On perverse homotopy t-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

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We study the category DM(S) of Beilinson motives (as described by Cisinski and Deglise) over a more or less general base scheme S, and establish several nice properties for a version thom(S) of the perverse homotopy t-structure (essentially defined by Ayoub) for it. thom(S) is characterized in terms of certain stalks of an S-motif H and its Tate twists at fields over S; it is closely related to certain coniveau spectral sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite type S-schemes. We conjecture that the heart of thom(S) is given by cycle modules over S (as defined by Rost); for varieties over characteristic 0 fields this conjecture was recently proved by Deglise. Our definition of thom(S) is closely related to a new effectivity filtration for DM(S) (and for the subcategory of Chow S-motives in it). We also sketch the construction of a certain Gersten weight structure for the category of S-comotives; this weight structure yields one more description of thom(S) and its heart.