Suppose $V, W, U$ are $Z_p$ module over a field $F$ of characteristic $p$ and $V=W \oplus U$. Is there a degree preserving surjective map from $F[V]^{Z_p}$ to $F[W]^{Z_p}$ ? In non-modular case the Reynold's operator does the job but in this case I don't have any clue.
1 Answer
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That holds more generally: If $k$ is any commutative ring with unit, $G$ any group and if $U,W$ are $k$-free $kG$-modules, $V = U \oplus W$, then there is a $k$-linear surjection $k[V]^G \to k[W]^G$:
Let $r: V \to W$ be the projection. Then $k[V]=k[W] \otimes k[U]$ and $r$ induces a $k$-algebra hom. $r: k[V] \mapsto k[W]$. Aussume for the moment we know that $r$ is compatible with the $G$ action, i.e. $r ( g \cdot v) = g \cdot r(v)$. Then $r$ preserves the invariants: $r: k[V]^G \to k[W]^G$.
Moreover, $r$ is surjective on the invariants. For, let $w \in k[W]^G$ and put $v := w \otimes 1$. Then $g \cdot v = (g\cdot w)\otimes (g\cdot 1) = w \otimes 1 = v$, i.e. $v \in k[V]^G$ and $r(v) = w$.
Now let's show that $r$ is compatible with the $G$-action. Note that we have an $k$-algebra homomorphism $\epsilon: k[U] \to k$ that satisfies $\epsilon(g \cdot u) = \epsilon(u)$: If $\lbrace x_1,...,x_n\rbrace$ is a $k$-basis of $U$ then $\epsilon$ is just the augmentation $k[x_1,...,x_n] \to k$.
Now $r$ is given by $r(w \otimes u) = \epsilon(u) \cdot w$ and we obtain $$r(g (w \otimes u)) = r(g w \otimes g u) = \epsilon(g u) (g w) =\epsilon(u) (gw) = g (\epsilon(u)w) = g\; r(w \otimes u).$$
BTW: To my understanding, the Reynold's operator yields $k[V]^H \to k[V]^G$ for $H \le G$ and adresses another problem than yours.
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$\begingroup$ Added: $r$ is degree-preserving. For let $u,w$ be monomials such that $r(u \otimes w) \neq 0$. Then $\deg u = 0$ and $\deg r(u \otimes w) = \deg(u \cdot w) = \deg w = \deg w + \deg u = \deg(w \otimes u)$. $\endgroup$– RalphMar 19, 2012 at 15:50
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$\begingroup$ @Ralph Your argument shows, still more generally, that if $U$ is a submodule of $V$ with quotient module $V/U \cong W$ then there is a $k$-algebra homomorphism $k[V] \rightarrow k[W]$ that preserves degrees and invariants. Is there any characterization of when the induced map $k[V]^G \rightarrow k[W]^G$ is surjective? (One small typo: the Reynolds averaging operator sends $k[V]^H$ to $k[V]^G$ for $H \le G$.) $\endgroup$ Mar 19, 2012 at 19:44
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$\begingroup$ I don't know, if there is such a characterization ($W$ $k$-free is sufficient, probably also $W$ $k$-projective). And thanks for pointing out the correct order of the Reynolds operator. $\endgroup$– RalphMar 20, 2012 at 1:45