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I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.

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 …

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 …

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

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 …

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 $S$. …

A stupid question: which statements in section 5 of BBD will fail if we replace $\overline{\mathbb{Q}_l}$-sheaves by just $\mathbb{Q}_l$-ones? I am especially interested in Proposition 5.1.15. BBD = …

In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}_l}$-)sheaves over a variety $X_0$ defined over a finite field $F$. Weights start to behave better when one extend …

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 …

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 …

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) Theore …

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 …

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 …

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 …

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 …