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WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

Part of: Homology and cohomology theories

Published online by Cambridge University Press:  15 March 2021

ALESSIO CIPRIANI  and
JON WOOLF
ALESSIO CIPRIANI
Affiliation:
Department of Mathematical Sciences, University of Liverpool, LiverpoolL69 7ZL, UK, e-mails: Alessio.Cipriani@liverpool.ac.uk, jonwoolf@liverpool.ac.uk
JON WOOLF
Affiliation:
Department of Mathematical Sciences, University of Liverpool, LiverpoolL69 7ZL, UK, e-mails: Alessio.Cipriani@liverpool.ac.uk, jonwoolf@liverpool.ac.uk
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Abstract

Let X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

MSC classification

Secondary: 55N33: Intersection homology and cohomology
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 185 - 196
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Assem, I., Skowronski, A. and Simson, D., Elements of the representation theory of associative algebras: techniques of representation theory, London Mathematical Society Student Texts, vol. 1 (Cambridge University Press, 2006).CrossRefGoogle Scholar
Bass, H., Algebraic K-theory (W. A. Benjamin, Inc., New York-Amsterdam, 1968).Google Scholar
Belinson, A., How to glue perverse sheaves, in K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), 42–51.Google Scholar
Belinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100 (Soc. Math. France, Paris, 1982), 5–171.Google Scholar
Belinson, A., Bezrukavnikov, R. and Mirković, I., Tilting exercises. Mosc. Math. J. 4(3) (2004) 547–557, 782.CrossRefGoogle Scholar
Belinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Am. Math. Soc. 9(2) (1996), 473–527.CrossRefGoogle Scholar
Braden, T., Perverse sheaves on Grassmannians, Canad. J. Math. 54(3) (2002), 493–532.CrossRefGoogle Scholar
Braden, T. and Grinberg, M., Perverse sheaves on rank stratifications, Duke Math. J. 96(2) (1999), 317–362.CrossRefGoogle Scholar
Cline, E., Parshall, B. and Scott, L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99.Google Scholar
Galligo, A., Granger, M. and Maisonobe, P., D-modules et faisceaux pervers dont le support singulier est un croisement normal, Ann. Inst. Fourier (Grenoble) 35(1) (1985), 1–48.CrossRefGoogle Scholar
Gel‘fand, S., MacPherson, R. and Vilonen, K., Perverse sheaves and quivers, Duke Math. J. 83(3) (1996), 621–643.Google Scholar
Goresky, M. and MacPherson, R., Intersection homology ii, Inventiones Mathematicae 72(1) (1983), 77–129.CrossRefGoogle Scholar
Kapranov, M. and Schechtman, V., Perverse sheaves over real hyperplane arrangements, Ann. Math. (2) 183(2) (2016), 619–679.CrossRefGoogle Scholar
Koenig, S. and Yang, D., Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.Google Scholar
Krause, H., Krull-Schmidt categories and projective covers, Expo. Math. 33(4) (2015), 535–549.CrossRefGoogle Scholar
MacPherson, R., Intersection homology and perverse sheaves, Unpublished notes (1990).Google Scholar
MacPherson, R. and Vilonen, K., Elementary construction of perverse sheaves, Invent. Math. 84(2) (1986), 403–435.CrossRefGoogle Scholar
Mirollo, R. and Vilonen, K., Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves, Ann. Sci. École Norm. Sup. (4) 20(3) (1987), 311–323.CrossRefGoogle Scholar
Polishchuk, A., Perverse sheaves on a triangulated space, Math. Res. Lett. 4(2–3) (1997), 191–199.CrossRefGoogle Scholar
Reich, R., Notes on Belinson’s “How to glue perverse sheaves” [mr0923134], J. Singul. 1 (2010), 94–115.CrossRefGoogle Scholar
Siebenmann, L., Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163.CrossRefGoogle Scholar
Treumann, D., Exit paths and constructible stacks, Compos. Math. 145(6) (2009), 1504–1532.CrossRefGoogle Scholar
Vilonen, K., Perverse sheaves and finite dimensional algebras, Trans. Am. Math. Soc. 341(2) (1994), 665–676.Google Scholar
Vybornov, M., Constructible sheaves on simplicial complexes and Koszul duality, Math. Res. Lett. 5(5) (1998), 675–683.CrossRefGoogle Scholar
Vybornov, M., Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr{O}$ , Invent. Math. 167(1) (2007), 19–46.CrossRefGoogle Scholar