Spectral Sequences Means of the Isomorphism of the Previous Step K to the Sub- C 2000 M

Home , Injective sheaf, Spectral sequence

p,q p,q Geometria Superiore Reference Cards for k > 3 we can choose Bk+1(E2 ) and Zk+1(E2 ) as the p,q p,q subobjects of Zk(E2 ) containing Bk(E2 ) and identiﬁed by Spectral Sequences means of the isomorphism of the previous step k to the sub- c 2000 M. Cailotto, Permissions on last. v0.0 objects im(dp−k,q+k−1) and ker(dp,q) of Ep,q respectively; in Send comments and corrections to [email protected] k k k this way we have Z (Ep,q)B (Ep,q) ' Ep,q . Generalities. k+1 2 k+1 2 k+1

Definitions Let A be an abelian category. Define Af the So we have a long chain of inclusions: quasi-abelian category of filtered objects of A: objects are ob- p,q p,q p,q p,q E = Z2(E ) ⊇ ··· ⊇ Zk(E ) ⊇ Zk+1(E ) ⊇ ··· ⊇ jects A of A with a (decreasing) filtration, i.e. a family of 2 2 2 2 subobjects (F p(A)) of A s.t. F p+1(A) ⊆ F p(A) for every ··· ⊇ B (Ep,q) ⊇ B (Ep,q) ⊇ ··· ⊇ B (Ep,q) = 0 p∈Z k+1 2 k 2 2 2 p ∈ Z; morphisms ϕ : A−→B are morphisms of A respecting the filtrations, i.e. ϕ(F p(A)) ⊆ F p(B). and the definition continues with: p,q p,q p (d) for every p,q ∈ Z two subobjects B∞(E2 ) and Z∞(E2 ) of The filtration (F (A)) of A is: p,q p,q p,q p∈Z E2 such that Z∞(E2 ) ⊇ B∞(E2 ) and for every k T p (i) separated if p∈ F (A) = 0, p,q p,q p,q p,q Z Zk(E2 ) ⊇ Z∞(E2 ) ⊇ B∞(E2 ) ⊇ Bk(E2 ) (ii) coseparated (or exhaustive) if S F p(A) = A, p∈ p,q p,q p,q Z and define: E = Z∞(E ) B∞(E ); p q ∞ 2 2 (iii) discrete if there exists p ∈ Z s.t. F (A) = 0 (so F (A) = 0 n (e) a family (E )n∈ of objects of A ; for every q > p), Z f ∼ p q (f) a family (βp,q) of isomorphisms βp,q : Ep,q→gr (Ep+q). (iv) codiscrete if there exists p ∈ Z s.t. F (A) = A (so F (A) = p,q∈Z ∞ p A for every q 6 p), Remarks on the definition of spectral sequences: (v) finite if it is discrete and codiscrete. (1) the construction of chains of subobjects in (c) can be per- Remark that discrete (resp. codiscrete) filtrations are sepa- formed for every r in the spectral sequence, defining for rated (resp. exhaustive). p,q p,q p,q every k > r the subobjects Bk(Er ) and Zk(Er ) of Er . p p+1 p,q For A ∈ Af : grp(A) := F (A) F (A) ∈ A, and if A admits (2) for a fixed n ∈ Z, the family {E∞ }p+q=n gives a measure L of the skips in the filtration of En: in fact, if the terms arbitrary direct sums, define: gr(A) := grp(A) ∈ A, the p∈Z L p,q∼ n exist, we have E∞ =gr(E ). associated graded object. This gives a functor gr : Af → A. p+q=n p,q n 0 A morphism of two spectral sequences E = (Er ,E ) and E = Let ϕ : A−→B be a morphism in Af ; then: 0p,q 0n p,q n (E r ,E ) is the datum of f = (fr ,f ) where: (1) if grϕ is a monomorphism, and the filtration of A is ex- (a) fp,q : Ep,q −→ E0p,q are morphisms of A, haustive and separated, then ϕ is a monomorphism; r r r (b) compatibility with the differentials: commutation of (2) if grϕ is an epimorphism, and the filtration of B is exhaus- p,q tive and discrete, then ϕ is an epimorphism; p,q fr 0p,q Er −−−−−−−−−−−→ E r (20) if grϕ is an epimorphism, the filtration of B is exhaustive p,q   0p,q dr y y d r and separated, the filtration of A induces a complete topol- p+r,q−r+1 0p+r,q−r+1 Er −−−−−−−→ E r ogy, then ϕ is an epimorphism; p+r,q−r+1 fr (3) if grϕ is an isomorphism, the filtration of A is exhaustive (c) compatibility with the isomorphisms: and separated, and the filtration of B is exhaustive and p,q p,q Z (E ) f¯p,q Z (E0 ) discrete, then ϕ is an isomorphism. r+1 r −−−−−→r r+1 r p,q 0p,q Br+1(Er ) Br+1(E r ) SpS(A): the category of spectral sequences of A. p,q   0p,q αr y y α r p,q n p,q 0p,q A spectral sequence E = (E ) r 2 ,(E ) in A is the E −−−−−−−−−→ E r > n∈Z r+1 p,q r+1 p,q∈Z f data of: r+1 ¯p,q p,q commutes, where the fr define the morphisms in cohomology; (a) a family (Er ) of objects of A defined for r > 2 and p,q ∈ Z, p,q p,q p,q p+r,q−r+1 (d) compatibility with the ∞ level (b) a family of morphisms (dr ) where dr : Er → Er fp,q B (Ep,q) ⊆ B (E0p,q), fp,q Z (Ep,q) ⊆ Z (E0p,q) p,q p−r,q+r−1 2 ∞ 2 ∞ 2 2 ∞ 2 ∞ 2 and dr ◦dr = 0 for all indices; in such a way that we induce morphisms: f¯p,q : Ep,q → E0p,q; p,q p,q ∞ ∞ ∞ so that we can define: Zr+1(Er ) = ker(dr ) and (e) fn : En → E0n morphisms of filtered objects; p,q p−r,q+r−1 Br+1(Er ) = im(dr ) for which p,q p,q p,q (f) finally the compatibility with the limit Br+1(Er ) ⊆ Zr+1(Er ) ⊆ Er and we require: f¯p,q p,q Ep,q −−−−−−−−−−−→∞ E0p,q (c) a family (αr ) of isomorphisms ∞ ∞ p,q   0p,q ∼ β β αp,q : Z (Ep,q)B (Ep,q) −→ Ep,q ; y y r r+1 r r+1 r r+1 p+q 0p+q grp(E ) −−−−−−→ grp(E ) . gr (f p+q ) we can define for every p,q ∈ Z two families of subobjects of p Ep,q, B (Ep,q) and Z (Ep,q), in such a way that Ep,q ' 2 k 2 k 2 k p,q p,q Remarks on the definition of morphisms of spectral sequences: Zk(E2 ) Bk(E2 ) by induction: p,q p,q p,q (1) if a morphisms f of spectral sequences has the property for k = 2 put B2(E ) = 0 and Z2(E ) = E , p,q 2 2 2 that for a fixed r all fr are isomorphisms, then by (c) for p,q for k = 3 already defined, all s > r the fs are also isomorphisms.

