Global dimension of real-exponent polynomial rings

The ring $R$ of real-exponent polynomials in $n$ variables over any field has global dimension $n+1$ and flat dimension $n$. In particular, the residue field $k<!--FUNCTION\; APPLICATION-->=R∕\mathfrak{𝔪}$ of $R$ modulo its maximal graded ideal $\mathfrak{𝔪}$ has flat dimension $n$ via a Koszul-like resolution. Projective and flat resolutions of all $R$-modules are constructed from this resolution of $k<!--FUNCTION\; APPLICATION-->$. The same results hold when $R$ is replaced by the monoid algebra for the positive cone of any subgroup of ${ℝ}^n$ satisfying a mild density condition.