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12.20 Spectral sequences

A nice discussion of spectral sequences may be found in [Eisenbud]. See also [McCleary], [Lang], etc.

Definition 12.20.1. Let \mathcal{A} be an abelian category.

  1. A spectral sequence in \mathcal{A} is given by a system (E_ r, d_ r)_{r \geq 1} where each E_ r is an object of \mathcal{A}, each d_ r : E_ r \to E_ r is a morphism such that d_ r \circ d_ r = 0 and E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r) for r \geq 1.

  2. A morphism of spectral sequences f : (E_ r, d_ r)_{r \geq 1} \to (E'_ r, d'_ r)_{r \geq 1} is given by a family of morphisms f_ r : E_ r \to E'_ r such that f_ r \circ d_ r = d'_ r \circ f_ r and such that f_{r + 1} is the morphism induced by f_ r via the identifications E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r) and E'_{r + 1} = \mathop{\mathrm{Ker}}(d'_ r)/\mathop{\mathrm{Im}}(d'_ r).

We will sometimes loosen this definition somewhat and allow E_{r + 1} to be an object with a given isomorphism E_{r + 1} \to \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r). In addition we sometimes have a system (E_ r, d_ r)_{r \geq r_0} for some r_0 \in \mathbf{Z} satisfying the properties of the definition above for indices \geq r_0. We will also call this a spectral sequence since by a simple renumbering it falls under the definition anyway. In fact, the cases r_0 = 0 and r_0 = -1 can be found in the literature.

Given a spectral sequence (E_ r, d_ r)_{r \geq 1} we define

0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1

by the following simple procedure. Set B_2 = \mathop{\mathrm{Im}}(d_1) and Z_2 = \mathop{\mathrm{Ker}}(d_1). Then it is clear that d_2 : Z_2/B_2 \to Z_2/B_2. Hence we can define B_3 as the unique subobject of E_1 containing B_2 such that B_3/B_2 is the image of d_2. Similarly we can define Z_3 as the unique subobject of E_1 containing B_2 such that Z_3/B_2 is the kernel of d_2. And so on and so forth. In particular we have

E_ r = Z_ r/B_ r

for all r \geq 1. In case the spectral sequence starts at r = r_0 then we can similarly construct B_ i, Z_ i as subobjects in E_{r_0}. In fact, in the literature one sometimes finds the notation

0 = B_ r(E_ r) \subset B_{r + 1}(E_ r) \subset B_{r + 2}(E_ r) \subset \ldots \subset Z_{r + 2}(E_ r) \subset Z_{r + 1}(E_ r) \subset Z_ r(E_ r) = E_ r

to denote the filtration described above but starting with E_ r.

Definition 12.20.2. Let \mathcal{A} be an abelian category. Let (E_ r, d_ r)_{r \geq 1} be a spectral sequence.

  1. If the subobjects Z_{\infty } = \bigcap Z_ r and B_{\infty } = \bigcup B_ r of E_1 exist then we define the limit1 of the spectral sequence to be the object E_{\infty } = Z_{\infty }/B_{\infty }.

  2. We say that the spectral sequence degenerates at E_ r if the differentials d_ r, d_{r + 1}, \ldots are all zero.

Note that if the spectral sequence degenerates at E_ r, then we have E_ r = E_{r + 1} = \ldots = E_{\infty } (and the limit exists of course). Also, almost any abelian category we will encounter has countable sums and intersections.

Remark 12.20.3 (Variant). It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accommodate this (and to get around certain technical issues) we introduce the following notion. Let \mathcal{A} be an abelian category. Let (T_ r)_{r \geq 1} be a sequence of translation or shift functors, i.e., T_ r : \mathcal{A} \to \mathcal{A} is an isomorphism of categories. In this setting a spectral sequence is given by a system (E_ r, d_ r)_{r \geq 1} where each E_ r is an object of \mathcal{A}, each d_ r : E_ r \to T_ rE_ r is a morphism such that T_ rd_ r \circ d_ r = 0 so that

\xymatrix{ \ldots \ar[r] & T_ r^{-1}E_ r \ar[r]^-{T_ r^{-1}d_ r} & E_ r \ar[r]^-{d_ r} & T_ rE_ r \ar[r]^{T_ r d_ r} & T_ r^2E_ r \ar[r] & \ldots }

is a complex and E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(T_ r^{-1}d_ r) for r \geq 1. It is clear what a morphism of spectral sequences means in this setting. In this setting we can still define

0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1

and Z_\infty and B_\infty (if they exist) as above.

[1] This notation is not universally accepted. In some references an additional pair of subobjects Z_\infty and B_\infty of E_1 such that 0 = B_1 \subset B_2 \subset \ldots \subset B_\infty \subset Z_\infty \subset \ldots \subset Z_2 \subset Z_1 = E_1 is part of the data comprising a spectral sequence!

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