SPECTRAL SEQUENCES AND AN APPLICATION TO THE SYMMETRY OF TOR Fan Zhou [email protected] In this expository note we will quickly develop spectral sequences and use them to prove the symmetry of the Tor functor. Contents 1. Bare Basics: Bi-things1 2. Exact Couples 2 3. Filtrations 4 4. Spectral Sequences5 5. Bicomplexes 6 6. Application to Tor 10 7. References 13 It seems most common in the literature for spectral sequences to begin on page 1. I suppose it doesn't really matter, but it seems as if by starting at page 0 many of the indices in the setup are greatly simplified. I therefore adopt this approach and hope that I do not mess up any indices in transitioning. We begin with some groundwork on spectral sequences; in the interest of length we will not prove any results about spectral sequences. We will however use them to prove the symmetry of the Tor functor, a property I used in M231a but which we did not prove. 1. Bare Basics: Bi-things Recall that the notion of a graded module: 0 Definition. A graded R-module is an indexed family B B M = (Mµ)µ2Z B B of R-modules. B B A graded map of degree a between graded modules M and N is B B B f : M −! N B B a family of maps B B f = fµ : Mµ −! Nµ+a @ µ2Z where we write deg f = a. These objects still admit index-wise notions of submodules, quotient modules, kernels, and images (and therefore exactness). Note as a minor detail that there would need to be some index correction f g sometimes; for example, A −! B −! C is exact when Img fµ−deg f = Ker gµ. 1 A (homological, i.e. decreasing index) chain complex can then be phrased as a graded module equipped with a map d such that deg d = −1 and d2 = 0. Recall that given such information we may then construct homology. Similarly one can define a bigraded module: 0 Definition. A bigraded R-module is an indexed family B M = (M ) B µ,ν µ,ν2Z B B of R-modules. B B A graded map of degree (a; b) between bigraded modules M and N is B B B f : M −! N B B a family of maps B B f = fµ,ν : Mµ,ν −! Nµ+a,ν+b @ µ,ν2Z where we write deg f = (a; b). Just as in the singly-graded case, this admits index-wise notions of submodules1 and quotient mod- ules and kernels and images and exactness. There is also an analogous notion of a chain complex, in the sense that here we may also take homology: 0 Definition. A differential bigraded module is (M; d) for a bigraded module M and a bigraded map B (the differential) d: M −! M with B 2 B d = 0: B B For deg d = (a; b), we may then construct homology in much the same manner: B @ Ker d Hµ,ν(M; d) := µ,ν=Img dµ−a,ν−b: 2. Exact Couples There is the notion of an exact couple: 0 Definition. An exact couple is (D; E; α; β; γ) where D; E are bigraded modules and α; β; γ are B bigraded maps between them fitting in a diagram B B α β γ α B D −! D −! E −! D −! D B B B which is exact, i.e. B B Ker α = Img γ; B B @ Ker β = Img α; Ker γ = Img β: The diagram that people usually draw is D α D γ β E 1 E.g. N ⊆ M if Nµν ⊆ Mµν for every index. 2 Given an exact couple (which we will denote with (D0;E0)), one can obtain others: 2 Proposition. Construct the first derived couple (D1;E1; α1; β1; γ1) by 6 1 0 0 6 D := Img α ⊆ D ; 6 6 1 0 0 6 E := H••(E ; d ); 6 6 0 : 0 0 0 0 6 d = β γ : E −! E ; 6 1 0 6 α := α j 1 ; 6 D 6 6 β1 := [β0(α0)◦−1 ]; 6 6 1 0 6 γ := γ ; 6 6 0 ◦−1 0;◦−1 0 6 where (α ) = α refers to the compositional inverse of α in 6 6 1 0 0 ◦−1 1 6 β (y) = [β (α ) (y)] for y 2 D 6 6 and 6 6 γ1[z] = γ0(z) for [z] 2 E1: 6 6 Note well that D1 ⊆ D0, so the indexing we are taking here is D1 := Img α . Note also 6 µ,ν µ−a1,ν−a2 6 that 6 6 1 0 6 deg α = deg α = (a1; a2); 6 6 1 0 0 6 deg β = deg β − deg α = (b1 − a1; b2 − a2); 6 6 1 0 6 deg γ = deg γ : 6 6 The content of the claim is that this is still an exact couple. 6 6 We can then iterate this construction to obtain (Dk;Ek; αk; βk; γk) the k-th derived couple, 6 6 defined recursively as the derived couple of the (k − 1)-th derived couple, which would be (we drop 6 6 the zero in α0 = α and similarly for other maps) 6 6 6 Dk = Img α◦k ⊆ D0 6 6 k 6 (Dµ,ν = Img αµ−a1,ν−a2 αµ−2a1,ν−2a2 ··· αµ−ka1,ν−ka2 ); 6 6 k k−1 k−1 6 E = H••(E ; d ); 6 6 k ◦−k 6 d = βα γ; 6 6 k 6 α = αjDk ; 6 6 βk = βα◦−k; 6 6 6 γk = γ; 6 6 6 with degrees 6 6 k 6 deg α = deg α; 6 6 deg βk = deg β − k deg α; 6 6 4 deg γk = deg γ: (wow this took up a whole page) 3 3. Filtrations Recall that a filtration of a module M is a family of submodules · · · ⊆ M µ−1 ⊆ M µ ⊆ M µ+1 ⊆ · · · : Recall that given such a filtration we could consider Grµ M := M µ=M µ−1 and M Gr M := Grµ M: µ Recall that given a complex C, we may consider filtrations: 0 Definition. A (increasing) filtration of a complex (C; d) is a family of graded submodules B µ B (F C)µ2Z B B with B B · · · ⊆ F µ−1C ⊆ F µC ⊆ F µ+1C ⊆ · · · B B B which is compatible with d: B µ µ B d: (F C)n −! (F C)n−1: B B A filtration is moreover said to be bounded if, for each n, there are a; b depending on n such that B B a @ F Cn = 0; b F Cn = Cn: µ µ µ µ µ µ We may shorten (F C)n to F Cn. We might also write C for F C, and C n for F Cn, in line with the philosophy that these things should be considered as filtrations of complexes rather than complexes of filtrations. Given a filtration, we may obtain an exact couple by looking at the long exact sequence of a pair. Namely, by considering ιµ−1 πµ 0 −! F µ−1C −! F µC −! F µC=F µ−1C −! 0 we obtain a long exact sequence α µ β µ µ−1 ··· Hn+1(F C) Hn+1(F C=F C) γ µ−1 µ µ−1 α=ι∗ µ β=π∗ µ µ−1 Hn(F C) Hn(F C) Hn(F C=F C) γ=δ µ−1 α Hn−1(F C) ··· 4 2 Proposition. Given a filtration of a complex, we may obtain an exact couple with 6 D = H (F µC); 6 µ,ν µ+ν 6 6 E = H (F µC=F µ−1C); 6 µ,ν µ+ν 6 µ−1 6 α = ι ; 6 ∗ 6 µ 6 β = π∗ ; 6 6 γ = δ; 6 6 6 so that the k-th derived exact couple is 6 µ−1 µ−k 6 (ι ···ι )∗ 6 Dk = Img H (F µ−kC) −! H (F µC) ; 6 µ,ν µ+ν µ+ν 6 6 k−1 k−1 6 Ek = Ker dµ,ν Img d ; 6 µ,ν = µ+k,ν−(k−1) 6 6 6 deg α = (1; −1); 6 6 deg β = (−k; k); 6 6 6 deg γ = (−1; 0); 6 4 deg dk = (−k − 1; k): Just so there's a picture, here is what this looks like at k = 0: α β ··· Dµ,ν+1 Eµ,ν+1 γ α β Dµ−1,ν+1 Dµ,ν Eµ,ν γ=δ α Dµ−1,ν ··· 4. Spectral Sequences Let us define what a spectral sequence is. Again you should be warned that people tend to start at page 1; because I like to be special I will start at page 02. 0 Definition. A spectral sequence is a sequence B k k B (E ; d )k≥0 B B B of differential bigraded modules such that @ k k−1 k−1 E = H••(E ; d ): Then, as we saw above, every filtration of a complex produces an exact couple whose derived exact couples give a spectral sequence. small If the filtration is moreover bounded (recall this means that for each n, eventually F Cn = 0 big and F Cn = Cn), then we may say something much better: 2Possibly I'll learn sometime later down the line that there's actually a generating function associated to these things and for that reason you should want to start at page 1, but whatever.