By Brian Conrad
Grothendieck's duality idea for coherent cohomology is a primary device in algebraic geometry and quantity thought, in components starting from the moduli of curves to the mathematics idea of modular types. offered is a scientific review of the complete concept, 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 regularly occurring foundational effects which aren't confirmed in present remedies of the topic, similar to the overall 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 need to appreciate the main points of this theory.
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Extra resources for Grothendieck Duality and Base Change
4) on the level of complexes in degree -diMhgfi without any intervention of signs. 6) in place such define as D (X) by wX1 y [nJ 0 f L wxly[n]. 4) Recall that in we [RD] use the in order without the intervention of isomorphism (gf) translations, triple composites, and to -+ L = the definition is modified definition D (Y) f an with respect to use the modified definition -a signs. 7) instead of the one in [RD]. 6)). 4). ), to be where 7 : -4 D+(X) (X, 61x) -+ flat map of ringed spaces. For example, if X* is of quasi-coherent injective sheaves on Y, then the (Y, f* ex) is the canonical bounded below complex complex of quasi-coherent a f* Ox -modules Je om y (f* 61X, X *) can be viewed as a complex of quasi-coherent on X, and this represents the complex f (j*).
3. Let X be a locally noetherian category of quasi-coherent sheaves in the category of Cx-modules. in the PROOF. 18], there is an on X. scheme, J an Then _0 is an injection i : J -4 injective object iniective object f for some quasi- / which is injective as an OX-module. r as an ex-module on j, this injection splits. Thus, Y is a direct summand of and so is injective as an OX-module. coherent Ox-module By this lemma, the injective objects in the category of quasi-coherent 6IXon a locally noetherian scheme X are exactly the injective ex-modules which are quasi-coherent as sheaves.
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.