Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$ when $i\neq j$.

In other words, $x_i^2$ does not vanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's syzygy theorem that can be doctored to apply to it?