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Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived categories of etale constructible sheaves?), or does one really need smoothness here? What additional assumptions (or corrections) are needed here, and what is the best reference (for local complete intersection morphisms)?

Upd. As Torsten Ekedahl states, this is wrong. Still, are their any special properties of cohomology of local complete intersection morphisms (that are common with smooth morphisms, and distinguish them from general finite type morphisms)?

edited Sep 22 2011 at 9:32

asked Sep 22 2011 at 7:12

Mikhail Bondarko
5,876●1●4●21

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This is false already for $Y$ a point and $X$ not a homology manifold. – Torsten Ekedahl Sep 22 2011 at 7:18

Thank you! Then could you tell me: do local complete intersection morphism have any 'special cohomological' properties (similar to those of smooth morphisms)? – Mikhail Bondarko Sep 22 2011 at 8:14

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For a variety $X$ with l.c.i. singularities the constant sheaf $\mathbb{Q}_{l,X}$ (with the appropriate shift) is perverse but I don't know if there is something one can say for arbitrary l.c.i. morphisms. – ulrich Sep 22 2011 at 8:57

Thanks!! Possibly this is all I need. – Mikhail Bondarko Sep 22 2011 at 9:36

This is obviously false in a very different situation than the one given by Torsten Ekedahl : Y is the affine line, X is the origin, f=i is the inclusion (a regular immersion), K is any complex on X-Y, j is the inclusion of X-Y in X, then i^!j_*K is always 0 but that's not true for i^*j_*K. On a less trivial note, if what you wanted were true just for the constant sheaf on Y, then it would mean that absolute purity holds for any regular closed immersion, not just for one between regular schemes. I'm a bit skeptical (though I can't give a counterexample right now). – Alex Sep 22 2011 at 22:56

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On ‘special properties’ of various ‘sheaf image’ functors for a local complete intersection morphism

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