Grothendieck duality and base change
type: Item
instance of: scholarly article
title: Grothendieck duality and base change
zbMATH Open document ID: 0992.14001
published in: Lecture Notes in Mathematics
publication date: 2000-12-19T00:00:00Z
full work available at URL: t1750.htm
review text: Grothendieck duality theory on noetherian schemes plays a crucial role in various branches of algebraic and arithmetic geometry, ranging from the study of moduli spaces of algebraic curves up to the arithmetic theory of modular forms. About fourty years ago, \textit{A. Grothendieck} initiated this theory, the main goal of which was to produce a certain ``trace map'' in the cohomology theory of coherent sheaves generalizing the classical Serre duality for smooth schemes over a field. The foundations of what is now called Grothendieck duality theory are worked out in \textit{R. Hartshorne}'s celebrated monograph ``Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.26101) published more than thirty-five years ago. The foundational framework developed in Hartshorne's book, based on residual complexes and the notion of a dualizing sheaf, makes this duality theory quite computable in terms of differential forms and residues, and this efficient computability has turned out to be extremely useful in many concrete situations in both algebraic and arithmetic geometry. However, in Hartshorne's construction of Grothendieck duality theory there are some assumptions on compatibility conditions, together with some explications of abstract results, which are not rigorously proven and are, in fact, quite difficult to verify. The hardest compatibiliy condition in the theory, and also one of the most important, is the base change compatibility of the trace map in the case of proper Cohen-Macaulay morphisms with pure relative dimension. For example, this important special case occurs in the study of flat families of semi-stable curves and their moduli. Although there are simpler methods for obtaining duality theorems in the projective Cohen-Macaulay case, which allow to ignore the base change compatibility problem [\textit{A. Altman} and \textit{S. Kleiman}, ``Introduction to Grothendieck duality theory'', Lect. Notes Math. 146 (1970; Zbl 0215.37201)], there remains the funda…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14A15
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 1544752
zbMATH Keywords: schemes
zbMATH Keywords: sheaves of differentials
zbMATH Keywords: dualizing sheaves
zbMATH Keywords: residues
zbMATH Keywords: duality theorems
zbMATH Keywords: base change theorems
zbMATH Keywords: Cohen-Macaulay maps
zbMATH Keywords: trace maps
zbMATH Keywords: algebraic curves
zbMATH Keywords: morphisms
MaRDI profile type: Publication
OpenAlex ID: W1548062843
DOI: 10.1007/B75857
author: Brian Conrad
description: scientific article; zbMATH DE number 1544752