On Dwork Cohomology and Algebraic D-modules

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. 1. Review of algebraic D-modules For the reader’s convenience, we recall here the notions and results from the theory of algebraic D-modules that we need. 1.1. Basic operations. Let X be a smooth algebraic variety over a field of characteristic zero, and let OX and DX be its structure sheaf and the sheaf of differential operators, respectively. Let Mod(DX) be the abelian category of left DX -modules, D (DX) its bounded derived category, and Dqc(DX) the full triangulated subcategory of D(DX) whose objects have quasi-coherent cohomologies. Let f : X −→ Y be a morphism of smooth algebraic varieties, and denote by DX→Y and DY←X the transfer bimodules. We use the following notation for the operations of tensor product, inverse image, and direct image for D-modules ⊗ : D(DX)× D (DX) −→ D (DX), (M,M ) 7→ (M⊗ OX DX)⊗ L DX M = M⊗ DX (DX ⊗OX M ), f ∗ : D(DY ) −→ D (DX), N 7→ DX→Y ⊗ L fDY fN , f+ : D (DX) −→ D (DY ), M 7→ Rf ∗(DY←X ⊗ L DX M), where in the first (resp. second) line of the formula, M⊗ OX DX (resp. DX ⊗OX M ) is given the natural stucture of left-right (resp. left-left) DX-bimodule, and ⊗ DX always uses up the “trivial” DX-module structure. These operations preserve quasicoherence. 2000 Mathematics Subject Classification. 32S40, 14F10. 1The standard published reference on algebraic D-modules is [4]. Bernstein’s lectures [3] were excellent, but unfortunately are still unpublished at the moment. An excellent reference, even though dealing with the analytic case, are Kashiwara’s books [6, 7].