Crossposted on MathOverflow.
The MathOverflow version of the question has been rewritten. For the sake of completeness, I pasted it here in a condensed form. I also deleted the old version.
Let $K$ be a commutative ring, and let $X_1,\dots,X_n$ be indeterminates. Here $n$ is an integer $\ge3$. For $1\le i < j\le n$ put $$ x_{ij}:=\frac{1}{X_i-X_j} $$ and let $Y_{ij}$ be an indeterminate. Let $I$ be the kernel of the $K$-algebra morphism $$ \varepsilon:K[(Y_{ij})]\to K[(x_{ij})],\quad Y_{ij}\mapsto x_{ij}. $$
Is $I$ finitely generated? If it is, can one give an explicit finite set of generators?
