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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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