duality

# Grothendieck duality

## Idea

Given a homomorphism $f$ of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor $f_\ast$ on quasicoherent sheaves has a (derived) right adjoint $f^!$. This is Verdier duality in a “Grothendieck context” of six operations.

## Statement

Suppose $f\colon X \to Y$ is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below right adjoint. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a natural isomorphism.

## References

• Robin Hartshorne, Residues and duality (Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.) Springer LNM 20, 1966 MR222093

• Domingo Toledo, Yue Lin L. Tong, Duality and intersection theory in complex manifolds. I., Math. Ann. 237 (1978), no. 1, 41–77, MR80d:32008, doi

• Mitya Boyarchenko, Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arxiv/1108.6020

• Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026

• Amnon Neeman, Derived categories and Grothendieck duality, in: Triangulated categories, 290–350, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010

• Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236, MR96c:18006, doi

• Brian Conrad, Grothendieck duality and base change, Springer Lec. Notes Math. 1750 (2000) vi+296 pp.

• Joseph Lipman, Notes on derived functors and Grothendieck dualitym in: Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math. 1960, Springer 2009, doi, draft pdf

• J. Lipman, Grothendieck operations and coherence in categories, conference slides, 2009, pdf

• Alonso Tarrío, Leovigildo; Jeremías López, Ana; Joseph Lipman, Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes, Contemporary Mathematics 244 Amer. Math. Soc. 1999. x+126L. MR2000h:14017; Duality and flat base change on formal schemes, Contemporary Math. 244 (1999), pp. 3–90.

• J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I., Astérisque No. 314 (2007), x+466 pp. (2008) MR2009h:14032; II. Astérisque No. 315 (2007), vi+364 pp. (2008) MR2009m:14007; also a file at K-theory archive THESE.pdf

• Amnon Yekutieli, James Zhang, Rigid dualizing complexes over commutative rings, Algebr. Represent. Theory 12 (2009), no. 1, 19–52, doi

• Amnon Yekutieli, The residue complex of a noncommutative graded algebra, J. Algebra 186 (1996), no. 2, 522–543; Smooth formal embeddings and the residue complex, Canad. J. Math. 50 (1998), no. 4, 863–896, MR99i:14004; Rigid dualizing complexes via differential graded algebras (survey), in: Triangulated categories, 452–463, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010, MR2011h:18015

• Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)

• Pieter Belmans, section 2.2 of Grothendieck duality: lecture 3, 2014 (pdf)

• Amnon Neeman, An improvement on the base-change theorem and the functor $f^!$, arXiv.

Revised on July 6, 2014 07:56:13 by Adeel Khan (77.9.210.144)