A TASTE OF SPECTRAL SEQUENCES ANDREW BAKER 1. Exact couples Suppose that A is an abelian category; for simplicity, we assume that its objects are sets and its morphisms are functions, but this is not really necessary since everything we do with elements can be veri¯ed only using morphisms. An exact sequence of morphisms of the form i / (1.1) D `@ D @@ ~~ @@ ~~ @@ ~~ k @ ~~ j E is called an exact couple. Notice that d = jk is a di®erential, i.e., it satis¯es d2 = dd = jkjk = 0. Therefore we can form the homology H(E; d) = ker d= im d: Now consider the sequence 0 0 i / 0 (1.2) D B` D BB || BB || 0 BB || 0 k B ~|| j E0 where D0 = iD = ker j; E0 = H(E; d); and i0 = i ; j0 = j i¡1; k0 = k : jD0 jD jker d Here the notation ( ) indicates passage to cosets. To see what k0 really does, let z 2 H(E; d). Then jk(z) = 0, so k(z) = i(z0) for some z0 2 D. But if w 2 D, then k(j(w)) = 0, so this is well-de¯ned on H(E; d). Notice also that the new di®erential d0 works as follows: using the same notation, we have d(z0) = 0, so [j(z0)] 2 H(E; d) and d0([z]) = [j(z0)]. Then (1.2) is also exact and is called the derived exact couple of (1.1). Iterating this we obtain a sequence of such exact couples i(r) (1.3) (r) / (r) D bE D EE yy EE yy EE yy k(r) E y| y j(r) E(r) in which E(r+1) = H(E(r); d(r)). In examples it is common to have gradings on things. For example, we might have integer bigradings on D and E for which i: Dp;q ¡! Dp+1;q¡1; j : Dp;q ¡! Ep+1;q¡1; k : Ep;q ¡! Dp¡1;q; d: Ep;q ¡! Dp;q¡1: Date: 20/01/2005. 1 2 ANDREW BAKER Then if we set Dr = D(r¡1);Er = E(r¡1); ir = i(r¡1); jr = j(r¡1); kr = k(r¡1); dr = d(r¡1); we ¯nd that r r r r r r r r r r r r i : Dp;q ¡! Dp+1;q¡1; j : Dp;q ¡! Ep+1;q¡1; k : Ep;q ¡! Dp¡1;q d : Ep;q ¡! Ep¡r;q+r¡1: 2. Grothendieck spectral sequences Let A ; B; C be abelian categories and suppose that A ; B have enough projectives, so objects admit resolutions by projective objects. Suppose that we have right exact covariant functors ©: A à B; £: B à C : Then the left derived functors Ls© and Ls£ exist for s > 0. Suppose also that for every projective object P of A we have Ls£(©(P )) = 0 (s > 1): Theorem 2.1 (Grothendieck spectral sequences). For every A in A there is a spectral sequence 2 Ep;q = Lp£Lq©(A) =) Lp+q(£©)(A): Furthermore, this spectral sequence is functorial in A. 2 This is a ¯rst quadrant spectral sequence since Ep;q = 0 unless p; q > 0. Example 2.2 (Change of rings for Tor). Suppose that R ¡! S is a ring homomorphism, and let N be a right S-module. Consider the functors © = ( ) ­R S : ModR à ModS; £ = ( ) ­S N : ModS à AbGps Then for every left R-module M there is a spectral sequence 2 S R R Ep;q = Torp (Torq (M; S);N) =) Torp+q(M; N): There are two standard ways to indicate such a ¯rst quadrant spectral sequence graphically. r In one we put Ep;q at position (p; q) in the plane representing the r-th page of the spectral sequence, then the di®erentials go r steps to the left and up by (r ¡ 1) steps. Notice that for r r(p;q) each (p; q) there is a number r(p; q) for which Ep;q = Ep;q when r > r(p; q); we write 1 r(p;q) Ep;q = Ep;q : Then for each n > 0, there is a ¯ltration R 0 = F¡1;n ⊆ F0;n ⊆ F1;n ⊆ ¢ ¢ ¢ ⊆ Fn;n = Torn (M; N) for which » 1 Fk;n=Fk¡1;n = Ek;n¡k (0 6 k 6 n): R Thus each Torn (M; N) arises from terms in the ¯rst quadrant lying on the line p + q = n and the ¯ltration increases with the p-value. Here the edge homomorphism 2 R 1 =» inc R E0;n = Torn (M; S) ­S N ¡! E0;n ¡! F0;n ¡¡! Torn (M; N) is the obvious homomorphism induced from the associativity homomorphism =» =» (M ­R S) ­S N ¡! M ­R (S ­S N) ¡! M ­R N: There is