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Spectral Sequences Means of the Isomorphism of the Previous Step K to the Sub- C 2000 M 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 p2Z k+1 2 k 2 2 2 p 2 Z; morphisms ' : A−!B are morphisms of A respecting the filtrations, i.e. '(F p(A)) ⊆ F p(B). and the definition continues with: p;q p;q p (d) for every p;q 2 Z two subobjects B1(E2 ) and Z1(E2 ) of The filtration (F (A)) of A is: p;q p;q p;q p2Z E2 such that Z1(E2 ) ⊇ B1(E2 ) and for every k T p (i) separated if p2 F (A) = 0, p;q p;q p;q p;q Z Zk(E2 ) ⊇ Z1(E2 ) ⊇ B1(E2 ) ⊇ Bk(E2 ) (ii) coseparated (or exhaustive) if S F p(A) = A, p2 p;q p;q p;q Z and define: E = Z1(E ) B1(E ); p q 1 2 2 (iii) discrete if there exists p 2 Z s.t. F (A) = 0 (so F (A) = 0 n (e) a family (E )n2 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 2 Z s.t. F (A) = A (so F (A) = p;q2Z 1 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 2 Af : grp(A) := F (A) F (A) 2 A, and if A admits (2) for a fixed n 2 Z, the family fE1 gp+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) 2 A, the p2Z L p;q∼ n exist, we have E1 =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 ' : 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 gr' 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 ' is a monomorphism; r r r (b) compatibility with the differentials: commutation of (2) if gr' is an epimorphism, and the filtration of B is exhaus- p;q tive and discrete, then ' is an epimorphism; p;q fr 0p;q Er −−−−−−−−−−−! E r (20) if gr' 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 ' is an epimorphism; p+r;q−r+1 fr (3) if gr' 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 ' 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 > n2Z r+1 p;q r+1 p;q2Z 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 2 Z, p;q p;q p;q p+r;q−r+1 (d) compatibility with the 1 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 1 2 1 2 2 1 2 1 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 1 1 1 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 −−−−−−−−−−−!1 E0p;q (c) a family (αr ) of isomorphisms 1 1 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 2 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 E1 = Er p;q > only if: the f2 are isomorphisms for all p;q 2 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 2 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) 1 2 k 2 1 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 = E1 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 Z1(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.
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