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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 the definition should be something like:

a perverse sheaf $P$ on a space (scheme) $S$ is supported on (or is it better to say 'at'?) a subspace $U$ if for any closed immersion $i:Z\to S$ such that $Z$ is disjoint from $U$ the zeroth (perverse) cohomology of both $Ri^*P$ and $Ri^!P$ is $0$. Does this make sense?

Upd. Possibly the word 'support' is not quite appropriate for me (could you suggest something better?:)), whereas the notion of support should be defined as YBL does below. For the intermediate extension $j_{!*}$ of a perverse sheaf $P$ from an (open) subscheme $U$ of $S$ to $S$ I would like to say that $j_{!*}P$ is supported on $U$. So, I want to say that a constant sheaf on a (connected) smooth variety is supported on any its dense subvariety (or at its generic point).

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A quick comment: the derived category of perverse sheaves is equivalent to the derived category of constructible sheaves (a theorem of Beilinson, or part of the Riemann-Hilbert theorem if you're in characteristic zero), so I think if your definition makes sense on the derived category perhaps one wants it just to be the support of the complex of constructible sheaves? I'm not sure what you mean by a mixed motivic sheaf though, so this may not have been a helpful thing to mention... –  Kevin McGerty Feb 4 '11 at 2:27
    
I probably do not want my notion to be 'derived', since I am thinking of intermediate images (and those are not exact). I realize once more that the word 'support' is not quite appropriate here. –  Mikhail Bondarko Feb 4 '11 at 9:59
    
The derived category of perverse sheaves is not equivalent to the derived category of constructible sheaves, it is equivalent to the full subcategory of the derived category of (not necessarily constructible) sheaves whose objects are complexes with constructible cohomology sheaves. That's in the classical setting, in the l-adic setting the triangulated category in which perverse sheaves live is obtained in a more complicated way (in the simplest cases, as a 2-limit of derived categories in which you then invert torsion, and maybe extend coefficients). Just thought I'd point this out. –  Alex Feb 5 '11 at 3:21
    
(continued) What the derived category of the category of constructible sheaves looks like is anybody's guess. See remark b) after theorem 1.3 of Beilinson's article "On the derived category of perverse sheaves". –  Alex Feb 5 '11 at 3:23
    
A comment about the proposed definition of support : if $X$ is an irreducible variety, if $U$ is a dense open subset, and if $K$ is the intermediate extension of a local system on $U$, then, for any proper closed subvariety $i:Z\rightarrow X$, the complex $i^*K$ is concentrated in perverse degree $<0$ and the complex $i^!K$ is concentrated in perverse degree $>0$. This is a consequence of lemma 4.3.2 of BBD and of the fact that $i^*$ is right $t$-exact for the perverse $t$-structure. (And is related to the fact that $K$ does not determine $U$, as any smaller nonempty open will do.) –  Alex Feb 5 '11 at 3:30

1 Answer 1

Let $j_i : U_i \hookrightarrow X$ open immersions and $j: U_1 \cup U_2 \hookrightarrow X$. If $j_1^*F = j_2^*F = 0$ then $j^*F = 0$ by considering a Mayer-Vietoris triangle. So there is a largest open set $U\subset X$ such that $j^*F = 0$. Define $supp(F)$ as the complementary closed subset.

The definition is autodual $supp(F) = supp(DF)$ since $j^*F = 0 \Leftrightarrow D(j^*F) = 0 \Leftrightarrow j^!DF = j^*DF = 0$ for an open immersion $j$.

Also $supp(F) \subset supp(F') \cup supp(F'')$ for a triangle $F'\to F\to F'' \to +1$. So if $t$ is a t-structure on $D(X)$ and $F\in D(X)$ is bounded for $t$ then $supp(F) \subset\bigcup_i supp({}^tH^i(F))$.

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Do you think that the support should always be a closed subscheme? I would like to say that the 'minimal support' of a constant sheaf on (a smooth irreducible variety) $X$ is the generic point of $X$ (or that the sheaf is supported at any open subvariety of $X$). –  Mikhail Bondarko Feb 3 '11 at 23:24
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That's just identifying a closed subset to the generic points of its irreductible components. –  YBL Feb 4 '11 at 13:03

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