By Brian Conrad
Grothendieck's duality idea for coherent cohomology is a primary device in algebraic geometry and quantity concept, in parts starting from the moduli of curves to the mathematics conception of modular types. offered is a scientific evaluation of the complete concept, together with many easy definitions and an in depth research of duality on curves, dualizing sheaves, and Grothendieck's residue image. alongside the way in which proofs are given of a few commonplace foundational effects which aren't confirmed in present remedies of the topic, corresponding to the final base swap 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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Additional resources for Grothendieck Duality and Base Change
2. 4). 2 by a factor of -1. For the benefit of the reader who prefers Deligne's convention, we will keep sign as it arises later on. 2). One reason not to get too worried about this issue (which will arise in exponent in this kind of sign will always involve relative dimensions or codimensions, so in the context of 6'xt sheaves and higher direct image sheaves, assertions such as compatibility with respect to other contexts later scheme on) is that the morphisms will always (since (_j)n(_j)rn will have the same = by such sign ambiguities explicit compatibility assertions turn out to be unaffected (-l)n+m).
4. 3. Both proofs require intricate various resolutions. The reader is strongly advised to skip manipulations with on a first reading (similar arguments will be used to prove Theorem below, whose proof the reader is also advised to skip on a first reading). 2 PROOF. 1) is equal to Trp(W*). 1) is described in terms of quasi-isomorphic to 01*, a bounded above while the definition of complex of flats which Trp(01*) is is described in terms of bounded below complex of quasi-coherent sheaves which is quasi-isomorphic to W*.
Let 1` and I/ denote the respective canonical truncations in rows < n. By the theory qc of injective resolutions in abelian categories, we can choose a map of double complexes pi lqc over f 016 - fOWq, and a map of double complexes in the tion of P2 : _ Kq*,* -+ Iq*c* over f001q*,. canonical truncations in rows Le ,t < p', and p2 denote the induced maps on the n. qc and consist of f,,-acyclics, applying f, to Tot (p2) yields a Since X and TotB(p) 2 quasi-isomorphism. Beware that applying f, to Tbt ED isomorphism.