General linear and functor cohomology over finite fields.

*(English)*Zbl 0952.20035This paper contains many definite results concerning the cohomology of families of representations of \(\text{GL}_n\) over a finite field \(\mathbb{F}_q\). A representation like the module \(\wedge^2 \mathbb{F}_q^n\) for \(\text{GL}_n(\mathbb{F}_q)\) makes sense for every \(n\). Such a representation ‘for all \(n\) simultaneously’ is an object of the category \(\mathcal F\), or \(\mathcal F(\mathbb{F}_q)\), of all functors from finite-dimensional \(\mathbb{F}_q\) vector spaces to \(\mathbb{F}_q\) vector spaces, studied extensively by Franjou, Lannes and Schwarz. Similarly we have the category \(\mathcal P\), or \(\mathcal P(\mathbb{F}_q)\), of ‘strict polynomial functors’ introduced by E. M. Friedlander and A. Suslin [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)]. An object of \(\mathcal P\) encodes the idea of having a polynomial representation of \(\text{GL}_n\) for all \(n\) simultaneously.

In the appendix Suslin shows that \(\text{Ext}^s_{\mathcal F(\mathbb{F}_q)}(P,Q)\) can be understood as the limit for \(n\to\infty\) of the \(\text{Ext}^s_{\text{GL}_n(\mathbb{F}_q)}(P(\mathbb{F}_q^n),Q(\mathbb{F}_q^n))\), if \(P\) and \(Q\) are functors of finite Eilenberg-MacLane degree that take values in finite-dimensional vector spaces.

In the main text the authors show that the category \(\mathcal P\) has great advantages for the computation of Ext groups. Indeed they determine many cohomology rings explicitly. If \(M^{(r)}\) denotes the \(r\)-th Frobenius twist of the representation \(M\), then they compute for instance the trigraded Hopf algebra \(\text{Ext}^*_{\mathcal P}(\Gamma^{*(j)},S^{*(r)})\) for \(0\leq j\), \(0\leq r\). Here it should be mentioned that the divided power functors \(\Gamma^m\) are projectives in \(\mathcal P\) and the symmetric power functors \(S^m\) are injectives.

Having obtained nearly total control over the homological algebra of \(\mathcal P\) they can then use it to get similar information in \(\mathcal F\). For this they develop a theory of comparison between \(\text{Ext}_{\mathcal P}\) and \(\text{Ext}_{\mathcal F}\), analogous to the main result of E. Cline, B. Parshall, L. Scott and W. van der Kallen [Invent. Math. 39, 143-163 (1977; Zbl 0346.20031)]. Such a comparison was independently achieved by N. J. Kuhn [Am. J. Math. 120, No. 6, 1317-1341 (1998; Zbl 0918.20035)]. But the results of the authors are much stronger and much more explicit than in these works.

In the appendix Suslin shows that \(\text{Ext}^s_{\mathcal F(\mathbb{F}_q)}(P,Q)\) can be understood as the limit for \(n\to\infty\) of the \(\text{Ext}^s_{\text{GL}_n(\mathbb{F}_q)}(P(\mathbb{F}_q^n),Q(\mathbb{F}_q^n))\), if \(P\) and \(Q\) are functors of finite Eilenberg-MacLane degree that take values in finite-dimensional vector spaces.

In the main text the authors show that the category \(\mathcal P\) has great advantages for the computation of Ext groups. Indeed they determine many cohomology rings explicitly. If \(M^{(r)}\) denotes the \(r\)-th Frobenius twist of the representation \(M\), then they compute for instance the trigraded Hopf algebra \(\text{Ext}^*_{\mathcal P}(\Gamma^{*(j)},S^{*(r)})\) for \(0\leq j\), \(0\leq r\). Here it should be mentioned that the divided power functors \(\Gamma^m\) are projectives in \(\mathcal P\) and the symmetric power functors \(S^m\) are injectives.

Having obtained nearly total control over the homological algebra of \(\mathcal P\) they can then use it to get similar information in \(\mathcal F\). For this they develop a theory of comparison between \(\text{Ext}_{\mathcal P}\) and \(\text{Ext}_{\mathcal F}\), analogous to the main result of E. Cline, B. Parshall, L. Scott and W. van der Kallen [Invent. Math. 39, 143-163 (1977; Zbl 0346.20031)]. Such a comparison was independently achieved by N. J. Kuhn [Am. J. Math. 120, No. 6, 1317-1341 (1998; Zbl 0918.20035)]. But the results of the authors are much stronger and much more explicit than in these works.

Reviewer: Wilberd van der Kallen (Utrecht)

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

18G05 | Projectives and injectives (category-theoretic aspects) |

18A22 | Special properties of functors (faithful, full, etc.) |

20J05 | Homological methods in group theory |

20G40 | Linear algebraic groups over finite fields |

14L15 | Group schemes |

18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |