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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 …

No. Let $G=SL_n$, acting on its defining representation $V$, with $n\geq2$. Let $R=\mathbb{Z}[X_1,\dots,X_n]$ be the obvious $\mathbb{Z}$-form of the ring of polynomial functions on $V$. Let $p$ be a …

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 …

Fix a prime $p$ and let $x$ be a $p$-adic integer that is not a rational number. Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is a $p$-adic integer. Then $M$ maps ont …

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 …

Yes. Both $R$ and $A=k+I$ are filtered by powers of $I$ and we may look at the associated graded rings. Now $I/I^2$ is a finitely generated $R/I$ module and $R/I$ is a finitely generated vector space, …

The answer is yes. Instead of showing that $M_n$ is finitely generated we may show it has the property that any ascending sequence of $A_0$-submodules stabilizes. If $N$ is an $A_0$-submodule of $M_n$ …

As Qing Liu explains there may be such nontrivial $e$. Suppose there was such an $e$. By Grothendieck's Generic Freeness Theorem, [Theorem 14.4 in David Eisenbud, Commutative algebra with a view tow …

When $B$ is étale over $C$ and $A$ or $B$ is finite over $C$, then the result is known by Theorem (2.4) in my paper Descent for the $K$-theory of polynomial rings.

Let $N$ be the $A$-module generated by $S$. Now $M$ is contained in $N+\mathfrak{m}M$, which is contained in $N+\mathfrak{m}(N+\mathfrak{m}M)$, hence in $N+\mathfrak{m}^2M$. Repeat.