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Spectral Sequences Spectral Sequences Daniel Murfet October 5, 2006 In this note we give a minimal presentation of spectral sequences following EGA. We cover essentially only that part of the theory needed in algebraic geometry. In Section 2 we start with a filtration of a complex, and show how the various pieces of a spectral sequence arise. Applying these observations in Section 3 to two natural filtrations of the total complex of a bicomplex, we deduce two spectral sequences and discover that their first three pages are not mysterious at all: they consist of very natural invariants of the bicomplex, arranged in the obvious way. With this background we can study the Grothendieck spectral sequence in Section 4. Contents 1 Definitions 1 2 The Spectral Sequence of a Filtration 3 2.1 First Quadrant Filtration ................................ 7 3 The Spectral Sequences of a Double Complex 8 4 The Grothendieck Spectral Sequence 12 4.1 Examples ......................................... 14 4.1.1 The Leray Spectral Sequence .......................... 14 4.1.2 The Local-to-Global Ext Spectral Sequence .................. 14 5 Hyperderived Functors 15 1 Definitions As usual we assume that our abelian categories all come with canonical structures, allowing us to define cokernels, kernels, images and direct sums in a canonical way. If we have two subobjects X, Y of an object A then X ⊆ Y means that X precedes Y as a subobject (i.e. the morphism X −→ A factors through Y −→ A). We use the notation and conventions of our notes on Derived Functors (DF) and Abelian Categories (AC). In particular we tend to denote the differential of any complex X by ∂n : Xn −→ Xn+1. The definitions in this section follow EGA III Ch.0 §11.1. Definition 1. Let A be an abelian category and X an object of A.A filtration (or decreasing filtration) or X is a sequence of subobjects of X · · · ⊇ F 0(X) ⊇ F 1(X) ⊇ · · · ⊇ F p(X) ⊇ · · · p p p If they exist, we write inf(F (X)) for the intersection ∩pF (X) and sup(F (X)) for the union p p ∪pF (X). We say that the filtration is separated if inf(F (X)) = 0 and coseparated or exhaustive if sup(F p(X)) = X. We say that the filtration is discrete if there exists p ∈ Z with F p(X) = 0, and codiscrete if there exists p ∈ Z with F p(X) = X. Definition 2. Let A be an abelian category and a ≥ 0 an integer. A spectral sequence in A starting on page a consists of the following elements: 1 pq (a) An object Er of A for every p, q ∈ Z and r ≥ a. pq pq p+r,q−r+1 p+r,q−r+1 pq (b) A morphism dr : Er −→ Er for p, q ∈ Z and r ≥ a such that dr dr = 0. pq pq pq p−r,q+r−1 If we set Zr+1(Er ) = Ker(dr ),Br+1(Er ) = Im(dr ) then pq pq pq Br+1(Er ) ⊆ Zr+1(Er ) ⊆ Er pq pq pq pq (c) An isomorphism αr : Zr+1(Er )/Br+1(Er ) −→ Er+1 for p, q ∈ Z and r ≥ a. Before continuing, let us introduce some notation. For k ≥ r + 1 one defines by recursion on pq pq pq k subobjects Bk(Er ) and Zk(Er ) of Er as the inverse image, by the canonical morphism pq pq pq pq Er −→ Er /Br+1(Er ), of the subobject of the quotient identified by αr with the subobjects pq pq pq pq Bk(Er+1) and Zk(Er+1) respectively. Clearly Bk(Er ) ⊆ Zk(Er ) and for k ≥ r + 1 we deduce a pullback pq pq Bk(Er ) / Bk(Er+1) pq pq Zk(Er ) / Zk(Er+1) with horizontal epimorphisms and vertical monomorphisms. From (AC,Lemma 35) we infer that pq pq pq pq the induced morphism on the cokernels Zk(Er )/Bk(Er ) −→ Zk(Er+1)/Bk(Er+1) is an isomor- phism. We can therefore recursively define a canonical isomorphism pq pq pq Zk(Er )/Bk(Er ) −→ Ek k ≥ r + 1, r ≥ a (1) pq pq pq If we set Br(Er ) = 0 and Zr(Er ) = Er then we have inclusions pq pq pq 0 = Br(Er ) ⊆ Br+1(Er ) ⊆ Br+2(Er ) ⊆ · · · pq pq pq pq (2) · · · ⊆ Zr+2(Er ) ⊆ Zr+1(Er ) ⊆ Zr(Er ) = Er We now return to enumerating the defining data of a spectral sequence: pq pq pq pq pq (d) Two subobjects B∞(Ea ) and Z∞(Ea ) of Ea such that B∞(Ea ) ⊆ Z∞(Ea ) and for k ≥ a pq pq pq pq pq pq pq we have Bk(Ea ) ⊆ B∞(Ea ) and Z∞(Ea ) ⊆ Zk(Ea ). We define E∞ = Z∞(Ea )/B∞(Ea ). n p n n (e) A family {E }n∈Z of objects