1 GeoSupRC - Spectral Sequences c 2000 by MC n−q(n),q(n) n−q(n),q(n) n−q(n),q(n) (2) a morphism of spectral sequences is an isomorphism if and Er for all s r, so that E∞ = Er p,q > only if: the f2 are isomorphisms for all p,q ∈ Z, in the if the spectral sequence is weakly convergent: the limit of the condition (d) equalities hold, and the fn are isomorphisms sequence can be read at level r. for all n ∈ Z. (2) if moreover the filtrations on the limit are exhaustive and sepa- p n Convergence and regularity conditions: rated, then they are finite and F (E ) = 0 for p > n−q(n) and p n n F (E ) = E for p 6 n−q(n); we have also an isomorphism (o) a spectral sequence is weakly convergent if n n−q(n),q(n) E ' Er . B (Ep,q) = S B (Ep,q) Z (Ep,q) = T Z (Ep,q) ∞ 2 k 2 ∞ 2 k 2 0 (0 ) (Deligne) a spectral sequence E is degenerated in Er0 , for r0 k>2 k>2 p,q > 2, if it is biregular and for any r > r0 we have dr = 0 for all p hold, i.e. if the data of (d) are determined by the previous p p+q p,q and q; then Er0 = E∞ and gr E = Er0 . points. A weakly convergent spectral sequence is: (10) in the category of R-modules of finite length with R a ring, (i) regular if P p,q n (o) p+q=n lgEr > lgE ; (1) for any p,q the decreasing sequence Z (Ep,q) becomes k 2 k (i) equality holds for r = r if and only if E degenerates at r ; stationary, 0 0 n n (ii) if E = 0 except for a finite number of n, then for all r > 2 (2) for any n the filtration of E is discrete and exhaustive. P p+q p,q P n n p,q we have p,q(−) lgEr = n(−) lgE . This implies that for a fixed pair (p,q) we have Z∞(E2 ) = Z (Ep,q) for k 0, but depending on p and q. k 2 Homological and Spherical Sp.Seq. (ii) co-regular if: (1) for any p,q the increasing sequence B (Ep,q) becomes Cohomological spectral sequences: E is cohomological if k 2 k p,q E2 = 0 whenever p < 0 or q < 0. Then under the same condi- stationary, p,q p,q n tions Er = 0 for all r, and E∞ = 0. Properties of the coho- (2) for any n the filtration of E is codiscrete mological spectral sequences: p,q This implies that for a fixed pair (p,q) we have B∞(E2 ) = p,q (1) if the spectral sequence is weakly convergent and the filtra- Bk(E2 ) for k 0, but depending on p and q. tions of the limit are exhaustive and separated, then from (iii) biregular if it is regular and co-regular, i.e. if: the isomorphisms βp,q for p < 0 or q < 0 we deduce that p,q p,q nEn if p 0 (1) for any p,q the sequences Z (E ) and B (E ) F p(En) = 6 k 2 k k 2 k 0 if p > n . became stationary, (2) for any n the filtration of En is finite. In particular we have an exact sequence p,q p,q 1,0 1 0,1 This implies that for a fixed pair (p,q) we have E = E for 0−→E∞ −→E −→E∞ −→0 ∞ k k 0, but depending on p and q. coming from 0 → F 1(E1) → E1 → E1/F 1(E1) → 0. Remarks on the regular spectral sequences: (2) In the previous situation we have the edge morphisms: p,q p,q n,0 n n 0,n (1) if E is regular, then E∞ = lim Er where the induc- Er −→E E −→Er −→r0 tive system is given by: given resp. by the compositions Ep,q ' Zp,qB (Ep,q)−→Zp,qB (Ep,q) ' Ep,q . n,0 n,0 n n n n r ∞ r 2 ∞ r+1 2 r+1 Er E∞ ' grn(E ) = F (E ) ,→ E p,q p,q n,0 n,0 The result follows from B∞ = lim Br(E2 ). where the first map comes from E∞ = lim Er ; and −→r0 −→r (2) dually, if E is co-regular, then Ep,q = lim Ep,q where n n 1 n n 0,n 0,n ∞ r E E /F (E ) = gr0(E ) ' E∞ ,→ Er ←−r0 the projective system is given by: 0,n 0,n T 0,n where the last map comes from E∞ = lim Er = Er . p,q p,q p,q p,q p,q p,q ←−r Er ' Zr(E2 ) B∞ −→Zr+1(E2 ) B∞ ' Er+1. r p,q p,q (3) There exists an exact sequence The result follows from Z∞ = lim Zr(E2 ). ←−r0 d0,1 0−→E1,0 −→E1 −→E0,1 −→2 E2,0 −→E2. (3) Let f be a morphism between spectral sequences E and 2 2 2 p,q 0 p,q E ; we have that f2 isomorphisms for all (p,q) implies f Homological spectral sequences: E is homological if E2 = is an isomorphism in the following two cases: 0 whenever p > 0 or q > 0. Then under the same conditions 0 p,q p,q (i) E and E are biregular; Er = 0 for all r, and E∞ = 0. Properties of the cohomologi- (ii) E and E0 are regular and A is an abelian category where cal spectral sequences: the inductive filtered limits exist and are exact. (1) if the spectral sequence is weakly convergent and the fil- Degeneration of spectral sequences: trations of the limit are exhaustive and separated, nEn if p n (0) (original sense: Godement, Grothendieck) a spectral sequence F p(En) = 6 0 if p > 0 . E degenerates if there exists an r > 2 s.t. for all n ∈ Z there n−q,q n−q,q exists q(n) ∈ Z with Er = 0 for all q 6= q(n) (so Es = 0 In particular we have an exact sequence for all s r and q 6= q(n)). Remarks on degenerate spectral > 0−→E0,−1 −→E−1 −→E−1,0 −→0. sequences: ∞ ∞ (1) if we have q(n+1) > q(n)−r +1 for all n (for example: q(n) = k (2) In the previous situation we have the edge morphisms: n−q(n),q(n) 0,−n −n −n −n,0 or q(n) = n+k with constant k) we can easily see that Es = Er −→E and E −→Er .

2 GeoSupRC - Spectral Sequences c 2000 by MC i,0 i i i−n,n (3) There exists an exact sequence Er −→E and E −→Er . d−2,0 −2 −2,0 2 0,−1 −1 −1,0 (4) There exists a long exact sequence E −→E2 −→ E2 −→E −→E2 → 0. i,0 i i−n,n i+1,0 i+1 ···−→Er −→E −→Er −→Er −→E −→··· . Spectral sequences with spherical base: E is said to have p,q spherical bases if there exists r and n > r for which Er = 0 if Spectral sequence of a filtered complex. p 6= 0,n. Properties of spherical base spectral sequences: (1) if the spectral sequence is weakly convergent and the filtra- Let C(A)f be the category of filtered complexes, i.e. complexes tions of the limit are exhaustive and separated, then the C· of objects of A endowed with a (decreasing) filtration F pC· filtrations are finite and compatible with the differentials (dF pCm ⊆ F pCm+1). Us- (Ei if p 0 ing F p(C·) = Cn ∩F p(C·) to define a filtration on every 6 n∈ F p(Ei) = n,i−n Z E∞ if 0 < p 6 n single object, we have an equivalence of categories with C(Af ). 0 if p > n. The filtration is regular (resp. strongly regular) if for every (2) in the previous situation for every i there is the exact se- n there exists an integer M(n) s.t. for p > M(n) we have quence: Hn(F pC) = 0 (resp. F pCn = 0). n,i−n i 0,i A homotopy between morphisms of filtered complexes is of or- 0−→E∞ −→E −→E∞ −→0 der less than or equal to k if it sends F p into F p−k. coming from the sequence 0 → F n(Ei) → Ei → Ei/F n(Ei) → i n i i 1 i i 0,i 0 considering E /F (E ) = E /F (E ) = gr0E ' E∞ . If (1) We have a functor from C(A)f to SpS(A), defined also for the in the definition we put n < r, then the spectral sequence is levels r = 0,1, for which: constant after the r level, and, under the usual hypothesis (r=0) Zp,q = F p(Cp+q), Zp+1,q−1 = F p+1(Cp+q) so that Ep,q = of weak convergence and filtrations of the limit exhaustive 0 −1 0 p p+q p+1 p+q p,q and separated, we can read the exact sequence with ∞ = r. F (C ) F (C ) with the differentials d0 induced by (3) we find also the morphisms those of the complex C by quotient. p,q n,i−n 0,i (r=1) E = H F p(Cp+q)F p+1(Cp+q) = Hp+q F p(C)/F p+1(C) E −→Ei and Ei −→E 1 r r p,q with the differentials d1 obtained by the short exact sequence given resp. by the compositions 0 → F p+1(C)/F p+2(C) → F p(C)/F p+2(C) → F p(C)/F p+1(C) → 0 n,i−n n,i−n i n i i Er E∞ ' grn(E ) = F (E ) ,→ E using the coboundary morphism: n,i−n n,i−n where the first morphism is given by E∞ = lim Er ; δp+q −→r Ep,q = Hp+q gr (C) −→ Hp+q+1 grp+1(C) = Ep+1,q . and 1 p 1 i i 1 i i 0,i 0,i E E /F (E ) = gr0(E ) ' E∞ ,→ Er (2) Construction of the functor: −∞ ∞ 0,i 0,i (a) Define F C = C and F C = 0. Consider the sequence of where the last morphism is given by E∞ = lim Er . ←−r indexes (p+r,p+1,p,p−r +1); from the exact sequence (4) There exists a long exact sequence 0 → F p+1CF p+rC → F pCF p+rC → F pCF p+1C → 0 n,i−n i 0,i n,i+1−n i+1 ···−→Er −→E −→Er −→Er −→E −→··· we define p,q p,q Zp,q(C) = imHp+q(F pC/F p+rC)−→Hp+q(F pC/F p+1C) In fact for r < n we can see that Er = Er+1 or all p and r q, so that we can suppose r = n; moreover =∼kerHp+q(F pC/F p+1C)−→Hp+q+1(F p+1C/F p+rC) ; E0,i ' ker(d0,i) ,→ E0,i ∞ n n then from the exact sequence n,i−n+1 E 0 → F pC F p+1C → F p−r+1C F p+1C → F p−r+1C F pC → 0 En,i−n+1 n ' En,i−n+1 . n 0,i ∞ im(dn ) we define p,q p+q−1 p−r+1 p p+q p p+1 Br (C) = im H (F C/F C)−→H (F C/F C) Spectral sequences with spherical fiber: E is said to have p,q =∼kerHp+q(F pC/F p+1C)−→Hp+q(F p−r+1C/F p+1C) ; spherical fiber if there exists r and n > r −1 for which Er = 0 if q 6= 0,n. Properties of spherical fiber spectral sequences: p,q p,q we find Br (C) ⊆ Zr (C) and we can use (1) if the spectral sequence is weakly convergent and the filtra- p,q p,q p,q tions of the limit are exhaustive and separated, then the Er (C) = Zr (C) Br (C). filtrations are finite and In particular Zp,q(C) = Hp+q(F pC/F p+1C) and Bp,q(C) = 0 (Ei if p i−n 1 1 6 (remark that the general definition does not work for r = 0). F p(Ei) = i−n,n E∞ if i−n < p 6 i 0 if p > i. (d) similarly (r = ∞) we define: p,q p+q p p+q p p+1 (2) in the previous situation for every i there is the exact se- Z∞ (C) = im H (F C)−→H (F C/F C) p+q p p+1 p+q+1 p+1 quence: =∼ker H (F C/F C)−→H (F C) p,q i,0 i i−n,n B (C) = imHp+q−1(C/F pC)−→Hp+q(F pC/F p+1C) 0−→E∞ −→E −→E∞ −→0 ∞ =∼kerHp+q(F pC/F p+1C)−→Hp+q(C/F p+1C) coming from the sequence 0 → F i(Ei) → Ei → Ei/F i(Ei) → i i i i i−n,n p,q 0 considering E /F (E ) = gri−nE ' E∞ . and E∞ (C) as the quotient; we have in fact (3) we find also the morphisms 0 = B1 ⊆ ··· ⊆ Bs ⊆ ··· ⊆ B∞ ⊆ Z∞ ⊆ ··· ⊆ Zr ⊆ ··· ⊆ Z1 = E1 .

