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ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS

Author(s)
Park, Junyeong
Issued Date
2024-06
DOI
10.1017/nmj.2023.32
URI
https://scholarworks.unist.ac.kr/handle/201301/68056
Citation
NAGOYA MATHEMATICAL JOURNAL, v.254, pp.420 - 461
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'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's comparison results in [19] for projective hypersurface complements to arbitrary projective complements.
Publisher
CAMBRIDGE UNIV PRESS
ISSN
0027-7630
Keyword (Author)
the Cayley tricktwisted de Rham complexesDwork cohomologyrigid cohomology
Keyword
ZETA-FUNCTIONFORMAL COHOMOLOGYEXPONENTIAL-SUMSHYPERSURFACE

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