Homotopy, Quasi-Isomorphism, and Coinvariants

LECTURE 10 Homotopy, Quasi-Isomorphism, and Coinvariants Please note that proofs of many of the claims in this lecture are left to Problem Set 5. Recall that a sequence of abelian groups with differential d is a complex if d2 = 0, f : X → Y is a morphism of chain complexes if df = f d, and h is a null-homotopy (of f ) if dh + hd = f , which we illustrate in the following diagram: X −1 ··· f Y −1 ··· d h d X0 d X1 f h f Y0 d Y1 ··· ··· . The invariants of a chain complex are the homology groups H i (X) := Ker(d : X i → X i+1 )/ Im(d : X i−1 → X i ), and for f, g : X ⇒ Y , we say that f ' g, that is, f and g are homotopic, if and only if there exists a null-homotopy of f − g, which by Lemma 9.10, forces f and g to give the same map on cohomology. For a finite group G and extension L/K of local fields with G = Gal(L/K), we have Ĥ 0 (G, L× ) = K × /NL× by definition. Our goal is to show that Ĥ 0 (G, L× ) ' Gab canonically, i.e., the abelianization of G. Our plan for this lecture will be to define the Tate cohomology groups Ĥ i for each i ∈ Z (which is more complicated for non-cyclic groups), and then use them to begin working towards a proof of this fact. Recall that out basic principle was that, given a homotopy h : f ' g, f and g are now indistinguishable for all practical purposes (which we will take on faith). An application of this principle is the construction of cones or homotopy cokernels: Claim 10.1. If f : X → Y is a map of complexes, then hCoker(f ) (a.k.a. Cone(f )), characterized by the universal property that maps hCoker(f ) → Z of chain complexes are equivalent to maps g : Y → Z plus a null-homotopy h of g ◦ f : X → Z, exists. Proof. We claim that the following chain complex is hCoker(f ): · · · → X 0 ⊕ Y −1 → X 1 ⊕ Y 0 → X 2 ⊕ Y 1 → · · · (10.1) with differential X i+1     x −dx d ⊕Y 3 7−→ ∈ X i+2 ⊕ Y i+1 , y f (x) + dy i which we note increases
the degree appropriately. We may summarize this differ0 ential as a matrix −d f d , and we note that it squares to zero as        −d 0 −d 0 d2 0 0 0 = = f d f d −f d + df d2 0 0 by the definition of a morphism of chain complexes and because both X and Y are complexes. We now check that this chain complex satisfies the universal property of hCoker(f ). So suppose we have a map hCoker(f ) → Z, so that the diagram ··· X i+1 ⊕ Y i X i+2 ⊕ Y i+1 ··· ··· Zi Z i+1 ··· commutes. If we write such a map as (x, y) 7→ h(x) + g(y), then this means dh(x) + dg(y) = d(h(x) + g(y)) = h(−dx) + g(f (x) + dy) = −h(dx) + gf (x) + g(dy). Taking x = 0 implies dg = gd, so we must have dh + hd = g ◦ f , hence h is a null-homotopy of g ◦ f , as desired.  Corollary 10.2. The composition X → Y → hCoker(f ) is canonically null-homotopic (as an exercise, construct this null-homotopy explicitly!). Example 10.3. Let X := (· · · → 0 → A → 0 → · · · ) and Y := (· · · → 0 → B → 0 → · · · ) for finite abelian groups A and B in degree 0, and let f : A → B. Then f hCoker(f ) = (· · · → 0 → A − → B → 0 → · · · ), with B in degree 0. Then we have H 0 hCoker(f ) = Coker(f ) and H −1 hCoker(f ) = Ker(f ), so we see that the language of chain complexes generalizes prior concepts. Notation 10.4. For a chain complex X, let X[n] denote the shift of X by n places, that is, the chain complex with X i+n in degree i, with the differential (−1)n d (where d denotes the differential for X). So for instance, X[1] = hCoker(X → 0). The content of this is that giving a null-homotopy of 0 : X → Y is equivalent to giving a map X[1] → Y . Lemma 10.5. For all maps f : X → Y , the sequence H i X → H i Y → H i hCoker(f ) is exact for all i. Proof. The composition is zero by Lemma 9.10 because X → Y → hCoker(f ) is null-homotopic. To show exactness, let y ∈ Y i such that dy = 0, and suppose that its image in H i hCoker(f ) is zero, so that        0 −d 0 α −dα = = y f d β f (α) + dβ for some α ∈ X i with dα = 0 and β ∈ Y i−1 . Then f (α) + dβ = y implies f (α) = y in H i Y , as desired.  