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Ind-abelian categories and quasi-coherent sheaves

Published online by Cambridge University Press:  02 October 2014

DANIEL SCHÄPPI
DANIEL SCHÄPPI*
Affiliation:
e-mail: [email protected]
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Abstract

We study the question of when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 391 - 423
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[BQ96]Borceux, F. and Quinteiro, C.A theory of enriched sheaves. Cahiers Topologie Géom. Différ. Catég. 37 (1996), no. 2, 145–162. MR 1394507 (97g:18008).Google Scholar
[Del90]Deligne, P.Catégories tannakiennes. The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87 (Birkhäuser Boston, Boston, MA, 1990), pp. 111–195. MR MR1106898 (92d:14002).Google Scholar
[DS86]Day, B. and Street, R.Categories in which all strong generators are dense. J. Pure Appl. Algebra 43 (1986), no. 3, 235–242. MR 868984 (88m:18013).Google Scholar
[DS97]Day, B. and Street, R.Monoidal bicategories and Hopf algebroids. Adv. Math. 129 (1997), no. 1, 99–157. MR MR1458415 (99f:18013).CrossRefGoogle Scholar
[Goe08]Goerss, P. G. Quasi-coherent sheaves on the moduli stack of formal groups. Preprint, arXiv:0802.0996v1 [math.AT]. (2008).Google Scholar
[GPS95]Gordon, R., Power, A. J., and Street, R.Coherence for tricategories. Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81. MR 1261589 (96j:18002).Google Scholar
[Gro10]Gross, P. Vector bundles as generators on schemes and stacks. Ph.D. thesis, Heinrich–Heine–Universität Düsseldorf (2010). http://reh.math.uni-duesseldorf.de/∼gross/docs/dissertation_PGross_July,26,2010.pdf (March 14, 2013).Google Scholar
[GU71]Gabriel, P. and Ulmer, F.Lokal präsentierbare Kategorien. Lecture Notes in Math. vol. 221. (Springer-Verlag, Berlin, 1971). MR 0327863 (48 #6205).Google Scholar
[Hov04]Hovey, M.Homotopy theory of comodules over a Hopf algebroid. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, Contemp. Math. vol. 346. (Amer. Math. Soc., Providence, RI, 2004), pp. 261–304. MR 2066503 (2005f:18011).Google Scholar
[HP02]Hyland, M. and Power, J.Pseudo-commutative monads and pseudo-closed 2-categories. J. Pure Appl. Algebra 175 (2002), no. 1–3, 141–185, Special volume celebrating the 70th birthday of Professor Max Kelly. MR 1935977 (2004b:18014).Google Scholar
[IK86]Im, G. B. and Kelly, G. M.A universal property of the convolution monoidal structure. J. Pure Appl. Algebra 43 (1986), no. 1, 75–88. MR 862873 (87m:18011).Google Scholar
[Joh75]Johnstone, P. T.Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7 (1975), no. 3, 294–297. MR 0390018 (52 #10845).Google Scholar
[Kel05]Kelly, G. M. Basic concepts of enriched category theory. Repr. Theory Appl. Categ. (2005), no. 10, vi + 137 pp. (electronic). Reprint of the 1982 original (Cambridge University Press, Cambridge; MR0651714). MR MR2177301.Google Scholar
[KS74]Kelly, G. M. and Street, R. Review of the elements of 2-categories. Category Seminar (Proc. Sem., Sydney, 1972/1973), (Springer, Berlin, 1974), pp. 75–103. Lecture Notes in Math., vol. 420. MR MR0357542 (50 #10010).Google Scholar
[Lac00]Lack, S.A coherent approach to pseudomonads. Adv. Math. 152 (2000), no. 2, 179–202. MR 1764104 (2001f:18017).Google Scholar
[LF11]Franco, I. L.Pseudo-commutativity of KZ 2-monads. Adv. Math. 228 (2011), no. 5, 2557–2605. MR 2838049.Google Scholar
[LF12]Franco, I. L. Tensor products of finitely cocomplete and abelian categories. Preprint, arXiv:1212.1545v1 [math.CT] (2012).Google Scholar
[McC00]McCrudden, P.Balanced coalgebroids. Theory Appl. Categ. 7 (2000), No. 6, 71–147 (electronic). MR MR1764504 (2001f:18018).Google Scholar
[Nau07]Naumann, N.The stack of formal groups in stable homotopy theory. Adv. Math. 215 (2007), no. 2, 569–600. MR 2355600 (2008m:55010).CrossRefGoogle Scholar
[OR70]Oberst, U. and Röhrl, H.Flat and coherent functors. J. Algebra 14 (1970), 91–105. MR 0257181 (41 #1835).Google Scholar
[Rum04]Rump, W.Triads. J. Algebra 280 (2004), no. 2, 435–462. MR 2089245 (2005g:18018).Google Scholar
[Sch11]Schäppi, D.The formal theory of Tannaka duality. Asterisque. Volume 357 (2013).Google Scholar
[Sch12]Schäppi, D. A characterization of categories of coherent sheaves of certain algebraic stacks. Preprint, arXiv: 1206.2764v1 [math.AG] (2012).Google Scholar
[Tho87]Thomason, R. W.Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes. Adv. Math. 65 (1987), no. 1, 16–34. MR 893468 (88g:14045).Google Scholar
Totaro, B.The resolution property for schemes and stacks. J. Reine Angew. Math. 577 (2004), 1–22. MR 2108211 (2005j:14002).Google Scholar