LECTURE 10 Homotopy, Quasi-Isomorphism, an Coinvariants Please note that proos o many o the claims in this lecture are let to Problem Set 5. Recall that a sequence o abelian groups with ierential is a complex i 2 0, : X Y is a morphism o chain complexes i, an h is a null-homotopy (o ) i h + h, which we illustrate in the ollowing iagram: X 1 X 0 X 1 h Y 1 Y 0 Y 1. The invariants o a chain complex are the homology groups H i (X) : Ker(: X i X i+1 )/ Im(: X i 1 X i ), an or, g : X Y, we say that g, that is, an g are homotopic, i an only i there exists a null-homotopy o g, which by Lemma 9.10, orces an g to give the same map on cohomology. For a inite group G an extension L/K o local iels with G Gal(L/K), we have Ĥ0 (G, L ) K /NL by einition. Our goal is to show that Ĥ0 (G, L ) G ab canonically, i.e., the abelianization o G. Our plan or this lecture will be to eine the Tate cohomology groups Ĥi or each i Z (which is more complicate or non-cyclic groups), an then use them to begin working towars a proo o this act. Recall that out basic principle was that, given a homotopy h: g, an g are now inistinguishable or all practical purposes (which we will take on aith). An application o this principle is the construction o cones or homotopy cokernels: Claim 10.1. I : X Y is a map o complexes, then hcoker() (a.k.a. Cone()), characterize by the universal property that maps hcoker() Z o chain complexes are equivalent to maps g : Y Z plus a null-homotopy h o g : X Z, exists. Proo. We claim that the ollowing chain complex is hcoker(): (10.1) X 0 Y 1 X 1 Y 0 X 2 Y 1 with ierential ( ( ) X i+1 Y i x x X y) i+2 Y i+1, (x) + y 42 h
10. HOMOTOPY, QUASI-ISOMORPHISM, AND COINVARIANTS 43 which we note increases the egree appropriately. We may summarize this ierential as a matrix ( ) 0, an we note that it squares to zero as ( ) ( ) ( ) ( ) 0 0 2 0 0 0 + 2 0 0 by the einition o a morphism o chain complexes an because both X an Y are complexes. We now check that this chain complex satisies the universal property o hcoker(). So suppose we have a map hcoker() Z, so that the iagram X i+1 Y i X i+2 Y i+1 Z i Z i+1 commutes. I we write such a map as (x, y) h(x) + g(y), then this means h(x) + g(y) (h(x) + g(y)) h( x) + g((x) + y) h(x) + g(x) + g(y). Taking x 0 implies g g, so we must have h + h g, hence h is a null-homotopy o g, as esire. Corollary 10.2. The composition X Y hcoker() is canonically null-homotopic (as an exercise, construct this null-homotopy explicitly!). Example 10.3. Let X : ( 0 A 0 ) an Y : ( 0 B 0 ) or inite abelian groups A an B in egree 0, an let : A B. Then with B in egree 0. Then we have hcoker() ( 0 A B 0 ), H 0 hcoker() Coker() an H 1 hcoker() Ker(), so we see that the language o chain complexes generalizes prior concepts. Notation 10.4. For a chain complex X, let X[n] enote the shit o X by n places, that is, the chain complex with X i+n in egree i, with the ierential ( 1) n (where enotes the ierential or X). So or instance, X[1] hcoker(x 0). The content o this is that giving a null-homotopy o 0: X Y is equivalent to giving a map X[1] Y. Lemma 10.5. For all maps : X Y, the sequence is exact or all i. H i X H i Y H i hcoker() Proo. The composition is zero by Lemma 9.10 because X Y hcoker() is null-homotopic. To show exactness, let y Y i such that y 0, an suppose that its image in H i hcoker() is zero, so that ( 0 y) ( 0 ) ( α β ) ( α ) (α) + β
44 10. HOMOTOPY, QUASI-ISOMORPHISM, AND COINVARIANTS or some α X i with α 0 an β Y i 1. Then (α) + β y implies (α) y in H i Y, as esire. Claim 10.6. There is also a notion o the homotopy kernel hker(), eine by the universal property that maps Z hker() are equivalent to maps Z X plus the ata o a null-homotopy o the composition Z X Y. In particular, hker() hcoker()[ 1]. Example 10.7. Let : A B be a map o abelian groups (in egree 0 as beore). Then hcoker() ( 0 A B 0 0 ) hker() ( 0 0 A B 0 ), where hker() 0 A. The homotopy cokernel also recovers the kernel an cokernel in its cohomology. Claim 10.8. The composition X Y hcoker() is null-homotopic, so there exists a canonical map X hker(y hcoker()), where we reer to the latter term as the mapping cyliner. This map is a homotopy equivalence. Deinition 10.9. A map : X Y is a homotopy equivalence i there exist a map g : Y X an homotopies g i X an g i Y, in which case we write X Y. It is a quasi-isomorphism i H i (): H i (X) H i (Y ) is an isomorphism or each i (i.e., X an Y are equal at the level o cohomology). Claim 10.10. I : X Y is a homotopy equivalence, then it is a quasiisomorphism. Proo. This is an immeiate consequence o Lemma 9.10, which ensures that an g are inverses at the level o cohomology. Corollary 10.11. Given : X Y, there is a long exact sequence H i 1 hcoker() H i X H i Y H i hcoker() H i+1 X. Proo. Letting g enote the map Y hcoker(), the composition Y g hcoker() hcoker(g) hker(g)[1] X[1] is null-homotopic by Corollary 10.2, an the homotopy equivalence is by Claim 10.8. So by Lemma 10.5, the sequence H i Y H i hcoker() H i X[1] H i+1 X is exact; a urther application o Lemma 10.5 shows the claim. Claim 10.12. Suppose i : X i Y i is injective or all i. Then hcoker() Y/X (i.e., the complex with Y i /X i in egree i) is a quasi-isomorphism.
10. HOMOTOPY, QUASI-ISOMORPHISM, AND COINVARIANTS 45 Example 10.13. I : A B is a map o abelian groups in egree 0, then the map hcoker() B/A looks like A B 0 0 B/A 0. It s easy to see that this is inee a quasi-isomorphism. Note that there is a ual statement, that i i is surjective in each egree, then the homotopy kernel is quasiisomorphic to the naive kernel. Remark 10.14. I A is an associative algebra (e.g. Z or Z[G]), then we can have chain complexes o A-moules X 1 X 0 X 1, where the X i are A-moules an is a map o A-moules. Here the cohomologies will also be A-moules. Now, our original problem was to eine Tate cohomology or a inite group G acting on some A. Note that Ĥ 0 (G, A) A G /N(A) Coker(N: A A G ). In act, we can o better than N: A A G ; the norm map actors through what we will call the coinvariants. Deinition 10.15. The coinvariants o A are A G : A / g G (g 1)A, which satisies the universal property that it is the maximal quotient o A with gx x holing or all x A an g G. Note that we can think o the invariants A G as being the intersection o the kernels o each (g 1)A, so it is the maximal submoule o A or which gx x hols similarly. Then the norm map actors as A N A G. N Our plan is now to eine erive (complex) versions o A G an A G N calle A hg A hg, an Tate cohomology will be the homotopy cokernel o this map. The basic observation is that Z is a G-moule (i.e. Z[G] acts on Z) in a trivial way, with every g G as the ientity automorphism. I M is a G-moule, then M G Hom G (Z, M) (because the image o 1 in M must be G-invariant an correspons to the element o M G ) an M G M Z[G] Z. Inee, let I A be an ieal acting on M. Then A/I A M M/IM by the right-exactness o tensor proucts. Here, Z Z[G]/I, where I is the augmentation ieal generate by elements g 1 an thereore M G M/I as esire. Now we have the general problem where A is an associative algebra an M an associative A-moule, an we woul like the erive the unctors A M an Hom A (M, ). These shoul take chain complexes o A-moules an prouce complexes o abelian groups, preserving cones an quasi-isomorphisms. We ll begin working on this in the next lecture. A G
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