Rational cohomology of an algebra need not 9 be detected by Frobenius kernels 0 0 2 n Wilberd van der Kallen a J 0 2 Abstract ] We record some cohomological computations in an example of T Friedrich Knop. Theexampleisapolynomialalgebra incharacteristic R two with an unusual SL action. . 2 h t a m 1 The example [ 1 In [4, section 5] Friedrich Knop gave an example of a transitive action of v the algebraic group SL on the afine plane A2 in characteristic two. At the 5 2 0 2002 CRM Workshop on Invariant Theory, cf. [5], he also explained that the 0 example has noteworthy properties in connection with Grosshans grading. 3 In this note we look at the rational cohomology and see that it is equally . 1 instructive. Thus consider the algebraic group G = SL defined over a field 0 2 9 k of characteristic two. The diagonal subgroup is T, the unipotent upper 0 triangular subgroup is U. Let N be the normalizer of T in G. The example : v is then G/N ∼= A2. As Knop observed, k[G/N] ֒→ k[G/T] is separable, i X k[G/T] has good filtration and k[G/N]U = k[G/T]U. Recall from [5, 2.3] r that the Grosshans graded grk[G/T], known as the ‘hull’ of grk[G/N], is a a purelyinseparableextensionofgrk[G/N]. Soinsomesensetheinseparability is a property of the Grosshans filtration, not of the ring extension. We will give cohomology computations that amplify these observations. a b We write the general matrix in G as , so that (cid:18)c d(cid:19) k[G] = k[a,b,c,d]/(ad−bc−1). Recall that k[U\G] = k[c,d], a polynomial ring in two variables. Here by k[U\G] we mean the ring of rational functions on G that are invariant under 1 left translation by elements of U. (One may check k[U\G] = k[c,d] using the U multiplicities in a good filtration of k[G] as G×G module.) Thus k[G/T] = T N U k[U\G] = k[cd] = k[U\G] = k[G/N] , a polynomial ring in one variable, written cd to indicate its image in k[G]. Note that ad ∈ k[G/T] satifies ad(ad−1) = abcd ∈ k[G/N], while no power (ad)2r is in k[G/N], because the 0 1 involution σ = ∈ N interchanges ad with bc, but (ad)2r−(bc)2r = 1. (cid:18)1 0(cid:19) It is easy to see that k[G/T] is generated by ad, ab, cd. Using the involution σ again, one sees that k[G/T] is a free k[G/N]-module with basis 1, ad. Thus k[G/N] ֒→ k[G/T] is separable, x2 − x − (ab)(cd) being the minimal polynomial of ad. One also finds that k[G/N] is a polynomial ring in two variables, called ab and cd. One may find a basis of k[G/T] and make its (good) Grosshans filtration explicit as a check of the above. TheGrosshansfiltrationofk[G/T]startswiththespanof1,thenthespan of 1, ab, ad, cd. Intersecting with k[G/N] one gets the span of 1 and the span of 1, ab, cd respectively. So the class of ad is an invariant in k[G/T]/k[G/N] and this defines an extension of k by k[G/N] that does not split because k[G/N]G = k[G/T]G = k. But ad − (ad)2r ∈ k[G/N] for all r ≥ 1, so for every such r the extension splits as an extension of modules for the Frobenius kernel Gr, with (ad)2r ∈ k[G/T]Gr as lift of the class of ad. We have shown: Proposition 1 H1(G,k[G/N]) is not the inverse limit of the H1(Gr,k[G/N]). Thisisincontrast withwhatoneknows forfinite dimensionalrepresentations [3, II 4.12]. ∗ One may go a little further and compute both H (G,k[G/N]) and ∗ H (Gr,k[G/N]). As k[G/T]/k[G/N] is a free k[G/N] module on one gener- ator, we get an the extension E : 0 → k[G/N] → k[G/T] → k[G/N] → 0. Further k[G/T] is a direct summand of the injective module k[G], so it is i easy to compute that H (G,k[G/N]) = k for all i ≥ 0. More specifically, if i f denotes the inclusion k ֒→ k[G/N], then H (G,k[G/N]) = k is spanned for i ≥ 1 by the Yoneda product of f and i copies of E. So, by [2, 3.2], ∗ H (G,k[G/N]) is a polynomial ring in one variable of degree one. Let r ≥ 1. Now k[G] is also Gr injective, by [3, I 4.12, 5.13]. But i E splits over Gr, so one gets H (Gr,k[G/N]) = 0 for i > 0. So the Gr 2 detect none of the higher cohomology of k[G/N]. One may also check that H0(Gr,k[G/N])(−r) ∼= k[G/N]. Of course H0(G,k[G/N]) = k is the inverse limit of the H0(Gr,k[G/N]). ∗ We leave it to the reader to compute the nontrivial H (G ,gr k[G/N]). 1 (Use [1, 3.10].) References [1] H. H. Andersen, J.-C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525. [2] D. J. Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition. Cam- bridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1998. [3] Jens Carsten Jantzen, Representations of algebraic groups. Second edi- tion. Mathematical Surveys and Monographs, 107. American Mathe- matical Society, Providence, RI, 2003. [4] Friedrich Knop, Homogeneous varieties for semisimple groups of rank one. Compositio Mathematica, 98 (1995), 77–89. [5] Wilberd van der Kallen, Cohomology with Grosshans graded coeffi- cients, In: Invariant Theory in All Characteristics, Edited by: H. E. A. Eddy Campbell and David L. Wehlau, CRM Proceedings and Lec- ture Notes, Volume 35 (2004) 127-138, Amer. Math. Soc., Providence, RI, 2004. 3