3 GeoSupRC - Spectral Sequences c 2000 by MC p,q p,q p+r,q−r+1 0 (b) the differentials dr : Er −→Er are defined by the compositions: ↓ 0 0 K(p+1,p+r) δr Zr/Br Zr/Zr+1 −→Br+1/Br ,→ Zr/Br ↓ ↓ ↓ ∼= 0 −→ K(p+r,p+r +1) −→ K(p,p+r +1) −→ K(p,p+r) −→ 0 ↓ ↓ ↓ so that we have kerd = Z /B and imd = B /B . r r+1 r r r+1 r 0 −→ K(p+1,p+r +1) −→ K(p,p+r +1) −→ K(p,p+1) −→ 0 p,q (c) the isomorphisms αr are induced by ↓ ↓ ↓ ∼ K(p+1,p+r) 0 0 H(Er) = Zr+1/Br Br+1/Br =Zr+1/Br+1 = Er+1 ; ↓ n 0 (e) we consider En(C) = H (C) the cohomology of C as a col- lection of objects with the filtrations defined by F pHn(C) = then constructing the commutative diagram n p n im(H (F (C))−→H (C)) with the morphisms induced in co- HnK(p+1,p+r) p  0 homology from the inclusions F (C) ,→ C; . y p,q ∼ p p+q n n+1 (f) the isomorphisms βn are induced by E∞ (C)=gr H (C) . H K(p,p+r) −−→ H K(p+r,p+r +1) % u  (x) the key lemma for the points (b), (c), (f) is: given a diagram y y HnK(p,p+r +1) −−→ HnK(p,p+1) −−→ Hn+1K(p+1,p+r +1) with exact row: 0 P u % c & d p,q 0 y and finally applying the square lemma to im(u) = Zr , im(u ) = L −−→ M −−→ N p,q p+r,q−r+1 0 p+r,q−r+1 a b Zr+1 and im() = Br+1 , im( ) = Br . then b induces an isomorphism im(c)/im(a) → im(b). The square In order to evaluate the terms E∞ we start with the exact of the lemma says that if we have a commutative diagram diagram of complexes 0 L 0 0  0 . y a ↓ ↓ c0 0 −→ K(p+1,∞) −→ K(p+1,∞) −→ 0 P −−→ M0 ↓ ↓ ↓ % c  b0 y y 0 −→ K(p,∞) −→ K(∞,−∞) −→ K(−∞,p) −→ 0 L −−→ M −−→ N a b ↓ ↓ ↓ with exact row and column, then we have the isomorphisms 0 −→ K(p,p+1) −→ K(−∞,p+1) −→ K(−∞,p) −→ 0 ↓ ↓ ↓ im(c)/im(a)∼im(bc) = im(b0c0)∼im(c0)/im(a0) = = 0 0 0 0 induced respectively from b and b . then construct the commutative diagram p q HnK(p+1,∞) (y) the key for the points (a), (d) is: let K(p,q) = F /F for p 6 q;  0 for (p1,q1) 6 (p2,q2), i.e. p1 6 p2 and q1 6 q2, we have the . y v commutative diagram u0 HnK(p,∞) −−→ HnK(−∞,∞) K(p ,q ) −−→ K(p ,q ) 2 2 1 2 % u   &  y y y y Hn−1K(−∞,p) −−→ HnK(p,p+1) −−→ HnK(−∞,p+1) K(p2,q1) −−→ K(p1,q1) v and calling the diagonal map u12, we have for (p2,q2) (p3,q3) p,q 6 and finally apply the square lemma to im(u) = Z∞ , im(v) = the compatibility u13 = u12 ◦u23. Moreover, for p1 6 p2 6 p3 Bp,q and im(u0) = F pEn, im(v0) = F p+1En, where n = p+q. we have the exact sequence ∞ (3) Starting with complexes with regular and exhaustive filtrations, 0−−→K(p2,p3)−−→K(p1,p3)−−→K(p1,p2)−−→0 if A admits exact filtered inductive limits, then we obtain reg- and the associated long exact sequence of cohomology. ular spectral sequences. (4) A family (Bp,q) of objects of A is convergent to a family of (z) The chain of inclusions in (d) can be seen using the sequences p,q∈Z n p,q p+q of index (p,p+1,p+r,p+r +1) and (p−r,p−r +1,p,p+1), and objects (D )n∈ , B =⇒ D , if there exists a complex X ∈ Z p,q the commutative diagrams C(A) with regular filtration s.t. E (X) ' Bp,q and Hn(X) ' n 2 ur+1 vr−1 D for every p,q,n ∈ . K(p,p+r +1) −−→ K(p,p+1) K(p,p+1) −−→ K(p−r +1,p+1) Z     (5) Given two morphisms of filtered complexes f,g : C → C0 with y y = vr y y ur = a homotopy s of order less than or equal than k, we have for K(p,p+r) −−→ K(p,p+1) , K(p−r,p+1) −−→ K(p−r,p+1) . the morphisms induced on the spectral sequences f,g : E(C) → E0(C0) that fn = gn for any n ∈ and also fp,q = gp,q for every In fact we have Z = im(Hnu ) ⊇ im(Hnu ) = Z and Z r r r r r+1 r+1 r > k. This is the main motivation for the definition of spectral B = ker(Hnv ) ⊆ ker(Hnv ) = B ; moreover considering r r−1 r r+1 sequence with constraint r 2, instead of r ∈ . the index sequence (p−r +1,p,p+1,p+s), the zero composi- > N tion (of coboundary morphisms) ∂ ∂ Sp. sequences of a double complex. Hn−1K(p−r +1,p)−−→r HnK(p,p+1)−−→s Hn+1K(p+1,p+s) Let C2(A) be the category of bicomplexes with objects in A, gives Zs = ker(∂s) ⊇ im(∂r) = Bs for any r,s, in particular for i.e. C(C(A)) or more explicitly: objects are the collections r = s = ∞. (Xn,m) ,(dn,m) ,(dn,m ) n,m∈Z IX n,m∈Z IIX n,m∈Z The basic isomorphism for the definition of dr can be see starting with the exact diagram of complexes where Xn,m ∈ obA,

4 GeoSupRC - Spectral Sequences c 2000 by MC dn,m p IX n+1,m n,m and F (X) codiscrete (resp. discrete) if there exists n0 (resp. Xn,m −→ Xn+1,m d ◦d = 0 II n,m IX IX 00 0 00 n,m d , n,m+1 n,m n ) s.t. for any n > n (resp. n 6 n ) we have X = 0; or if Xn,m −→IIX Xn,m+1 d ◦d = 0 00 0 00 0 IIX IIX there exists m (resp. m ) s.t. for any m 6 m (resp. m > m ) we have Xn,m = 0. n+1,m n,m n,m+1 n,m ·· and dIIX ◦dIX = dIX ◦dIIX . Morphisms between X 2 n,m (5) We define two functors C (A)−→SpS(A): the two spectral ·· n,m f n,m and Y are collections X −→ Y for n,m ∈ Z s.t. sequences associated to a bicomplex X are the spectral se- dn,m ◦fn,m = fn+1,m ◦dn,m quences associated to the simple complex using the two filtra- IY IX tions: I E(X) and II E(X) respectively. They are the same n,m n,m n,m+1 n,m dIIY ◦f = f ◦dIIX . limit (the total cohomology) and we can calculate the first terms: Remark that C2(A) has an internal automorphism (exchange I Ep,q(X) I Ep,q+1(X) II Eq,p(X) II Eq,p+1(X) of the indexes). Standard notions on the category of bicom- 0 0 0 0 k k k k plexes: Xp,q −−−−−−−−−→ Xp,q+1 Xp,q −−−−−−−−−→ Xp+1,q I p,q p,q II q,p p,q (1) Cocycles, coboundary and cohomology operators: d0 =dII d0 =dI ·,m ·,m n,· n,· I Ep,q(X) I Ep+1,q(X) II Eq,p(X) II Eq+1,p(X) ZII := ker(dII ) ZI := ker(dI ) 1 1 1 1 ·,m ·,m−1 n,· n−1,· k k k k BII := im(dII ) BI := im(dI ) p,q p+1,q p,q p,q+1 ·,m ·,m ·,m n,· n,· n,· H (X) −−−−−−−→ H (X) H (X) −−−−−−−−→ H (X) H := Z B H := Z B II I p,q p,q II I II q,p p,q I II II II I I I d1 =dI d1 =dII I p,q p q II q,p q p these structures are complexes w.r.t. the pointed index (with E2 (X) = H H (X) E2 (X) = H H (X) . ·,m n,· I II II I differentials resp. d and d ) so that we define also: I II (6) Two homotopic morphisms of double complexes induce the n m n,m n+1,m m n n,m n,m+1 same morphism on the associated spectral sequence. ZI HII := ker(HII → HII ) ZII HI := ker(HI → HI ) n m n−1,m n,m m n n,m−1 n,m BI HII := im(HII → HII ) BII HI := im(HI → HI ) (7) Given a bicomplex X, if one of the following conditions hold: HnHm := ZnHm BnHm Hm Hn := Zm HnBm Hn . I II I II I II II I II I II I (i) there exists n0 and m0 s.t. for n < n0 or m < m0 (or: n > n or m > m ) we have Xn,m = 0; (2) a morphism f of bicomplexes X and Y is homotopic to zero if 0 0 there exist a pair of families of morphisms for n,m ∈ (ii) there exists n0 and n1 (or: m0 and m1) s.t. for n < n0 or Z n,m sn,m : Xn,m → Y n−1,m n > n1 (or: m < m0 or m > m1) we have X = 0; tn,m : Xn,m → Y n,m−1 then the two associated spectral sequences are biregular (in particular, for k > n −n (or: k > m −m ) the sequences are with the commutation properties 1 0 1 0 constant). sn,m+1 ◦dn,m = dn−1,m ◦sn,m IIX IIY (8) Given a bicomplex X in A category with exact filtrant inductive n+1,m n,m n,m−1 n,m t ◦dIX = dIY ◦t limits, then: n,m and reconstructing f: f is (i) if there exists n0 s.t. for n > n0 (or: there exists m0 s.t. n,m I sn+1,m ◦dn,m +dn−1,m ◦sn,m +tn,m+1 ◦dn,m +dn,m−1 ◦tn,m. for m < m0) we have X = 0 then the sequence E(X) IX IY IIX IIY is regular; Obviously s (resp. t) induces homotopies for the morphism (ii) if there exists n s.t. for n < n (or: there exists m s.t. ·,m ·,m 0 0 0 induced by f between the complexes H (X) and H (Y ) n,m II II II for m > m0) we have X = 0 then the sequence E(X) n,· n,· is regular. (resp. HI (X) and HI (Y )) and so f induces the null map n m n m between the cohomologies H H (X) and H H (Y ) (resp. (9) (Weyl’s lemma) Given a bicomplex C·· with non negative de- m n m n I II I II HII HI (X) and HII HI (Y )). Two maps of bicomplexes are grees (Cn,m = 0 if n < 0 or m < 0) with exact rows and columns, homotopic if the difference is homotopic to zero. except for the first, there exists a canonical isomorphism be- (3) Under one of the following hypothesis: tween the cohomology of the first row and that of the first (i) A admits inductive limits; column. (ii) for objects of C2(A) s.t. for any k the cardinality of {(n,m)| n+m = k, Xn,m 6= 0} is finite; Sp. seq. of a functor w.r.t. a finitely filtered object. we have a functor s : C2(A)−→C(A) acting on the objects by s(X)k = LXn,m and dk : s(X)k −→s(X)k+1 Let A be the abelian category of finitely filtered objects of s(X) ff n+m=k A, i.e. objects of A endowed with a finite filtration; morphisms in this category are the morphisms of A (without reference to where the differentials are induced by Xn,m −→s(X)k+1 de- the filtrations). If A has sufficiently many injective objects, fined for n+m = k as ιk+1 ◦dn,m +(−)nιk+1 ◦dn,m us- · n+1,m I n,m+1 II we have a functor of “filtered resolution”: Aff −→ C(I)f with k+1 i,j k+1 the properties that: ing ιi,j : X −→s(X) for i+j = k +1. By definition the (total) cohomology of a bicomplex is the cohomology of the (1) A· is an injective resolution of A, associated simple complex. (2) F i(A·) is an injective resolution of F i(A), (4) We have two different filtrations on s(X) given by (3) F i(A·) = 0 (resp. = A) if F i(A) = 0 (resp. = A), F p s(X)k = LXn,m and F p s(X)k = LXn,m ; I II (4) given F : A → B and two morphisms u·,v· : A·−→B· exten- n+m=k n+m=k · · n>p m>p sions of f then they are homotopic: u ∼ v , p (5) given f : A → B for which f F i(A) ⊆ F i+s(B) for any i ∈ they are exhaustive and separated; we use the symbols FI (X) p p · · · · i · i+s · and FII (X). Moreover: FI (X) is discrete (resp. codiscrete) Z then f : A → B verifies f F (A ) ⊆ F (B ).

