The homotopy Leray spectral sequence

Abstract

In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its E 2 -page. Our description of the E 2 -page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work. Contents 2 All objects in T (S) are obtained by taking extensions of arbitrary coproducts. Equivalently, an object K of T (S) is zero if and only if: ∀X/S smooth,(n, i) ∈ Z 2 , Hom T (S) M S (X)(i)[n], K = 0. 3 Recall that this expresses the compatibility of T with projective limits. 4 This condition is automatically verified (see [BD17, 3.2.13]) if T satisfies absolute purity and the following vanishing statement holds: ∀fields E, ∀n > m, H n,m (Spec(E), T ) = 0. 5Under two geometric assumptions, this means that:• In case (S1), we will require that, for p the characteristic exponent of k, T is Z[1/p]linear.• In case (S2), we require that T is Q-linear.