ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS

Abstract

For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork&apos;s theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky-Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}<^>n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$, which computes the rigid cohomology of $\mathbb {P}<^>n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}<^>n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz&apos;s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.