## Grothendieck Duality and Base Change, Issue 1750 (Google eBook)Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory. |

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5-functorial acyclic admits a dualizing algebraically closed base change base change maps bounded below complex bounded complex canonical isomorphism canonical map canonical truncation Cartan-Eilenberg resolution closed immersion closed point codimension codimension filtration coherent coherent sheaves cohomology commutative diagram complex of injectives complex of quasi-coherent construction define definition denote derived category derived functors diagram of complexes differentials direct limit double complex duality theory dualizing complex explicit ffx modules fibers finite Krull dimension finite map finite type flat base change follows Gorenstein Grothendieck spectral sequence Grothendieck's homotopy injective resolution intervention of signs invertible sheaf Koszul Krull dimension lci map Lemma Let f locally free locally noetherian scheme map f map of complexes map with pure morphism natural map noetherian ring pure codimension pure relative dimension quasi-coherent sheaves quasi-isomorphism residual complex residually stable resp respect right column separated smooth smooth map surjective trace map truncation in rows x-module