have two lifts of \beta: M\to N say \alpha and \gamma. Then \alpha - \gamma lifts the zero map. i.e. we need only show that any lifting of 0 is homotopic to 0. Suppose then that \eta is a lifting of zero. We need that \eta_i=h_{i-1}\phi_i + \psi_{i+1}h_i for some h_i: F_i\to G_{i+1}. Note that \eta_0 : coker \ \phi_1\to coker \ \psi_1 takes F_0\to im\psi_1. So we lift to h_0: F_0\to G_1 such that \psi_1 h_0=\eta_0. But now \psi_1(h_0\phi_1-\eta_1)=\eta_0\phi_1-\psi_1\eta_1=0. Thus h_0\phi_1-\eta_1 maps into ker \ \psi_1=im \ \psi_2. But F_1 is projective so we can lift to h_1: F_1\to G_2. Repeat this process.

I highly recommend just doing the diagram chasing yourself. This is sort of a mess to read, and so should only be used as a sort of guideline if you get stuck somewhere.

Hmm…what else did I say I would do? Oh right. If you’ve seen homological constructions, then you can probably guess that there is a connecting homomorphism type theorem. i.e. Something that is phrased, “a short exact sequence of complexes induces a long exact sequence in homology.” So this trick tends to be useful in actually calculations of your derived functor. I won’t go through it, since it is just your standard “Snake Lemma” construction.

When I said there was extra structure, I was thinking about going into putting a product on the whole thing to make it into a graded ring, but I’ve decided that that is getting a little far afield for now. This may be the end of my ramblings on derived things for awhile.

The other thing I thought I should mention was my confusion on what this blog has become. I’m at a sort of turning point. I’m not sure if I should eliminate the non-math/mathematical physics and turn it into a blog just on that stuff (it sort of accidentally has shifted to that unofficially), or if I should make a conscious effort to balance things more. There are positives and negatives to both in my mind. Actually changing to a more focused blog would help draw and keep readers that actually care about the stuff I’ve been doing recently. On the other hand, it sort of goes against everything I believe in. But how its been going now, I’ve probably alienated all readers that used to read for philosophy or random art things, and so randomly posting on those things seems sort of pointless if I’ve lost those people and it will just serve to confuse and possibly alienate people only interested in the math side.

No immediate decisions will be made, so there is some time.


One thought on “Derived Functors II

  1. Well, I belong to the “other readers” camp and admit to skimming all the posts that contain pictures of math, as it were. I liked the arty philosophy posts, which is how I came to read your blog in the first place.

    Why not just have two blogs, a work blog and a “there’s more to my life than my work” blog? You could always mention here that you’ve posted something on the other blog, and vice versa, in order to maintain a balance.

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