Claim 10.6. There is also a notion of the homotopy kernel hKer(f ), defined by the universal property that maps Z → hKer(f ) are equivalent to maps Z → X plus the data of a null-homotopy of the composition Z → X → Y . In particular, hKer(f ) = hCoker(f )[−1]. Example 10.7. Let f : A → B be a map of abelian groups (in degree 0 as before). Then f hCoker(f ) = (· · · → 0 → A − → B → 0 → 0 → ···) f hKer(f ) = (· · · → 0 → 0 → A − → B → 0 → · · · ), where hKer(f )0 = A. The homotopy cokernel also recovers the kernel and cokernel in its cohomology. Claim 10.8. The composition f X− → Y → hCoker(f ) is null-homotopic, so there exists a canonical map X → hKer(Y → hCoker(f )), where we refer to the latter term as “the mapping cylinder.” This map is a homotopy equivalence. Definition 10.9. A map f : X → Y is a homotopy equivalence if there exist a map g : Y → X and homotopies gf ' idX and f g ' idY , in which case we write X 'Y. ∼ It is a quasi-isomorphism if H i (f ) : H i (X) − → H i (Y ) is an isomorphism for each i (i.e., X and Y are equal at the level of cohomology). Claim 10.10. If f : X → Y is a homotopy equivalence, then it is a quasiisomorphism. Proof. This is an immediate consequence of Lemma 9.10, which ensures that f and g are inverses at the level of cohomology.  Corollary 10.11. Given f : X → Y , there is a long exact sequence · · · → H i−1 hCoker(f ) → H i X → H i Y → H i hCoker(f ) → H i+1 X → · · · . Proof. Letting g denote the map Y → hCoker(f ), the composition g Y − → hCoker(f ) → hCoker(g) = hKer(g)[1] ' X[1] is null-homotopic by Corollary 10.2, and the homotopy equivalence is by Claim 10.8. So by Lemma 10.5, the sequence H i Y → H i hCoker(f ) → H i X[1] = H i+1 X is exact; a further application of Lemma 10.5 shows the claim.  Claim 10.12. Suppose f i : X i ,→ Y i is injective for all i. Then hCoker(f ) → Y /X (i.e., the complex with Y i /X i in degree i) is a quasi-isomorphism. Example 10.13. If f : A ,→ B is a map of abelian groups in degree 0, then the map hCoker(f ) → B/A looks like ··· A B 0 ··· ··· 0 B/A 0 ··· . It’s easy to see that this is indeed a quasi-isomorphism. Note that there is a dual statement, that if f i is surjective in each degree, then the homotopy kernel is quasiisomorphic to the naive kernel. Remark 10.14. If A is an associative algebra (e.g. Z or Z[G]), then we can have chain complexes of A-modules d d · · · → X −1 − → X0 − → X1 → · · · , where the X i are A-modules and d is a map of A-modules. Here the cohomologies will also be A-modules. Now, our original problem was to define Tate cohomology for a finite group G acting on some A. Note that Ĥ 0 (G, A) = AG /N(A) = Coker(N : A → AG ). In fact, we can do better than N : A → AG ; the norm map factors through what we will call the coinvariants. P Definition 10.15. The coinvariants of A are AG := A g∈G (g − 1)A, which satisfies the universal property that it is the maximal quotient of A with gx = x holding for all x ∈ A and g ∈ G. Note that we can think of the invariants AG as being the intersection of the kernels of each (g − 1)A, so it is the maximal submodule of A for which gx = x holds similarly. Then the norm map factors as A N AG N AG . N Our plan is now to define derived (complex) versions of AG and AG called AhG − → AhG , and Tate cohomology will be the homotopy cokernel of this map. The basic observation is that Z is a G-module (i.e. Z[G] acts on Z) in a trivial way, with every g ∈ G as the identity automorphism. If M is a G-module, then M G = HomG (Z, M ) (because the image of 1 in M must be G-invariant and corresponds to the element of M G ) and MG = M ⊗Z[G] Z. Indeed, let I ⊆ A be an ideal acting on M . Then A/I ⊗A M = M/IM by the right-exactness of tensor products. Here, Z = Z[G]/I, where I is the “augmentation ideal” generated by elements g − 1 and therefore MG = M/I as desired. Now we have the general problem where A is an associative algebra and M an associative A-module, and we would like the “derive” the functors − ⊗A M and HomA (M, −). These should take chain complexes of A-modules and produce complexes of abelian groups, preserving cones and quasi-isomorphisms. We’ll begin working on this in the next lecture.