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.
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.
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
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.
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 }.
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.
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