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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, but probably the difference from the 'topological case' is not very large here) i.e. I want to use the corresponding left and right $t$-exactnesses of $f_\ast$, $f^\ast$, $f_!$ and $f^!$ when $f$ is smooth or affine; constant sheaf over a local complete intersection variety (shifted by the dimension) is perverse; etc. It seems that there are two possibilities.

  1. Reduce the situation to $\mathbb{Z}/l\mathbb{Z}$-sheaves (and consider the corresponding derived categories). Will the 'usual' properties of the perverse $t$-structure (for $\mathbb{Q}$ or $\mathbb{Q}_l$-sheaves) hold in this setting?

  2. Consider arbitrary torsion sheaves and use only those properties of the 'obvious perverse' $t$-structure that are still true in this setting (cf. sections 3.3 and 4.1 of BBD).

My questions are:

  1. Are the properties of $\mathbb{Z}/l\mathbb{Z}$-sheaves really parallel to those of $\mathbb{Q}_l$-ones?

  2. What are the most appropriate references for possibilities 1 and 2?

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1 Answer 1

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If you want only $\mathbb{Z}/\ell\mathbb{Z}$ coefficients (not general $\mathbb{Z}/\ell^m\mathbb{Z}$), then there is only one middle perverse t-structure, which is good. The way the exactness properties of the 4 operations is proved in BBD is to reduce to $\mathbb{Z}/\ell/\mathbb{Z}$ coefficients (see 4.0), so the answer to 1 is "obviously yes" and the answer to 2 is "BBD 4.1".

As for the fact that the shifted constant sheaf on a lci variety is perverse, it only uses the results of BBD 4, and again, these are all true for $\mathbb{Z}/\ell\mathbb{Z}$ coefficients as explained in BBD 4.0. (The problems start appearing when the coeffients are not a field.) So in that case too the answer to 1 is "yes" and the answer to 2 is "BBD 4".

NB : Here "BBD" refers to the book "Analyse et topologie sur les espaces singuliers I", by Beilinson-Bernstein-Deligne (aka Asterisque 100).

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  • $\begingroup$ Do you think that the situation changes when one replaces $Z/lZ$-derived categories with (say) $Z/l^2Z$-ones? It seems that the 'canonical' $t$-structure for $Z/l^2Z$-modules is self-dual, so that there should exist a self-dual $t$ also. Something like this is said in \S4.0 of BBD; yet I am not sure that I understood this French text correctly. $\endgroup$ May 7, 2012 at 20:05
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    $\begingroup$ Yes, when I wrote my answer I had not realized that $\mathbb{Z}/\ell^m\mathbb{Z}$ coefficients work just like $\mathbb{Z}/\ell\mathbb{Z}$ for many things. There is a self-dual t-structure, BBD 4.1 is still fine I think, though you have to be careful if you want to take tensor products. $\endgroup$
    – Alex
    May 8, 2012 at 22:31

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