Cohomology of projective schemes: From annihilators to vanishing

This article comes from our quest for bounds on the Castelnuovo-Mumford regularity of schemes in terms of their “defining equations”, in the spirit of [BM], [BEL], [CP] or [CU]. The references [BS], [BM], [V] or [C] explains how this notion of regularity is a mesure of the algebraic complexity of the scheme, and provides several computational motivations. It was already remarked by several authors (see for instance [M], [MV], [NS1] or [NS2]) 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. Such a connection is a particular case of Lemma 2.0, which shows the way but will not be used in the sequel, because some natural killers have stronger annihilation properties that leads to sharper estimates. In connection with our previous joint work with Philippon, we introduce partial annihilators of modules (i.e. elements that annihilates in some degrees) and prove that uniform (partial)annihilators of Koszul homology modules give rise to (partial)annihilators of Čech cohomology modules of the quotient by a sequence of parameters (Proposition 2.1). Combined with Proposition 2.3, this leads to our key for passing from annihilators to vanishing: Proposition 2.4. The third section gathers results on uniform annihilators that have two main sources: tight closure and liaison. For the applications to Castelnuovo-Mumford regularity, the key is to determine an annihilator of the cohomology modules on which we have a control in terms of degrees of generators. The Jacobian ideal, which kills phantom homology by a theorem of Hochster and Huneke, and the ideal we construct via liaison are the only ones that we are able to control today, it would be interesting to have such a control on other natural annihilators (e. g. the parameter test ideal). 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 of [CU] for the unmixed part (even if not completely comparable). Also, they do not rely on Kodaira vanishing. The main result of a new type is the following (Theorem 4.4),