The Second Cohomology of Small Irreducible Modules for Simple Algebraic Groups
نویسنده
چکیده
Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p > 0, and let V be a rational G-module such that dim V ≤ p. According to a result of Jantzen, V is completely reducible, and H(G, V ) = 0. In this paper we show that H(G, V ) = 0 unless some composition factor of V is a non-trivial Frobenius twist of the adjoint representation of G.
منابع مشابه
The second cohomology of simple SL2-modules
Let G be the simple algebraic group SL2 defined over an algebraically closed field K of characteristic p > 0. In this paper, we compute the second cohomology of all irreducible representations of G.
متن کاملThe second cohomology of simple SL3-modules
Let G be the simple, simply connected algebraic group SL3 defined over an algebraically closed field K of characteristic p > 0. In this paper, we find H(G, V ) for any irreducible G-module V .
متن کاملOn the Associated Primes of the generalized $d$-Local Cohomology Modules
The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary generalized local cohomology modules. Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$ are finitely generated $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)...
متن کاملThe Cohomology of Small Irreducible Modules for Simple Algebraic Groups
LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebra...
متن کاملUPPER BOUNDS FOR FINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES
Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properti...
متن کامل