1. Introduction

We construct a series of spectral sequences for the hypercohomology of a filtered complex. The basic constructions used are the shift (dècalée) operation of Deligne and its inverse [3].

Theorem 1.1. Let D : CC' be a left exact functor between abelian categories. Assume that C has enough injectives. For any good filtered complex (K,F) of objects in C we have natural spectral sequences for each r > 1.

For r = 1 this coincides with the spectral sequence for the hypercohomology of a filtered complex ([3]; section 1.4.5). For r = 2 and the trivial filtration F this coincides with the Leray spectral sequence for hypercohomology (see [3]; section 1.4.7).

Here ⁱ’s denote the hyperderived functors associated with D and E_r-1,p,q denotes the complex of E_r-1 terms of the spectral sequence (see [3] section 1.3.1) for a filtered complex which contains E_r^p,q as the (p + q)-th term.

As applications, we provide proofs of some facts which are apparently well known to experts but are not well-documented in the literature (see however [7],[4]; the proof of the latter two applications is attributed to P. Deligne in [4] and [1]). The first is the existence of a Leray spectral sequence for de Rham cohomology. The second is the comparison of this spectral sequence with the Leray spectral sequence for singular cohomology. The third is the isomorphism of the Bloch-Ogus-Gersten spectral sequence with the Leray spectral sequence from E₂ onwards.

In Section 2 we recall some facts and definitions due to Deligne ([3] sections 1.3 and 1.4). In Section 3 we generalise some of these ideas to get the main results. The applications are elementary corollaries of the lemmas from section 3 and are proved in Section 4. In appendix A we recapitulate the required material from [3] and in appendix B we redo the main result using the language of derived categories.

I am grateful to V. Srinivas for advice and encouragement. I followed up his suggestion that the problem of the Bloch-Ogus spectral sequence and that of the Leray spectral sequence for de Rham cohomology could be related. J. Biswas and Balaji Srinivasan have had to listen to half-baked versions of the results and also had to correct my wrong impressions about the existence of connections for proper submersions.