Rigid Cohomology over Laurent Series Fields

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In this monograph, the authors introduce a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic, building on Berthelot's rigid cohomology. They establish key properties of these cohomology groups, including finite dimensionality and cohomological descent, while also providing interpretations through Monsky-Washnitzer cohomology and Le Stum's overconvergent site. The theory's applications to arithmetic questions, such as l-independence and the weight monodromy conjecture, are explored. This construction of cohomology groups, akin to Galois representations for varieties over local fields in mixed characteristic, addresses a significant gap in the study of arithmetic cohomology theories over function fields. By broadening existing methodologies, the findings represent a foundational step toward a more comprehensive theory of p-adic cohomology over non-perfect ground fields. This work will be a valuable resource for those interested in the arithmetic of varieties over local fields of positive characteristic. The inclusion of appendices on essential background topics like rigid cohomology and adic spaces ensures the monograph is as self-contained as possible, making it an ideal starting point for graduate students aiming to delve into the classical theory of rigid cohomology and pursue future research in the field.