Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do

The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, of finite type over a field) then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characterize semi-simple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $S$-motives (that was introduced previously and independently by D. Hebert and the author; weight structures and their transversality with t-structures were also defined by the author in recent papers). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow-weight spectral sequences for 'perverse 'etale homology' (that we prove unconditionally); this statement also yields the existence of the Chow-weight filtration for such (co)homology that is strictly restricted by ('motivic') morphisms.Comment: a few minor corrections mad

Relative continuous K-theory and cyclic homology

We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to the cyclic chain pro-complex "lim"CC(A_i) (subject to minor conditions on A). This identification is a continuous version of the classical Goodwillie isomorphism. The work comes from an attempt to understand the article of Bloch, Esnault, and Kerz "p-adic deformations of algebraic cycle classes".Comment: 26 pages. The section about pro-spectra is corrected. To appear in the Munster Journal of Mathematic