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They must be extremely nice, as the projection of the cusp $y^2=x^3$ onto the $x$-axis is already a counterexample. You want the varieties normal?

The answer to the first question is positive. One does not need normality of the generic fiber. By the universal coefficient theorem [Prop I 4.18 (a) in Jantzen Representations of algebraic groups], …

No. Take for $S$ the ring of real power series in $t$ for which the coefficients of $t$ and $t^3$ both vanish. Take $f:=X^2-t^6$, $g:=t^5+t^2X$. We must check that there is no $h$ so that $gh$ is …

Let $k$ be a commutative ring and let $G$ be a flat affine algebraic group scheme over $k$. Let $G$ act by algebra automorphisms on the commutative $k$-algebra $A$. So $G(R)$ acts by $R$-algebra auto …

In positive characteristic the only connected groups of finite homological dimension are the tori. We need the following result from Jantzen, Representations of algebraic groups. [J, I 5.13], [J, I …

We offer two facts and a Theorem. Let $S$ be a commutative noetherian ring containing $\mathbb Z$ and let $G=G_S$ be reductive over $S$ in the sense of SGA3. That is, $G$ is smooth over $S$ with …

It is true. The standard reference is the Book by Jantzen, Representations of Algebraic Groups, Second edition. In particular we need the Appendix `Chapter B', and the base change Proposition in part …

The linkage bound is 2. If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma. Lemma. If $V$, $W$ are finite dimensional, there is an $m$ dependin …