\centerline{\bf Marc Chardin}
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\centerline{\bf Cohomology of projective schemes: From annihilators to
vanishing}
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It was already remarked by several authors (for instance Miyazaki, Nagel,
Schenzel and Vogel) that one may bound the Castelnuovo-Mumford regularity
of a Cohen-Macaulay projective scheme in terms of its $a$-invariant and
the power of the maximal ideal that kills all but the top local
cohomology modules. We exploit this type of results in a slightly more
general setting.
In connection with our previous joint work with Philippon, we
introduce partial annihilators of modules ({\it i.e.} elements that
annihilates in some degrees) and prove that uniform
(partial)annihilators of Koszul homology modules give rise to
(partial)annihilators of \v Cech cohomology modules of the quotient by a
sequence of parameters. This leads to our key for passing from
annihilators to vanishing.
We then collect results on uniform annihilators that
have two main sources: tight closure and liaison.
The results on regularity are combinations of the two preceeding
ingredients: control of annihilators and passing from annihilators to
vanishing. They hold in any characteristic and, in positive
characteristic, improves the ones we proved with Ulrich for the unmixed
part (even if not completely comparable). Also, they do not rely on
Kodaira vanishing.
It is a very first step in the direction of bounding the regularity of all
the isolated components of a scheme in terms of degrees of generators.
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