  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 =
R/m of R modulo its maximal graded ideal 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. The same results hold when R is replaced
by the monoid algebra for the positive cone of any subgroup of $\mathbb{R}^n$
satisfying a mild density condition.