5 GeoSupRC - Spectral Sequences c 2000 by MC Given an additive functor F : A−→B between abelian cate- (2) Given a complex X· of F -acyclic objects (i.e. RqF (Xi) = gories, the first with suff. many injectives, and an object A 0 for every q > 0) then RpF (X·) ' HpF (X·) and moreover · p q · p+q · of Aff , we take the filtered injective resolution A and we R F H (X ) =⇒ H F (X ). In fact the first sp.seq. de- define the spectral sequence of F w.r.t. A as the spectral se- generates in I Ep,0(X·) = Hp R0F (X·) = HpF (X·). quence associated to the filtered complex F (A·) with filtration 2 F i F (A·) = F F i(A·). Limit for this sp.seq. is the coho- (3) By the previous two points, we can calculate the derived functors of F using F -acyclic resolutions of the objects of A. mology Hp+q F (A·) = Rp+qF (A) with the filtration (4) Considering two functors F : A−→B and G : B−→C and A ∈ i p+q p+q i · p+q · F R F (A) = im H F F (A ) −→H F (A ) A, if A has sufficiently many injectives we can resolve A with = imRp+qF F i(A)−→Rp+qF (A) I· ∈ C(I), then apply the spectral sequences of G w.r.t. F (I·); = kerRp+qF (A)−→Rp+qF A/F i(A) . we have: p,q I E (FI·) = Hp RqG(FI·) II Ep,q(FI·) = RpGHq(FI·) The terms in the first level are given by: 2 2 = Hp RqG◦F (I·) = RpG(RqF (A)) Ep,q = Hp+q grp F (A·) = Rp+qF (grp(A)) . 1 = Rp (RqG◦F )(A) = (RpG◦RqF )(A) Hypercohomology. sequences with the same limit: the hypercohomology of G◦F w.r.t. A, R·GF (I·). (0) Given a complex C· in C(A), a Cartan-Eilenberg resolution of (5) Leray sp.seq.: if F sends injective objects of A into G-acyclic C· is a bicomplex L·· with L·j = 0 for j < 0 with a morphism objects of B and G is left exact, then R·GF (I·) ' Rp(G◦ of complexes C· −→ L·0 s.t. F )(A) and RpG◦RqF =⇒ Rp+q(G◦F ). In fact the first sp.seq. i (i) for any i the sequence 0−→Ci −→ Li· is exact and induce I p,0 · p 0 p degenerates in E2 (FI ) = R R G◦F (A) = R (G◦F )(A) exact sequences by the coboundary, cocycles and cohomol- by the acyclicity assumption. ogy functors; (6) In the general situation of (4) we can write the edge mor- ·j (ii) for any j the simple complex L admits split short exact phisms: sequences Rp R0G◦F (A)−→RpGF (I·)−→R0 (RpG◦F )(A) 0−→Bi(L·j)−→Zi(L·j)−→Hi(L·j)−→0 i ·j i,j i+1 ·j 0−→Z (L )−→L −→B (L )−→0 (RpG◦R0F )(A)−→RpGF (I·)−→(R0G◦RpF )(A) for any i. and by composition (first up and second down) we have a i,j The resolution is injective if every object L is (so that by the canonical morphism: second condition, also the coboundary, cocycles and cohomol- Rp R0G◦F −→R0G◦RpF. ogy objects are). (1) If A has enough injectives then every complexes admits an in- If G is an exact functor, then this is an isomorphism and we jective Cartan Eilenberg resolution and a morphism between write Rp (G◦F ) ' G◦RpF . In fact the exactness of G implies f · the degeneracy of both sp.seq. and the edge morphisms are complexes C· −→ C0 can be lift to a morphism between the isomorphisms. F ·· resolutions L·· −→ L0 , so that we obtain an exact “resolu- 2 (7) If F is an exact functor, without hypothesis on G, then the tion functor” CE : C(A) → C (I). Moreover homotopic mor- II p,0 · p phisms go to homotopic morphism between resolutions. Ob- second sp.seq. degenerates in E2 (FI ) = (R G◦F )(A) ' viously, bounded complexes admit resolutions bounded in the RpGF (I·) and we have Rp (RqG◦F )(A) =⇒ (Rp+qG◦F )(A) same way. so that by the first sp.seq. which is cohomological we have the (2) Let F : A−→B be a covariant functor between abelian cate- exact sequence of the low degree terms: gories; we (partially) define two functors C(A)−→SpS(B) in 0 → R1(R0G◦F ) → R1G◦F → R0(R1G◦F ) → R2(R0G◦F ) → R2G◦F the following way: starting with a complex C· we take an injective CE resolution L··, if there exists the simple complex as- but nothing more. sociated to F (L··) ∈ C2(B) we use the two spectral sequences (8) Let X· ∈ C(A) and I·· ∈ C2(A) be a resolution of X· with the associated to the bicomplex: they are the spectral sequences second degree positive, not necessarily CE, with F -acyclic ob- of hypercohomology of F w.r.t. the complex C·. The limit of jects of A for a functor F : A → B. We define a bicomplex · nXp se q = 0 the spectral sequences is the hypercohomology of F w.r.t. C X·· ∈ C2(A) by Xp,q = . The first spectral se- indicated as RnF (C·) = Hn F (L··) and the terms in the first 0 se q 6= 0 level are: quences associated to X·· and I·· give I p,q ·· p q · II p,q ·· p q · p,q q nH0 (Ip,·) ' Xp if q = 0 p,q E2 (FL ) = H R F (C ) E2 (FL ) = R F H (C ) I E (I··) = H (Ip,·) = II ' I E (X··) 1 II 0 if q 6= 0 1 where RqF (C·) = RqF (Ci) . i∈Z wence the canonical X·· → I·· gives isomorphisms on the limit: (3) the first spectral sequence does not require a CE resolution, n · n ·· n ·· n ·· · H (X ) = Htot(X ) ' Htot(I ) = H s(I ) so that X and I p,q ·· q p and it starts with E1 (FL ) = R F (C ). s(I··) are quasi-isomorphic. Moreover RF (X·) ' RF s(I··) ' ·· ·· n · n ·· Standard results: F s(I ) ' s F (I ) and also H RF (X ) ' Htot F (I ) . · (1) Given A ∈ A and an arbitrary resolution C· (i.e. a positive For the first sp.seq. associated to CE F (X ) we have (terms degree complex s.t. 0 → A → C· is exact) then RpF (C·) ' at level one) RqF (Xp) =⇒ Rp+qF (X·), and for the first sp.seq. p p q · p+q ·· I p,q ·· R F (A) and H R F (C ) =⇒ R F (A). In fact the sec- associated to the bicomplex F (I ) we have E1 F (I ) = ond sp.seq. degenerates: II Ep,q(C·) = RpF Hq(C·) = 0 if q p,· q p 2 HII F (I ) ' R F (X ) and so we have a canonical mor- II p,0 · p 0 · p p+q · p+q ·· p+q · q > 0 and E2 (C ) = R F H (C ) ' R F (A). phism at the limit: R F (X ) ' Htot F (I ) ' H RF (X )

6 GeoSupRC - Spectral Sequences c 2000 by MC (we are used RF (X·) in the sense of derived categories, and the In case that M is R-flat we deduce that symbol ' means quasi-isomorphic, i.e. isomorphism in the de- S ∼ S M ⊗R Torq (T,N)=Torp (M ⊗R T,N); rived category). and if N is S-flat we have Tricomplexes and Hypercohomology of bicomplexes. R ∼ R Torp (M,T ⊗S N)=Torp (M,T )⊗S N. Let A be an abelian category and X··· ∈ C3(A) (category of tricomplexes of A). Under the usual hypothesis we have three In particular if we have a morphism R → S, we can choose ways to reduce X··· to a bicomplex, associating the indices T = S and we have the base change spectral sequence i,j,k as i+j = m, k = n or i+k = m, j = n or i = m, j +k = n. S R R Torp (Torq (M,S),N) =⇒ Torp+q(M,N) . Accordingly, we have six sp.seq. converging to the same limit Exchanging the roles of R and S we can start with N ∈ mod-S, (the total cohomology): T ∈ S-mod-R, M ∈ R-mod; from the associativity property for m n ···  HI,II HIII (X ) N ⊗S T ⊗R M we obtain two spectral sequences m n ···  HIII HI,II (X )   TorS(N,TorR(T,M)) Hm Hn (X···)  p q · · I,III II m+n ··· R S =⇒ Hp+q(N ⊗R T ⊗S M ) m n ··· =⇒ Htot (X ) . Torp (Torq (N,T ),M) HII HI,III (X ) m n ···  HI HII,III (X ) In case that N is S-flat we deduce that m n ···  HII,III HI (X ) R ∼ R N ⊗S Torq (T,M)=Torp (N ⊗S T,M); Let F : A → B an additive functor between abelian categories, and if M is R-flat we have A having sufficiently many injectives. Then we define the hy- S ∼ S perderived functors of F on a bicomplex X·· ∈ C2(A) as the Torp (N,T ⊗R M)=Torp (N,T )⊗R M. total cohomology of the tricomplex F (I···) where I··· is an in- ·· In particular if we have a morphism R → S, we can choose jective CE-resolution of X . We have the following spectral T = S and we have the base change spectral sequence sequences with limit the hypercohomology of F on X··: TorS(N,TorR(S,M)) =⇒ TorR (N,M) . Hm(RnF (X··))  p q p+q m n ··  R F (Htot(X )) Kunneth-dual spectral sequences. Let L ∈ C(mod-A) and m n ··  R F (HII (X ))  m+n ·· M ∈ C(A-mod); then we have two spectral sequences m n ·· =⇒ R F (X ) . H (R F (X )) p q00 ) II I LExt (H 0 (L),H (M)) m n ··  A q H (R F (X )) 0 00 A I II  q +q =q =⇒ Extp+q(L,M) m n ··  p q RII F (HI (X )) Htot(ExtA(L,M)) Suppose now one has G : B → C another functor, with B having and in case L is projective or M is injective we have enough G-acyclic objects. We start with a complex X· ∈ C(A), ·· M p q00 p we take an injective CE-resolution I and calculate the hyper- Ext (Hq0 (L),H (M)) =⇒ H (Hom(L⊗A M)) . ·· 2 A tot cohomology of G on the bicomplex F (I ) ∈ C (B). We have q0+q00=q the following sp.seq. For M ∈ R-mod, T ∈ S-mod-R, N ∈ S-mod, from the adjoint m n · R (R G◦F )(X )  property Hom (M,Hom (T,N))∼Hom (T ⊗ M,N) we ob- m n · R S = S R R G(R F (X ))  tain two spectral sequences m n ·  R G(R F (X ))  m+n ·· p q =⇒ R G(FI ) . Ext (M,Ext (T,N)) p+q · ∼ Rm(RnG◦F )(X·) R S H (HomR(M·,HomS(T,N )))= p R =⇒ ∼ p+q · Hm(Rn G(FI··))  Ext (Tor (T,M),N) =H (HomS(M· ⊗R T,N )) . I II  S q Rm F (Hn(I··)) II I In case that M is R-projective we deduce that Suppose that F sends injective objects of A to G-acyclic objects q ∼ q of B; then we obtain the following situation: HomR(M,ExtS(T,N)=ExtS(T ⊗R M,N); RmG(RnF (X·))  and if N is S-injective we have RmG(RnF (X·))  p ∼ R =⇒ Rm(R0G◦F )(X·) . Ext (M,HomS(T,N)=HomS(Torp (T,M),N) . Rm(G◦F )Hn(X·) R Hm(Rn(G◦F )(X·)) In particular if we have a morphism R → S, we can choose T = S and we have the Hom-base change spectral sequence Extp (TorR(S,M),N) =⇒ Extp+q(M,N) . Algebraic spectral sequences. S q R Exchanging the roles of R and S we can start with N ∈ S-mod, Kunneth spectral sequences. Let L ∈ C(mod-A) and M ∈ T ∈ R-mod-S, M ∈ R-mod; from the usual adjoint property ∼ C(A-mod); then we have two spectral sequences HomS(N,HomR(T,M))=HomR(T ⊗S N,M) we obtain two spec- 0 00 tral sequences LTorA(Hq (L),Hq (M))) p p q q0+q00=q =⇒ TorA (L,M) Ext (N,Ext (T,M)) Hp+q(Hom (N ,Hom (T,M·)))=∼ p+q S R =⇒ S · R p A p S =∼Hp+q(Hom (T ⊗ N ,M·)) . Htot(Torq (L,M)) ExtR(Torq (T,N),M) R S · and in case L or M is flat we have In case that N is S-projective we deduce that M 0 00 p A q q q p Torp (H (L),H (M)) =⇒ Htot(L⊗A M) . ∼ HomS(N,ExtR(T,M)=ExtR(T ⊗S N,M); 0 00 q +q =q and if M is R-projective we have For M ∈ mod-R, T ∈ R-mod-S, N ∈ S-mod, from the associa- p ∼ S Ext (N,HomR(T,M))=HomR(Torp (T,N),M) . tivity property for M ⊗R T ⊗S N we obtain two spectral se- S quences In particular if we have a morphism R → S, we can choose R S T = S and we have the Hom-base change spectral sequence Torp (M,Torq (T,N)) · · S R =⇒ Hp+q(M ⊗R T ⊗S N ) p q p+q Torp (Torq (M,T ),N) ExtS(N,ExtR(S,M)) =⇒ ExtR (N,M) .

7 GeoSupRC - Spectral Sequences c 2000 by MC Duality spectral sequences. In the situation M ∈ R-mod, Spectral sequences in sheaf theory. T ∈ R-mod-S, N ∈ mod-S, from the canonical morphism Cohomology and Hypercohomology of (global sections HomS(T,N)⊗R M −−→HomS(HomR(M,T ),N) of) Sheaves: we obtain canonical morphisms (1) Hn(X,F ) is the n-th cohomology group of the sheaves F : TorR(Hom (T,N),M)−−→Hom (Ext (M,T ),N) . S S R Hn(X,−) = RnΓ(X,−): Ab(X)−→Ab Let be R left noetherian, M of finite type over R; then if N · is injective, the morphisms are iso. In particular we have the (2) the hypercohomology groups of the global sections on K : following spectral sequences Hn(X,K·) = RnΓ(X,−)(K·) TorR (Extq (T,N),M) p+q · ∼ −q S H (HomS(T,N )⊗R M·)= is a complex for which the two sequences p −q =⇒ ∼ p+q · =H (HomS(HomR(M·,T ),N )) . ExtS(ExtR (M,T ),N) I p,q p q · II p,q p q · E2 = H R Γ(X,−)(K ) E2 = R Γ(X,−) H (K ) Other spectral sequences. In the case M ∈ R-mod, T ∈ = Hp Hq(X,K·) = Hp X,Hq(K·) R-mod-S, N ∈ S-mod, we have a canonical morphism I p,q p q p HomR(M,T )⊗S N −−→HomR(M,T ⊗S N) converge (the first starts with: E1 = C (H (X,K ))). and, in case R is left noetherian and M of finite type over R, we can write the spectral sequences Cohomology and Hypercohomology of Coverings: TorS(Extq (M,T ),N) p+q ∼ (0) Given a topological space X, a presheaf F on X and an open p R Htot (HomR(M·,T )⊗S N·)= q+1 =⇒ covering of X U = (Ui)i∈I , or I = (I0,···,Iq) ∈ I we will Extp (M,TorS(T,N)) ∼ p+q R q =Htot HomR(M·,T ⊗S N·) q write U = T U and also ϕ ∈ F (U ). The complex of alter- Spectral sequences of group cohomology. I I` I I `=0 Let G be a group and M ∈ Z[G]-mod (resp. M ∈ mod-Z[G]). nating q-cochains with values in F is given by: The cohomology (resp. homology) groups of G with values in ( ) [G] M is Hn(G,M) = Extn ( ,M) (resp. H (G,M) = TorZ (M, )). Cq(U,F ) = ϕ ∈ QF (U )| ϕ = sgn(σ)ϕ ∀I,∀σ ∈ Σ(I) [G] Z n n Z I I σ(I) Z q+1 n n I∈I We have that H (G,M) = H (Hom [G](B·(Z[G],Z),M)) and Z q q q+1 Hn(G,M) = Hn(M ⊗ [G] B·(Z[G],Z)) where B·(Z[G],Z) is the with differentials d : C (U,F )−→C (U,F ) sending ϕ = Z q q+2 standard resolution of Z as (trivial) Z[G]-module, i.e. Bi(Z[G],Z) = (ϕI )I∈Iq+1 in d (ϕ) defined for J ∈ I by ⊗i+2 Z⊗ [G] Z[G] with projection : B0(Z[G],Z)−→Z. q+1 Z q P ` d (ϕ)J = (−) ϕ |U The cohomology of G with values in M is determined also (J0,···,Jˆ`,···,Jq+1) J as Hn(G,M) = Hn(C·(G,M)) where C·(G,M) is the complex `=0 n n given by C (G,M) = Hom(G ,M) with differentials If I is ordered we can write Cq(U,F ) = QF (U ). (I0,...,Iq ) n I <···

