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This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and daunti …

This is a continuation of various questions about Chevalley groups over rings, cf these two questions (and a rather bad question of mine here). Consider a semisimple Lie algebra $\mathfrak g$ over $\m …

Background: Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be …

BACKGROUND Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity …

NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague. The first question is: Has anyone developed a theory of Frobenius splitting for ind-vari …

Background Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be …

Setup Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the positi …

Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ t_1 …

I don't have a complete answer, but let me just note that your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diag …

Let $G$ be an algebraic group over an algebraically closed field $k$ of arbitrary characteristic and let $U$ be the hyperalgebra of $G$. Recall that $U$ is defined as the subspace of the full linear d …

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a para …

Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular affi …

Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel subgro …

Let $G$ be a semisimple algebraic group over an algebraically closed field of positive characteristic $p$ and let $B \subseteq G$ be a Borel subgroup. Set $X := G/B$, the flag variety of $G$. Also let …

Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $ …