By Brian Conrad

ISBN-10: 354040015X

ISBN-13: 9783540400158

ISBN-10: 3540411348

ISBN-13: 9783540411345

Grothendieck's duality thought for coherent cohomology is a primary device in algebraic geometry and quantity concept, in components starting from the moduli of curves to the mathematics conception of modular kinds. provided is a scientific assessment of the total thought, together with many simple definitions and an in depth research of duality on curves, dualizing sheaves, and Grothendieck's residue image. alongside the best way proofs are given of a few commonly used foundational effects which aren't confirmed in present remedies of the topic, akin to the final base switch compatibility of the hint map for correct Cohen-Macaulay morphisms (e.g., semistable curves). this could be of curiosity to mathematicians who've a few familiarity with Grothendieck's paintings and want to appreciate the main points of this theory.

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**Sample text**

14) is to try to eliminate all references to lq*c* and to reduce to a commutativity assertion involving flat sheaves, which we will then check locally. 14) by a more tractable diagram. 2. We noted above that the quasi-isomorphism -+ al* has a homotopy inverse, so applying f 0 to this shows that there is a double is * 2. BASIC COMPATIBILITIES 48 complex map o` 1*0 : qc -+ I** between Cartan-Eilenberg resolutions which is complex homotopy inverse to p. From the construction of (p-1 as in the homotopies between (p-' o (p and 1, as well as between 1 to be conap atible -,,vith respect'to vertical canonical be and o Vcan chosen 1, V on canonical truncations in truncation.

Beware that applying f, to Tbt ED isomorphism. The quasi-isomorphism f,,(Tot p') 2 3p, is probably not a fits into the bottom quasirow of 2. 4) Tot" lqc f* TotC) I'qc e f. (Ciqc) f* TOtEo Tot'l Tbt' O 1,0 lqc* Poc* f* (P2 generally bounded below. 4) are quasi-isomorphisms. 4) consist of acyclics. 4) are quasi-isomorphisms follows from the fact that f* has finite cohomological dimension on the category of quasi-coherent 6X-modules. 4. 7) f* 019+n f* ("'Ojo+n (, We Rnf*(W) [0] 0 0? 6) so that it respects the natural quasi-isomorphism Let us from f 0010 to each side (this forces 0 to be a quasi-isomorphism).

A The [RD] proceeds by developing a theory of smooth (resp. finite) map f and then 'gluing' duality in f (resp. f ) for a I these to define f for more general maps. The first step in this procedure is to construct general isomorphisms relating canonical bundles for smooth maps and normal bundles for local complete intersection maps. This section reviews some and we of these initial constructions in [RD] and their relation with the errors correct some sign along way. 2. SMOOTH AND FINITE MAPS 29 for any scheme maps f : X -4 Y, g : Y -+ Z such that each of g, f, and gf is either a separated smooth map or a local complete intersection (Ici) map.

### Grothendieck Duality and Base Change by Brian Conrad

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