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Dwork cohomology and algebraic $$\mathcal D$$-modules. (English) Zbl 0985.14007
Using local cohomology and algebraic $${\mathcal D}$$-modules, the authors generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology. More precisely, let $$X$$ and $$S$$ be smooth complex algebraic varieties, $$p:X\to S$$ a map, $$\pi: V\to X$$ an algebraic vector bundle with rank $$r$$, $$s:X \to V^*$$ a section of the dual vector bundle and $$Y=s^{-1} (0)_{\text{red}}$$. Let $$M^\bullet$$ be a bounded complex of quasi-coherent left $${\mathcal D}_X$$-modules and denote by $$(\pi^*M^\bullet)_s$$ the complex of $${\mathcal D}$$-modules with twisted action of $${\mathcal D}_V$$ by $$s$$. Let $$\pi_+$$ denote the direct image of algebraic left $${\mathcal D}$$-modules.
One proves the following comparison result:
Theorem. There is a canonical isomorphism in the derived category of left $${\mathcal D}_X$$-modules ${\mathbf R}\Gamma_Y M^\bullet[r] =\pi_+ \bigl((\pi^* M^\bullet)_s\bigr).$

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F40 de Rham cohomology and algebraic geometry
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