From Sheaf Cohomology to the Algebraic de Rham Theorem (original) (raw)
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Nagoya Mathematical Journal, 1997
Consider the complex of differential forms on an open aίfine subvariety U of A^ with differential ω »-• dω + φ Λ ω, where d is the usual exterior derivative and φ is a fixed 1-form on U. For certain U and φ, we compute the cohomology of this complex. Recent work on this problem has been done by Kita [KI] and Aomoto-Kita-Orlik-Terao [AKOT], to which we refer for further background and applications along the above lines. We take a somewhat different approach here. In [DW1], Dwork introduced a p-adic cohomology theory for varieties over finite fields, which is also often referred to as "twisted de Rham cohomology." Dwork's definition is algebraic and makes sense over any field of characteristic zero. The connection between Dwork's theory and classical de Rham cohomology was studied by Katz [KI, K2], who introduced an algebraic notion of "Laplace transform" to connect the two theories. This theory of the Laplace transform was developed further by Dwork [DW2 chapters 10 and 11] (see also Batyrev [B section 7]).
1999
differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a
De Rahm Cohomology is a powerful tool which allows one to extract purely topo-logical information about a manifold, essentially by doing algebra on its cotangent bundle. A particularly useful method of computing de Rham cohomology groups was discovered by Austrian mathematicians Mayer and Vietoris, and involves partitioning a manifold into subspaces, ideally ones whose cohomology groups are already known. In this paper, we briefly cover the prerequisite homological algebra, introduce de Rham cohomology, then proceed to prove the invariance of cohomology under smooth homotopy which allows for a simple proof of the Poincaré lemma. We then develop the Mayer Vietoris sequence, perform a few computations, including a simple homological proof of the Brouwer fixed-point theorem, then conclude with an introduction to co-homology with compact support, followed by a discussion and proof of a version of the Poincaré Duality Theorem, which links the dual notions of homology and cohomology.
On the Cohomology of Regular Differential Forms and Dualizing Sheaves
Proceedings of the American Mathematical Society
Let Y be an integral Noetherian scheme. Let f:X→Y be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension d and with geometrically connected fibres. Assume that X is reduced. Let ω X/Y d be the sheaf of relative regular differential forms. The aim of the paper is to study when the direct images of this sheaf are torsion-free. The question being local, it is assumed that Y=SpecD for some complete discrete valuation ring D with residue class field k algebraically closed and X=ProjS for some graded reduced D-algebra S. The authors give a set of equivalent conditions for R i f * ω X/Y d to be torsion-free. Using them the following results are proved. (1) Suppose that X/Y is arithmetically S k+2 (Serre’s condition). Then R d-i f * ω X/Y d is torsion-free and commutes with base change for all i≤k. (2) If R d-i f * ω X/Y d is torsion-free for all i≤k then R i f * O X is torsion-free for all i≤k. (3) Suppose that X/Y is globally a complete inters...