The Stacks project

Lemma 15.90.21. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet $ be a filtered complex of $A$-modules. There exists a canonical spectral sequence $(E_ r, \text{d}_ r)_{r \geq 1}$ of bigraded derived complete $A$-modules with $d_ r$ of bidegree $(r, -r + 1)$ and with

\[ E_1^{p, q} = H^{p + q}((\text{gr}^ pK^\bullet )^\wedge ) \]

If the filtration on each $K^ n$ is finite, then the spectral sequence is bounded and converges to $H^*((K^\bullet )^\wedge )$.

Proof. By Lemma 15.90.10 we know that derived completion is given by $R\mathop{\mathrm{Hom}}\nolimits _ A(C, -)$ for some $C \in D^ b(A)$. By Lemmas 15.90.20 and 15.67.2 we see that $C$ has finite projective dimension. Thus we may choose a bounded complex of projective modules $P^\bullet $ representing $C$. Then

\[ M^\bullet = \mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , K^\bullet ) \]

is a complex of $A$-modules representing $(K^\bullet )^\wedge $. It comes with a filtration given by $F^ pM^\bullet = \mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , F^ pK^\bullet )$. We see that $F^ pM^\bullet $ represents $(F^ pK^\bullet )^\wedge $ and hence $\text{gr}^ pM^\bullet $ represents $(\text{gr}K^\bullet )^\wedge $. Thus we find our spectral sequence by taking the spectral sequence of the filtered complex $M^\bullet $, see Homology, Section 12.24. If the filtration on each $K^ n$ is finite, then the filtration on each $M^ n$ is finite because $P^\bullet $ is a bounded complex. Hence the final statement follows from Homology, Lemma 12.24.11. $\square$


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