2
votes
1answer
154 views

Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$

Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$, $$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$ That is, the ...
2
votes
2answers
59 views

Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$

Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$. It is clear that $$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$ ...
1
vote
0answers
61 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...