Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply‐connected, semisimple algebraic group over k , an algebraically closed field of characteristic zero. Let Oε[ G ] be the quantized function algebra of G at a primitive l th root of unity ε, and letO ϵ [ G ] ¯ be the ‘restricted’ quantized function algebra at ε, a finite‐dimensional k ‐algebra obtained from O ε [ G ] by factoring out a centrally generated ideal. It is known thatO ϵ [ G ] ¯ is a Hopf algebra. We study the cohomology ring ExtO ϵ [ G ] ¯ ∗ ( k , k ) , a graded commutative algebra, and, for any finite‐dimensionalO ϵ [ G ] ¯ ‐module M , the ExtO ϵ [ G ] ¯ ∗ ( k , k ) ‐module ExtO ϵ [ G ] ¯ ∗ ( k , M ) . We prove that for sufficiently large l there is an isomorphism of graded algebrasExtO ϵ [ G ] ¯ ∗ ( k , k ) ≅ k [ X 1 , … , X 2 N ] , where each X i is homogeneous of degree 2 , and 2 N equals the number of roots associated to G . Moreover we show that in this case ExtO ϵ [ G ] ¯ ∗ ( k , M ) is a finitely generated ExtO ϵ [ G ] ¯ ∗ ( k , k ) ‐module. We also show under much less restrictive conditions on l that ExtO ϵ [ G ] ¯ ∗ ( k , k ) continues to be a finitely generated graded commutative algebra over which ExtO ϵ [ G ] ¯ ∗ ( k , M ) is a finitely generated module. 1991 Mathematics Subject Classification : 16W30, 17B37, 17B56.