of A, each with a filtration {F (E )}p∈Z. We define grp(E ) = F p(En)/F p+1(En) for p, n ∈ Z. pq pq p+q (f) For each pair (p, q) ∈ Z × Z an isomorphism β : E∞ −→ grp(E ). n The family of objects {E }n∈Z, without the associated filtrations, is called the limit of the spectral pq n sequence. We usually just write E or (Er ,E ) for the spectral sequence, with all other data pq p+q implicit, and write Er ⇒ E to represent the fact that the spectral sequence converges to the n family {E }n∈Z. pq n Definition 3. Let A be an abelian category. A spectral sequence E = (Er ,E ) is weakly pq pq pq pq convergent if B∞(Ea ) = supk(Bk(Ea )) and Z∞(Ea ) = infk(Zk(Ea )). We say that the spectral sequence is regular if it is weakly convergent and also pq (1) For each pair (p, q) the decreasing sequence {Zk(Ea )}k≥a stabilises: that is, we have pq pq pq Zk(Ea ) = Zk+1(Ea ) for all sufficiently large k. In this case it follows that Z∞(Ea ) = pq Zk(Ea ) for all sufficiently large k (the bound depending on p, q). p n (2) For each n ∈ Z the filtration {F (E )}p∈Z is discrete and exhaustive. We say that the spectral sequence E is coregular if it is weakly convergent and also pq (3) For each pair (p, q) the increasing sequence {Bk(Ea )}k≥a stablises. In this case it follows pq pq that B∞(Ea ) = Bk(Ea ) for all sufficiently large k (the bound depending on p, q). 2 p n (4) For each n ∈ Z the filtration {F (E )}p∈Z is codiscrete. We say that E is biregular if it is both regular and coregular, or equivalently if the following conditions are satisfied pq pq (a) For each pair (p, q) the sequences {Zk(Ea )}k≥a and {Bk(Ea )}k≥a both stabilise and there- pq pq pq pq fore Z∞(Ea ) = Zk(Ea ) and B∞(Ea ) = Bk(Ea ) for all sufficiently large k (therefore pq ∼ pq E∞ = Ek ). p n (b) For each n ∈ Z the filtration {F (E )}p∈Z is discrete and codiscrete, and therefore finite. Most spectral sequences we will encounter will be biregular. Remark 1. Let E be a spectral sequence, and suppose that for some r ≥ a and p, q ∈ Z we pq have Er = 0. It follows from (1) and (2) that the entry of every subsequent page of the spectral pq sequence is also zero: that is, Ek = 0 for k ≥ r. Further, if we fix some s ≥ a then for all pq pq pq sufficiently large k we have Zk(Es ) = Bk(Es ) = 0. In particular we must have E∞ = 0. Definition 4. Let A be an abelian category and E a biregular spectral sequence in A starting on pq page a. We say that E degenerates on page r ≥ a if for every p, q ∈ Z the morphism dr is zero. That is, all the morphisms on the rth page are zero. It follows that we have a chain of canonical isomorphisms pq ∼ pq ∼ pq ∼ Er = Er+1 = Er+2 = ··· pq and since E is biregular this eventually stabilises to E∞ . That is, for every k ≥ r we have an pq ∼ pq isomorphism Ek = E∞ . Finding a degenerate page of the spectral sequence is one of the most common ways to extract useful information. 2 The Spectral Sequence of a Filtration Throughout this section let A be an abelian category. Let C be a complex in A and suppose we p have a decreasing filtration {F (C)}p∈Z of C. That is, a sequence of subobjects · · · ⊇ F p−1(C) ⊇ F p(C) ⊇ F p+1(C) ⊇ · · · For p, q ∈ Z we place the object F pCp+q = F p(C)p+q at position (p, q) to form the following commutative diagram in A F 0C2 F 0C1 F 1C2 F 0C0 F 1C1 F 2C2 in which all diagonal morphisms are monomorphisms. In other words, we “spread out” the object Cn along the diagonal p + q = n. To make the construction of the spectral sequence more transparent, we will allow ourselves to informally talk about “elements” of Cn and its filtrations F pCn. The reader may be comforted, however, by our assurance that any occurrences of such informal notation will be accompanied by the equivalent formal statement. For each n ∈ Z we have a filtration · · · ⊇ F p−1Cn ⊇ F pCn ⊇ F p+1Cn ⊇ · · · of the object Cn, and we think of elements of Cn further down the filtration as being “closer to zero”. The 3 filtration of the complex C then allows us to talk about elements x ∈ Cn which are “close to being cocycles”.
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