Auslander-Reiten duality for Grothendieck abelian categories

Krause, Henning

doi:10.1090/tran/7379

Auslander-Reiten duality for Grothendieck abelian categories
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Trans. Amer. Math. Soc. 371 (2019), 2455-2472 Request permission

Abstract:

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor $\mathrm {Ext}^1(C,-)$ into modules over the endomorphism ring of $C$ admits a partially defined right adjoint when $C$ is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.

References

Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York, 1978, pp. 1–244. MR 0480688
Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
Maurice Auslander and Idun Reiten, Stable equivalence of dualizing $R$-varieties, Advances in Math. 12 (1974), 306–366. MR 342505, DOI 10.1016/S0001-8708(74)80007-1
Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. III. Almost split sequences, Comm. Algebra 3 (1975), 239–294. MR 379599, DOI 10.1080/00927877508822046
Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. MR 1832549, DOI 10.1017/S0024609301008104
A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519–541. MR 1039961, DOI 10.1070/IM1990v035n03ABEH000716
Siegfried Breitsprecher, Lokal endlich präsentierbare Grothendieck-Kategorien, Mitt. Math. Sem. Giessen 85 (1970), 1–25 (German). MR 262330
Xiao-Wu Chen and Jue Le, A note on morphisms determined by objects, J. Algebra 428 (2015), 138–148. MR 3314289, DOI 10.1016/j.jalgebra.2015.01.010
Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI 10.1007/BFb0078849
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
Henning Krause, A short proof for Auslander’s defect formula, Linear Algebra Appl. 365 (2003), 267–270. Special issue on linear algebra methods in representation theory. MR 1987342, DOI 10.1016/S0024-3795(02)00481-0
Henning Krause, Morphisms determined by objects and flat covers, Forum Math. 28 (2016), no. 3, 425–435. MR 3510823, DOI 10.1515/forum-2014-0115
Henning Krause, Deriving Auslander’s formula, Doc. Math. 20 (2015), 669–688. MR 3398723, DOI 10.1016/s2212-5671(15)00121-5
Henning Krause and Jue Le, The Auslander-Reiten formula for complexes of modules, Adv. Math. 207 (2006), no. 1, 133–148. MR 2264068, DOI 10.1016/j.aim.2005.11.008
Helmut Lenzing and Rita Zuazua, Auslander-Reiten duality for abelian categories, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 2, 169–177 (2005). MR 2135956
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
F. Oort, Natural maps of extension functors, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 559–566. MR 0156881
I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366. MR 1887637, DOI 10.1090/S0894-0347-02-00387-9
Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
J. Šťovíček, On purity and applications to coderived and singularity categories, arXiv:1412.1615.