# Bad behaviour of perverse sheaves over 'general' bases?

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 (i.e. left and right $t$-exactness of various image functors, mostly those coming from affine morphisms) be similar to those described in BBD? Are any of these properties difficult to prove/ wrong in general? What could cause more difficulties: a 'large' base scheme, or a mixed characteristic one?

Upd. Any evidence or references concerning this question would be very interesting. In particular, did anybody study perverse sheaves (or something similar) over $\mathbb{Z}_p$ (cf. the discussion in comments)?

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Great question. Other than base=k case, is there any normalization condition (or expectation) that you want this theory to satisfy? (Of course they 'should be' l-adic avatar of motivic sheaves... but any more concrete expectations?) Maybe the 1st step is to treat the case when the base is a complete DVR. I could be wrong, but Gabber's results seem to be more 'absolute', rather than having a general base, so shall we also expect a theory of perv sheaves in the base-free case? Finally, the question might be related to recent work of Cisinski and Frédéric, and if so, I hope experts can explain. –  shenghao Feb 18 '11 at 0:03
Sorry; what do you mean by a normalization condition? Currently, I just want the functoriality properties of the perverse t-structure with respect to (affine) morphisms of schemes. Certainly, there are also some more (complicated) statements that would be interesting to check (and that would follow if we new that we have a motivic t-structure for Cisinski-Deglise's motives that is compatible with the perverse t-structure for cohomology). An yes, a complete DVR seems to be a very interesting case; yet I was not able to find anything about perverse sheaves in this situation. –  Mikhail Bondarko Feb 18 '11 at 6:31
Perhaps, I should ask about perverse sheaves over complete DVR in a separate question. –  Mikhail Bondarko Feb 18 '11 at 6:37
By a 'normalization condition' I was thinking of $H^*(pt)=Z$ for a cohom theory; I meant some condition like "in particular, when ... (e.g. when the base is ...), it returns something already known. But since the whole theory is built over a field, I don't know what to expect in general. For instance, shall we define the perv t-stru using the relative dim of the support again, or is there a def of totally different nature? DVR is definitely an interesting case; at least when passing from $\mathbb C$ to $\mathbb F$ in BBD, DVR is only a bridge for $D^b_c,$ not perv sheaves. –  shenghao Feb 18 '11 at 22:54
Well, I don't think that it would be difficult to prove that $H^*_S(S)=\mathbb{Q}_{l,S}$. Besides, I suspect that there could be more than one 'nice' \etale cohomology theory for motives (one could try to define one either by considering an arbitrary 'fixed' base scheme, or by passing to the limit for finitely generated subschemes of definition). –  Mikhail Bondarko Feb 18 '11 at 23:21

The answer is very likely "yes", but you will need to put together some technical articles (and unpublished results) that may not have yet been put together. Here are the key ingredients, as I see it :

(1) General definition of perverse t-structure, not using stratifications (using stratifications is probably a bad idea in general), not using finiteness conditions, no need for a base field either : see Gabber's article "Note on some t-structures". Section 8 is of particular interest, since this is where he defines perverse t-structures on étale n-torsion sheaves on a noetherian scheme X, provided that X admits a dualizing complex (a priori in the sense of SGA 5 I). He also gives a condition on the perversity that guarantees that the perverse truncation functors will respect constructibility.

(2) So you want dualizing complexes. Deligne has proved that they exist if X is of finite type over a regular basis S of dimension $\leq 1$ (SGA 4 1/2, [Th finitude], by the way the dualizing complex is what you might expect i.e. the exceptional inverse image of the constant sheaf on S). More generally, Gabber's recent (mostly unpublished) results include the existence of a dualizing complex over any noetherian excellent scheme that admits a dimension function (where "dualizing complex" has a slightly weaker definition that in SGA 5 I, but I don't think it matters for the application to (1)). A remark : for the first case (X of finite type of S regular of dimension $\leq 1$, the dimension function on X would simply be Artin's rectified dimension, if I remember well. In general, I think that the existence of a dimension function on X is equivalent to the fact that X is universally catenary (which is part of the definition of "excellent", so I am confused).

(3) In any case, suppose you're okay with (1) and (2), so you have your dualizing complex, your perverse t-structure, you chose the perversity function so the truncation functors are compatible with constructibility, and let's say you even chose the self-dual perversity, that is, the perversity is related to the dimension function in the usual way. Then your category of constructible perverse sheaves should be stable by duality, and artinian (use theorem 8.3 of the article of Gabber I cited in (1)).

(4) Now you want to study the exactness properties of the 4 operations, so probably you'll be happy to use Gabber's finiteness theorems so constructible things stay constructible, so you'll restrict to morphisms of finite type (and the schemes already have to be noetherian excellent). The exactness properties of direct images by affine maps also follow from results of Gabber (he generalized SGA 4 XIV; if you're of finite type over a trait, the result was proved by Gabber too, but much earlier). I think these are the central results, the t-exactness of shifted $f^*$ for $f$ smooth should just come from duality.

(5) And finally, you want to go $\ell$-adic. Then I don't know anything better than Ekedahl's results (maybe they could be generalized to other schemes using Gabber's finiteness theorem ? I haven't tried so I can't be sure). So you'll finally need to restrict to the case where X is of finite type over S regular of dimension $\leq 1$. Note that in this case the existence of the dualizing complex and the finiteness results are already in SGA 4 1/2, the construction of the t-structures is in a published paper of Gabber, so the only possible problem is affine Lefschetz (the exactness properties of affine direct images), which as far as I understand has been known for more than 15 years in this case, but I'm unable to cite a reference right now.

(6) Well, I guess you'd probably be interested in the nearby and vanishing cycles functors next. Over a dvr or a general base ? ;) Good luck anyway, I don't think I know any reference (published or not).

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