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I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would l …

Q1. So, I suggest you the following plan of the proof. Note that any Verdier localization of a triangulated category possessing a differential graded enhancement possesses a differential graded enha …

I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux perv …

Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of she …

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (p …

I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, bu …

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases? Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to define the p …

Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties …

In her paper "Mixed perverse sheaves for schemes over number fields" A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $S$ using …

It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially compli …

Let $j:U\to S$ be an (open) immesrion; let $P_U$ be a perverse (\'etale, though my question makes sense in the topological setting also) sheaf on $U$. Then I would like to say that the intermediate e …

I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter? I suspect that …

Which statement is usually called the Decomposition Theorem (for perverse sheaves)? Is this (roughly): a proper pushforward of an intersection complex could be decomposed into a direct sum of (shifted …

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. …