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.
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机译：假设$ X $是在一个Artinian交换环$ R_0 $上的投影方案，并且让$ \ Cal {F} $是$ \ Cal {O} _X $-模块的连贯捆。我们对所谓的同调缺陷函数$ \ Delta ^ i_ {X，\ Cal {F}} $以及$（对）的同调假设数$ \ nu ^ i_ {X，\ Cal {F}} $给出界限。 X，\ Cal {F}）。$作为边界不变量，我们使用“同调对角线” $ \ big（h ^ j_ {X，\ Cal {F}}（-j）\ big）_ {j \ le i} $及其以下级别的$ i $和对中的$（X，\ Cal {F}）$对中的第$ i $个“同伦希尔伯特多项式” $ p ^ i_ {X，\ Cal {F}} $。边界是Severi \，-\，Enriques \，-\，Zariski \，-\ Serre消失定理的定量和扩展版本。
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