Download e-book for iPad: Grothendieck Duality and Base Change by Brian Conrad

Algebraic Geometry

By Brian Conrad

Grothendieck's duality conception for coherent cohomology is a basic instrument in algebraic geometry and quantity conception, in parts starting from the moduli of curves to the mathematics concept of modular types. awarded is a scientific assessment of the total concept, together with many simple definitions and an in depth examine of duality on curves, dualizing sheaves, and Grothendieck's residue image. alongside the way in which proofs are given of a few accepted foundational effects which aren't confirmed in current remedies of the topic, resembling 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 need to appreciate the main points of this theory.

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A dx,, of We will write dx for the free generator dxl A wxly on codimension with lci m near y E Y, local equations U. Pry over an open U C X around i(y). We get from this an ordered is 6tale. ... basis of i * (Jy / Jy2) We denote basis. If f is smooth a Y, and denote by tj Un on by t' the free generator tv, ..... A ... tm' the corresponding dual tvm of wylx over U n Y. A section i, we can regard local equations local coordinates relative to f around i(y). Note separated map with a relative to i around y C- Y as that if f and g are smooth (resp of local coordinates lci) maps, we can easily view an ,'ordered union' equations) for f near a point x and local coor(resp.

4-17) is a dt9om (f*9*,f0W*) quasi-isomorphism.

Acyclic, coherent ex-mo'dules. Thus, f,, (Tot9 1**) represents Rf, (f #W*). 3. 7) is R so If ]Rnf pp;ly) (UP;/y Rnf 0gr isomorphism. 1). r), arbitrary ey-module. 5) on 01*. 6) is a quasi-isomorphism. 2. 1). 1). ) as 0 for i :/ n, there is wp / y [n]. 8); compute we Rf* (wp; / y [n]) = f*(jo+n), Rnf * (wp;/Y) = H ny*jo). 1) that this con is that when J* is 2. BASIC COMPATIBILITIES 36 of wp /y[n] this double as required in the definition of Thus, the Trp. 1) is Trp(ey). 4), by a and if sign we had used (-I)'.

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