Bounds for Cohomological Deficiency Functions of Projective Schemes
Over Artinian Rings
M. Brodmann, C. Matteotti, and N.D. Minh
Abstract. Let $X$ be a projective scheme
over an artinian commutative ring $R_0$ and let $\Cal{F}$ be a coherent
sheaf of $\Cal{O}_X$modules. We give bounds on the so called cohomological
deficiency functions $\Delta^i_{X, \Cal{F}}$ and the cohomological postulation
numbers $\nu^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ As bounding invariants
we use the "cohomology diagonal" $ \big( h^j_{X, \Cal{F}}(j) \big)_{j\le
i}$ at and below level $i$ and the $i$th "cohomological Hilbert polynomial"
$p^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ Our bounds present themselves
as a quantitative and extended version of the vanishing theorem of Severi\,\,Enriques\,\,Zariski\,\,Serre.
