The fixed subalgebra of a finitely generated algebra

Question

Let $k$ be a field, $A$ a finitely generated $k$-algebra, put $A^{G}:=\{a \in A \mid g(a)=a ~ \mbox{for all}~g \in G\}$, where $G$ is a finite group of automorphisms of $A$. If

(1) the order of $G$ is not divisible by the characteristic of $k$,

then $A^{G}$ is a finitely generated $k$-algebra.

I saw this statement in I. R. Safarevich's Basic Algebraic Geometry. But I don't know where (1) is used. In in Atiyah Macdonald's Commutative Algebra (exercise 5 of Chapter 7), Condition (1) is omitted.

I wonder what the correct statement is.

Answer 1

The condition on the characteristic is not necessary (nor is the condition that $k$ be a field, though it should be noetherian); see for instance Corollary 1.19 of http://people.fas.harvard.edu/~amathew/chgraded.pdf

Essentially, the point is that $A$ will be integral over $A^G$, and $A^G$ is "sandwiched" between $k, A^G, A$. From these facts, the result is not too complicated to prove.

The result that I only know under conditions on the characteristic is the strengthening to the case of $G$ a reductive algebraic group, acting (algebraically) on a finitely generated $k$-algebra: then $k$ is required to be of characteristic zero. The reason is that the proof uses semisimplicity of the category of $G$-representations, which is only true in characteristic zero.

Answer 2

Indeed the result is true with no restrictions on the characteristic: see $\S 14.6$, Theorem 339 of my commutative algebra notes. As I mention there, the result was first proved by Hilbert in characteristic zero and then in 1928 by Emmy Noether in arbitrary characteristic.

Answer 3

As pointed out above, we don't need $k$ to be a field; nor do we need $|G|$ to be invertible in $k$. Let $A=k[a_1,\ldots, a_m]$, and notice $a_i$ satisfies the polynomial $\prod_{g\in G}(X-g(a_i))\in A^G[X]$. Taking all of these coefficients, as $i$ varies from 1 to $m$, gives us $m|G|$ elements which generate a $k$-algebra $Y$. Clearly $Y\subset A^G$ and $Y$ is finitely generated as a $k$-algebra, thus by Hilbert's Basis Theorem, $Y$ is Noetherian. We have $k\subset Y\subset A^G\subset A$.

Now, $A$ is a finitely generated integral $Y$-algebra, because all the $a_i$ are integral over $Y$, and is thus a finitely generated module over $Y$. (This is a general fact that can be easily proven by induction on the number of generators needed to display $A$ as a $Y$-algebra). Since $Y$ is Noetherian this implies $A^G$ is also a finitely generated $Y$-module.

Finally, $A^G$ is a finitely generated module over a finitely generated $k$-algebra and is thus itself a finitely generated $k$-algebra, as desired.