On Dwork cohomology for singular hypersurfaces (original) (raw)
On Dwork cohomology and algebraic D-modules
Geometric Aspects of Dwork Theory, 2004
After works by Katz, Monsky, and Adolphson-Sperber, a comparison theorem between relative de Rham cohomology and Dwork cohomology is established in a paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We propose here an alternative proof of this result. The use of Fourier transform techniques makes our approach more functorial.
A study of the cohomological rigidity property
arXiv: Commutative Algebra, 2020
In this paper, motivated by a work of Luk and Yao, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersection rings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections.
Journal of Algebra, 1998
We construct a new Koszul complex that computes local cohomology for a quasi-coherent module on an affine scheme with supports in the closed subset defined by a finitely generated ideal.
On ppp-adic absolute Hodge cohomology and syntomic coefficients. I
Commentarii Mathematici Helvetici, 2018
We interpret syntomic cohomology defined in [49] as a p-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson [8] and generalizes the results of Bannai [6] and Chiarellotto, Ciccioni, Mazzari [15] in the good reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for projective and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated subcategory of the derived category of potentially semistable Galois representations. Along the way, we obtain p-adic realizations of mixed motives including p-adic comparison isomorphisms. We apply this to the motivic fundamental group generalizing results of Olsson and Vologodsky [55], [69]. 2. A p-adic absolute Hodge cohomology, I 2.1. The derived category of admissible filtered (ϕ, N, G K)-modules.
On the zeta function of a projective complete intersection
Illinois Journal of Mathematics, 2008
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
2006
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, due to the short span of lectures. However, details and examples will be worked out and, possibly, a few arguments too. 10. * Considerations about a formula of Kleiman-Plücker-Teissier for the degree (class) of the dual variety to a complete intersection.