## Duke Mathematical Journal

### Bifunctor cohomology and cohomological finite generation for reductive groups

#### Abstract

Let $G$ be a reductive linear algebraic group over a field $k$. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. Invariant theory states that the ring of invariants $A^G=H^0(G,A)$ is finitely generated. We show that in fact the full cohomology ring $H^*(G,A)$ is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed in [22]. We also continue the study of bifunctor cohomology of $\Gamma^*(\mathfrak{gl}^{(1)})$

#### Article information

Source
Duke Math. J. Volume 151, Number 2 (2010), 251-278.

Dates
First available in Project Euclid: 14 January 2010

https://projecteuclid.org/euclid.dmj/1263478512

Digital Object Identifier
doi:10.1215/00127094-2009-065

Mathematical Reviews number (MathSciNet)
MR2598378

Zentralblatt MATH identifier
1196.20053

Subjects
Primary: 20G10: Cohomology theory

#### Citation

Touzé, Antoine; Van Der Kallen, Wilberd. Bifunctor cohomology and cohomological finite generation for reductive groups. Duke Math. J. 151 (2010), no. 2, 251--278. doi:10.1215/00127094-2009-065. https://projecteuclid.org/euclid.dmj/1263478512

#### References

• K. Akin, D. A. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), 207--278.
• D. J. Benson, Representations and Cohomology, II: Cohomology of Groups and Modules, 2nd ed., Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, Cambridge, 1998.
• H. Borsari and W. Ferrer Santos, Geometrically reductive Hopf algebras, J. Algebra 152 (1992), 65--77.
• H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, 1956.
• E. Cline, B. Parshall, L. Scott, and W. Van Der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143--163.
• L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224--239.
• Y. FéLix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Grad. Texts in Math. 205, Springer, New York, 2001.
• V. Franjou and E. M. Friedlander, Cohomology of bifunctors, Proc. Lond. Math. Soc. (3) 97 (2008), 514--544.
• E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209--270.
• F. D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math. 107 (1992), 127--133.
• —, Algebraic Homogeneous Spaces and Invariant Theory, Lecture Notes in Math. 1673, Springer, Berlin, 1997.
• J. C. Jantzen, Representations of Algebraic Groups, Math. Surveys Monogr. 107, Amer. Math. Soc., Providence, 2003.
• S. Mac Lane, Homology, reprint of 1975 edition, Classics Math., Springer, Berlin, 1995.
• W. S. Massey, Products in exact couples, Ann. of Math. (2) 59 (1954), 558--569.
• O. Mathieu, Filtrations of $G$-modules, Ann. Sci. École Norm. Sup. (4) 23 (1990), 625--644.
• M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3 (1963/1964), 369--377.
• E. Noether, Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik $p$, Nachr. Ges. Wiss. Göttingen (1926), 28--35.
• V. L. Popov, On Hilbert's theorem on invariants (in Russian), Dokl. Akad. Nauk SSSR 249 (1979), 551--555.; English translation in Soviet Math. Dokl. 20 (1979), 1318--1322.
• T. A. Springer, Invariant Theory, Lecture Notes in Math. 585, Springer, Berlin, 1977.
• V. Srinivas and W. Van Der Kallen, Finite Schur filtration dimension for modules over an algebra with Schur filtration, Transform. Groups 14 (2009), 695--711.
• A. Suslin, E. M. Friedlander, and C. P. Bendel, Infinitesimal $1$-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), 693--728.
• A. Touzé, Universal classes for algebraic groups, Duke Math. J. 151 (2010), 219--250.
• W. Van Der Kallen, Cohomology with Grosshans graded coefficients'' in Invariant Theory in All Characteristics (Kingston, Ontario, Canada), CRM Proc. Lecture Notes 35 (2004), Amer. Math. Soc., Providence, 2004, 127--138.
• —, A reductive group with finitely generated cohomology algebras'' in Algebraic Groups and Homogeneous Spaces (Mumbai, 2004), Tata Inst. Fund. Res. Studies in Math., Narosa, New Delhi, 2007.
• W. C. Waterhouse, Geometrically reductive affine group schemes, Arch. Math. (Basel) 62 (1994), 306--307.