Annals of K-Theory

Colocalising subcategories of modules over finite group schemes

Dave Benson, Srikanth Iyengar, Henning Krause, and Julia Pevtsova

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The Hom closed colocalising subcategories of the stable module category of a finite group scheme are classified. This complements the classification of the tensor closed localising subcategories in our previous work. Both classifications involve π-points in the sense of Friedlander and Pevtsova. We identify for each π-point an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding minimal colocalising subcategory.

Article information

Ann. K-Theory, Volume 2, Number 3 (2017), 387-408.

Received: 14 April 2016
Accepted: 29 August 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16G10: Representations of Artinian rings
Secondary: 18E30: Derived categories, triangulated categories 20C20: Modular representations and characters 20G10: Cohomology theory 20J06: Cohomology of groups

cosupport stable module category finite group scheme colocalising subcategory


Benson, Dave; Iyengar, Srikanth; Krause, Henning; Pevtsova, Julia. Colocalising subcategories of modules over finite group schemes. Ann. K-Theory 2 (2017), no. 3, 387--408. doi:10.2140/akt.2017.2.387.

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  • M. Auslander, “Functors and morphisms determined by objects”, pp. 1–244 in Representation theory of algebras (Philadelphia, 1976), edited by R. Gordon, Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978.
  • D. Benson and H. Krause, “Pure injectives and the spectrum of the cohomology ring of a finite group”, J. Reine Angew. Math. 542 (2002), 23–51.
  • D. Benson, S. B. Iyengar, and H. Krause, “Local cohomology and support for triangulated categories”, Ann. Sci. Éc. Norm. Supér. $(4)$ 41:4 (2008), 575–621.
  • D. Benson, S. B. Iyengar, and H. Krause, “Stratifying triangulated categories”, J. Topol. 4:3 (2011), 641–666.
  • D. J. Benson, S. B. Iyengar, and H. Krause, “Colocalizing subcategories and cosupport”, J. Reine Angew. Math. 673 (2012), 161–207.
  • D. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, “Stratification for module categories of finite group schemes”, preprint, 2016.
  • D. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, “Stratification and $\pi$-cosupport: finite groups”, Math. Z. (online publication February 2017).
  • E. H. Brown, Jr., “Abstract homotopy theory”, Trans. Amer. Math. Soc. 119 (1965), 79–85.
  • J. F. Carlson, “The varieties and the cohomology ring of a module”, J. Algebra 85:1 (1983), 104–143.
  • J. F. Carlson, Modules and group algebras, Birkhäuser, Basel, 1996.
  • H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, 1956.
  • E. Cline, B. Parshall, and L. Scott, “Finite-dimensional algebras and highest weight categories”, J. Reine Angew. Math. 391 (1988), 85–99.
  • W. Crawley-Boevey, “Tame algebras and generic modules”, Proc. London Math. Soc. $(3)$ 63:2 (1991), 241–265.
  • W. Crawley-Boevey, “Modules of finite length over their endomorphism rings”, pp. 127–184 in Representations of algebras and related topics (Kyoto, 1990), edited by H. Tachikawa and S. Brenner, London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press, 1992.
  • E. M. Friedlander and J. Pevtsova, “Representation-theoretic support spaces for finite group schemes”, Amer. J. Math. 127:2 (2005), 379–420. Correction in 128:4 (2006), 1067–1068.
  • E. M. Friedlander and J. Pevtsova, “$\Pi$-supports for modules for finite group schemes”, Duke Math. J. 139:2 (2007), 317–368.
  • D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge Univ. Press, 1988.
  • M. Hochster, “Prime ideal structure in commutative rings”, Trans. Amer. Math. Soc. 142 (1969), 43–60.
  • J. C. Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs 107, American Mathematical Society, Providence, RI, 2003.
  • H. Krause, “A short proof for Auslander's defect formula”, Linear Algebra Appl. 365 (2003), 267–270.
  • A. Neeman, “The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel”, Ann. Sci. École Norm. Sup. $(4)$ 25:5 (1992), 547–566.
  • A. Neeman, “The Grothendieck duality theorem via Bousfield's techniques and Brown representability”, J. Amer. Math. Soc. 9:1 (1996), 205–236.
  • A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton Univ. Press, 2001.
  • A. Neeman, “Colocalizing subcategories of $\mathbf D(R)$”, J. Reine Angew. Math. 653 (2011), 221–243.
  • A. Skowroński and K. Yamagata, Frobenius algebras, I: Basic representation theory, European Mathematical Society, Zürich, 2011.
  • A. Suslin, “Detection theorem for finite group schemes”, J. Pure Appl. Algebra 206:1–2 (2006), 189–221.
  • A. Suslin, E. M. Friedlander, and C. P. Bendel, “Support varieties for infinitesimal group schemes”, J. Amer. Math. Soc. 10:3 (1997), 729–759.
  • W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics 66, Springer, 1979.