also another edge homomorphism R =» 1 inc 2 S Torn (M; N) ¡! Fn;n=Fn¡1;n ¡! En;0 ¡¡! En;0 = Torn(M ­R S; N) agreeing with the obvious map. r An alternative is to plot points in the (s; t)-plane where we put Ep;q at position (p + q; p); then the non-trivial terms appear only in the region where s > t > 0. This time the di®erential r R d goes 1 to the left and down by r. Then Torn (M; N) arises from terms on the t = n column and the ¯ltration increases with the t-value. A TASTE OF SPECTRAL SEQUENCES 3 There are variations of the Grothendieck spectral sequence 2.1 involving left (or a mixture of right and left) exact functors. Example 2.3 (Change of rings for Ext). Suppose that R ¡! S is a ring homomorphism, and let L be a right S-module. Consider the functors © = HomR(S; ): ModR à ModS; £ = HomS(L; ): ModS à AbGps Then for every right R-module M there is a third quadrant spectral sequence 2 ¡p ¡q ¡p¡q Ep;q = ExtS (L; ExtR (S; M) =) ExtR (L; M): For each (p; q) there is an r(p; q) for which r r(p;q) 1 Ep;q = Ep;q = Ep;q: For each n > 0, there is a ¯ltration n 0;n 1;n n;n n+1;n ExtR(L; M) = F ¶ F ¶ ¢ ¢ ¢ ¶ F ¶ F = 0 for which k;n k+1;n » 1 F =F = E¡k;k¡n (n > k > 0): Example 2.4 (Change of rings for Ext). Suppose that R ¡! S is a ring homomorphism, and let N be a right S-module. Consider the functors © = ( ) ­R S : ModR à ModS; £ = HomS( ;N): ModS à AbGps Then for every right R-module M there is a third quadrant spectral sequence 2 ¡p R ¡p¡q Ep;q = ExtS (Tor¡q(M; S);N) =) ExtR (M; N): For each (p; q) there is an r(p; q) for which r r(p;q) 1 Ep;q = Ep;q = Ep;q: For each n > 0, there is a ¯ltration n 0;n 1;n n;n n+1;n ExtR(M; N) = F ¶ F ¶ ¢ ¢ ¢ ¶ F ¶ F = 0 for which k;n k+1;n » 1 F =F = E¡k;k¡n (n > k > 0): Remark 2.5. By switching signs, such a spectral sequencecan be regraded as a cohomological p;q p;q p+r;q¡r+1 spectral sequence (Er ; dr) with dr : Er ¡! Er . Example 2.6 (Lyndon-Hochschild-Serre spectral sequence). Let G be a ¯nite group and N/G. Consider the functors N G=N © = ( ) : ModZ[G] à ModZ[G=N]; £ = ( ) : ModZ[G=N] à AbGps Then for every Z[G]-module M there is a ¯rst quadrant cohomological spectral sequence p;q p q p+q E2 = H (G=N;H (N; M)) =) H (G; M): 3. A spectral sequence for Tor and Ext of a limit Recall that in the category of right R modules, for any small ¯ltered category, every functor F : Iop à Mod (often called an inverse system in Mod ) there is an inverse limit lim F . This R R á I de¯nes a left exact covariant functor op lim: ModI à Mod á R R I on the abelian category of inverse systems in ModR; this category has enough injectives. How- ever, there is a weaker kind of injective object, called flasque or flabby; this is an object J for which every short exact sequence 0 ! J ¡! A ¡! B ! 0 4 ANDREW BAKER gives rise to an short exact sequence 0 ! lim J ¡! lim A ¡! lim B ! 0: á á á I I I For every inverse system F there is a flabby resolution 0 ! F ¡! J ² which can be de¯ned as follows. For each s > 0, de¯ne J s by Y s J (i) = F (is); 6= 6= 6= is!is¡1¡!¢¢¢¡!i0¡!i where the product is taken over all diagrams in I where no arrow is the identity except perhaps 0 is ! is¡1. Then there is a map F ¡! J de¯ned from the product of all the maps induced by diagrams j ¡! i, Y F (i) ¡! J 0(i) = F (j): j!i 0 Notice that this map is injective and it cokernel is J de¯ned by 0 Y J (i) = F (j): 6= j¡!i Iterating this we obtain a flabby resolution 0 ! F ¡! J ². Then for s > 0 we have Rs lim F = limsF = Hs(lim J ²; ±); á á á I I I the cohomology of the resulting cochain complex 0 ! lim J 0 ¡!± lim J 1 ¡!¢± ¢ ¢ ¡!± lim J s ¡!¢± ¢ ¢ á á á I I I Theorem 3.1.