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Post Undeleted by Hugo Chapdelaine
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Hugo Chapdelaine
Hugo Chapdelaine
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Algebraic Commutative algebraic groups endowed with a ring action

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of abelian k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rightarrow 1 $$ where $H$ is an abelian variety and $K$ is a (commutative) linear algebaic group. Assume moreover (if it helps) that $$ 1\rightarrow Lie(K)\rightarrow Lie(G) \rightarrow Lie(H)\rightarrow 1 $$ is exact. Let $R$ be a ring and assume that we have an injection $\rho:R\rightarrow End_k(G)$.

Q: Does $\rho$ necessarily restrict to $K$?

Algebraic groups endowed with a ring action

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of abelian k-algebraic connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rightarrow 1 $$ where $H$ is an abelian variety and $K$ is a (commutative) linear algebaic group. Assume moreover (if it helps) that $$ 1\rightarrow Lie(K)\rightarrow Lie(G) \rightarrow Lie(H)\rightarrow 1 $$ is exact. Let $R$ be a ring and assume that we have an injection $\rho:R\rightarrow End_k(G)$.

Q: Does $\rho$ necessarily restrict to $K$?

Commutative algebraic groups endowed with a ring action

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rightarrow 1 $$ where $H$ is an abelian variety and $K$ is a (commutative) linear algebaic group. Assume moreover (if it helps) that $$ 1\rightarrow Lie(K)\rightarrow Lie(G) \rightarrow Lie(H)\rightarrow 1 $$ is exact. Let $R$ be a ring and assume that we have an injection $\rho:R\rightarrow End_k(G)$.

Q: Does $\rho$ necessarily restrict to $K$?

Source Link
Hugo Chapdelaine
Hugo Chapdelaine
  • 7.6k
  • 2
  • 28
  • 70

Algebraic groups endowed with a ring action

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of abelian k-algebraic connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rightarrow 1 $$ where $H$ is an abelian variety and $K$ is a (commutative) linear algebaic group. Assume moreover (if it helps) that $$ 1\rightarrow Lie(K)\rightarrow Lie(G) \rightarrow Lie(H)\rightarrow 1 $$ is exact. Let $R$ be a ring and assume that we have an injection $\rho:R\rightarrow End_k(G)$.

Q: Does $\rho$ necessarily restrict to $K$?