8 GeoSupRC - Spectral Sequences c 2000 by MC · · (3) Cartan Leray sp.seq.: take K = C (X,F ), for example the canon- Modules; for the complex HomOX (L·,G) of OX Modules we ical flabby resolution; then we have H·(U,K·) ' H·(X,F ) and p p have Ext (F,G) = H HomO (L·,G) . The second sp.seq. moreover OX X of hypercohomology of Γ(X,−) gives p q p+q H (U,h (X,F )) =⇒ H (X,F ) p,q q II E = Hp X,Hq(Hom (L·,G)) = Hp X,Ext (F,G) ; 2 OX OX (Cp (U,hq(X,F )) at level one) where hq(X,F ) = Hq (−,F ) is n n the presheaf sending U in Hq (U,F | ). It is the Leray sp.seq. by biregularity we have Ext (F,G) ' H X,Hom (L·,G) U OX OX for the the composed functor as limit, and the first sp.seq. starts with p,q · 0 I E = Hp Hq(X,Hom (L ,G)) . C (U,−) H (−) 2 OX · Ab(X) −−−−−→ C+(Ab) −−−−−→ Ab . (3) With the same notations Extn (F,G) ' Hn Hom (L ,C·(X,G)) tot OX · · 0 · 0 OX If K is a resolution for F , we have h (K ) = F , so h (X,F ) = F where C·(X,−) is the canonical flabby resolution. This can be and we have the edge morphisms Hp(U,F )−→Hp(X,F ) which · seen using the bicomplex of OX Modules C X,HomOX (L·,G) ' are isomorphisms if the Cartan Leray sp.seq. degenerate; for · p q+1 HomOX L·,C (X,G) whose global sections give a bicomplex this is sufficient that H (UI ,F |UI ) = 0 for every p > 0, I ∈ I (for any q), i.e. by definition if the covering U is acyclic w.r.t. · · of modules C X,HomOX (L·,G) ' HomOX L·,C (X,G) and F . we can use the second sp.seq. starting with II p,q p q · (4) by using inductive limits on the coverings ordered by fineness, E = H H HomO L·,C (X,G) ˇ 2 II I X we can write the Cech spectral sequence: = Hp Extq F,C·(X,G) p,q II OX E = Hˇ p (U,hq(X,F )) =⇒ Hp+q(X,F ). p q 2 = H ΓX,Ext F,C·(X,G) II OX p q (5) Let K· be a complex of sheaves; then we have functorially two = H ΓX,C·X,Ext (F,G) II OX sp.seq. starting with = Hp X,Extq (F,G) . OX p,q M k q i E1 = C (U,h (X,K )) p,q p q · E1 = C U,h (X,K ) The other sp.seq. gives i+j=p p,q p q · I p,q p q · p,q p q · E2 = H U,h (X,K ) E = H H C X,Hom (L ,G) E = H (U,h (X,K )) 2 I II OX · 2 p q = H H X,HomOX (L·,G) . where hq(X,K·) is the complex of presheaves whose value in U is Hq(U,K·), hq(X,K·) by definition is the q-th presheaf of hy- percohmology of K·, i.e. U 7→ Hq(U,K·), and having limit the Cohomology and Hypercohomology of direct images: hypercohomology Hn(X,K·). The key point is: take a Cartan (0) Given a left exact functor T from an abelian category A, with Eilenberg resolution of L·· of K·, then make the tricomplex sufficiently many injectives, to Ab(X), the functor RnT on A C·(U,L··) = Ck(U,Li,j) of cochains. By the reduction to a is obtained as the sheaf associated to the presheaf sending U bicomplex by p = i+j and q = k, the first sp.seq. degenerates: to Rn (Γ(U,−)◦T )(A) where Γ(U,−)◦T is a functor A → Ab. n0 if q > 0 I Ep,q = Hq C·(U,Li,j) = f 1 Γ(X,Li,j) i q = 0 In particular, given a continuous map X −→ Y and a sheaf on q X, R f∗(F ) is the sheaf associated to the presheaf sending an I p,0 p ·· p · open set V of Y to Hn(f←V,F ). So flabby sheaves are acyclic so that E2 = Htot Γ(X,L ) = H (X,K ) and by regular- p · ·· p · for the direct image functors. ity Htot C (U,L ) ' H (X,K ). Taking the second sp.seq. II p,q p q · p+q · f we obtain also E2 = H U,h (X,K ) =⇒ H (X,K ). (1) For X −→ Y and F sheaf on X, the Leray sp.seq. for the By the reduction to a bicomplex by p = i+k and q = j, the first composite Γ(Y,−)◦f∗ = Γ(X,−) gives p,q sp.seq. starts with I E = Hq Ck(U,Li,·) = Ck U,hq(X,Ki) p q p+q 1 H (Y,R f∗(F )) =⇒ H (X,F ) so that I Ep,q = Hp C·U,hq(X,K·) = Hp U,hq(X,K·). 2 tot (Leray sp.seq. for a continuous map). (6) If U is a covering acyclic for the sheaves of K· then we have n · n · f g a canonical isomorphism H (U,K ) ' H (X,K ). In fact in (2) For X −→ Y −→ Z, the Leray sp.seq. for g∗ ◦f∗ = (g ◦f)∗ gives · nK if q = 0 p q p+q this case hq(X,K·) = on the open sets involved (R g∗ ◦R f∗)(F ) =⇒ R (g ◦f)∗(F ). 0 i q 6= 0 in U, and the degeneration of the previous sp.seq. The edge morphisms are given by p p R g∗(f∗F ) −→ R (g ◦f)∗(F ) Cohomology and Hypercohomology of Hom functors: p p R (g ◦f)∗(F ) −→ g∗ (R f∗(F )) . (0) Let I be an injective sheaf; then HomX (F,I) is a flabby sheaf for any F , so acyclic for the global sections Γ(X,−). (3) The hypercohomology of the functor f∗ w.r.t. a complex of · n · n · sheaves K on X is R f∗(K ) = H f∗ CE(K ) , and the (1) Let F and G be sheaves on X; then we have a regular sp.seq. tot sp.seq. converging to it are: with I p,q p q · II p,q p q · p,q p q n E2 = H R f∗(K ) E2 = R f∗ H (K ) . E2 = H X,ExtX (F,G) =⇒ ExtX (F,G). (4) Given an open covering U of X we can use the bicomplex of It is the Leray sp.seq. for the composition Γ(X,HomX (F,G)) = · · Hom (F,G), i.e. Γ(X,−)◦Hom (F,−) = Hom (F,−). More- sheaves C (U,K ) giving two regular spectral sequences with X X X n · · over we have the edge morphisms limit Htot C (U,K ) . They start with p p p,q H (X,HomX (F,G)) −→ Ext (F,G) I p,q q · p II E = HqCp(U,K·) X E1 = H C (U,K ) 1 p 0 p q p p q · ExtX (F,G) −→ H X,ExtX (F,G) = H (U,K ) = C U,h (K ) I p,q p q · II p,q p · q · (2) Another computation of this sp.seq. in the case of a ringed E2 = H H (U,K ) E2 = H C U,h (K ) p q · space can be done by a left resolution L· of F with locally free = H U,h (K )

9 GeoSupRC - Spectral Sequences c 2000 by MC where hq(−) is the presheaf sending U to Hq(Γ(U,K·)). If the covering is acyclic for the sheaves in K· we can write I p,q p q · II p,q p q · E2 = H H (X,K ) E2 = H X,H (K ) where Hq(X,−) is the sheaf associated to the presheaf hq(X,−). Comparing the ﬁrst two sp.seq. we see that n · · n · Htot f∗ C (U,K ) ' R f∗(K )

II p,q p q · and moreover we can write the terms E1 as C U,pf h (K ) . (5) Let K· be a complex of sheaves; there exist functorially two sp.seq. starting at level two with Hp(U,hq(f,K·)), where hq(f,K·) is the complex of presheaves whose value in U is Rqf K· ) (sheaves on Y ), and Hp U,hq(f,K·) where hq(f,K·) |U∗ |U by definition is the presheaf U 7→ Rqf K· ), and having |U∗ |U q · limit R f∗(K ). The key point is: take a Cartan Eilenberg resolution of L·· of K·, then make the tricomplex C·(U,L··) = Ck(U,Li,j) of sheaves of cochains. By the reduction to a bicomplex with p = i+j and q = k, we see the limits and the second sp.seq.; by the reduction to a bicomplex by p = i+k and q = j, we see the first sp.seq. References. [CE] Cartan and Eilemberg, Homological Algebra. [T] Grothendieck, Sur quelques points d’algèbre homologique. [EgaIII] Grothendieck, Eléments de GéométrieAlgébrique,III. [G] Godement, Topologie Algébriqueet Théoriedes faisceaux. [B] Bourbaki, Algèbre,X: AlgèbreHomologique.

Copyright c 2000 M. Cailotto, July 2000 v0.0 Dip. di Matematica Pura ed Applicata, Univ. Padova (Italy) Thanks to TEX, a trademark of the American Mathematical Society, and DEK. Permission is granted to make and distribute copies of this card provided the copyright notice and this permission notice are preserved on all copies.

10 GeoSupRC - Spectral Sequences c 2000 by MC