Gorenstein injective sheaves

Abstract

We define Gorenstein injective quasi-coherent sheaves, and prove that the notion is local in case the scheme is Gorenstein. We also give a new characterization of a Gorenstein scheme in terms of the total acyclicity of every acyclic complex of injective quasi-coherent modules.

Keywords

Gorenstein injective / Noetherian scheme

References

[1] Christensen L, Jorgensen D. Tate (co)homology via pinched complexes. Trans Amer Math Soc, 2014, 366: 667–689

[2] Conrad B. Grothendieck Duality and Base Change. Lecture Notes in Math, Vol 1750. Berlin: Springer-Verlag, 2000

[3] Enochs E E, Estrada S. Relative homological algebra in the category of quasi-coherent sheaves. Adv Math, 2005, 194: 284–295

[4] Enochs E E, Estrada S, García-Rozas J R. Gorenstein categories and Tate cohomology on projective schemes. Math Nachr, 2008, 281: 525–540

[5] Enochs E E, Estrada S, Iacob A. Balance with unbounded complexes. Bull Lond Math Soc, 2012, 44: 439–442

[6] Enochs E E, Jenda O. Relative Homological Algebra. de Gruyter Exp Math, Vol 30. Berlin: Walter de Gruyter, 2000

[7] Gillespie J. Kaplansky classes and derived categories. Math Z, 2007, 257: 811–843

[8] Hartshorne R. Residues and Duality. Lecture Notes in Math, Vol 20. Berlin: Springer-Verlag, 1966

[9] Holm H. Gorenstein homological dimensions. J Pure Appl Algebra, 2004, 189: 167–193

[10] Krause H. The stable derived category of a Noetherian scheme. Compos Math, 2005, 141(5): 1128–1162

[11] Murfet D. Modules over a scheme

[12] Murfet D, Salarian S. Totally acyclic complexes over noetherian schemes. Adv Math, 2011, 226: 1096–1133

[13] Neeman A. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J Amer Math Soc, 1996, 9: 205–236

[14] Tarrío L A, López A J, Rodríguez M P, Gonsalves M J V. The derived category of quasicoherent sheaves and axiomatic stable homotopy. Adv Math, 2008, 218(4): 1224–1252

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg