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Comparison of spectral sequences involving bifunctors. (English) Zbl 1180.18007

Suppose given two abelian categories \({\mathcal A}\) and \({\mathcal A}'\) with objects \(X\) and \(X'\) in \({\mathcal A}\) and \({\mathcal A}'\), respectively, and furthermore let \({\mathcal B}\) and \({\mathcal C}\) be two other abelian categories. Given two additive functors \(F:{\mathcal A}\times{\mathcal A}'\rightarrow{\mathcal B}\) and \(G:{\mathcal B}\rightarrow{\mathcal C}\) such that \(F(X,-), F(-,X')\) and \(G\) are left exact there exist a Grothendieck spectral sequence for the composition of functors \({G\circ F(X,-)}\) whose \(E_2\)-term evaluated at \(X^\prime\) is isomorphic to \((R^iG)(R^jF)(X,X')\) and which converges to \({R^{i+j}(G\circ F)(X,X')}\). Similarly for the composition of functors \(G\circ F(-,X')\) we get another Grothendieck spectral sequence with the same \(E_2\)-term and converging to the same target.
The main theorem of this well-written paper states that under some additional assumptions on the objects \(X, X'\) the above spectral sequences are isomorphic. This is proved by directly comparing the underlying double complexes of the spectral sequences via a chain of homotopy equivalences.
The author finally discusses several interesting applications. In particular, he gives a new proof of a result by F. R. Beyl [Bull. Sci. Math., II. Ser. 105, 417–434 (1981; Zbl 0465.18009)] on the isomorphism between the Grothendieck spectral sequence and the Lyndon-Hochschild-Serre spectral sequence. Finally he derives comparison results for change-of-rings spectral sequences and Ext-of-sheaves spectral sequences.

MSC:

18G40 Spectral sequences, hypercohomology

Citations:

Zbl 0465.18009
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