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.

中文翻译：

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实指数多项式环的全局维数

戒指$右$实指数多项式的$n$任何字段上的变量都具有全局维度$n+1$和平面尺寸$n$。特别是残渣场$k<!--FUNCTION\; APPLICATION-->=右∕\mathfrak{𝔪}$的$右$以其最大分级理想为模$\mathfrak{𝔪}$具有平面尺寸$n$通过类似科苏尔的决议。所有投影和平面分辨率$右$-模块是根据这个分辨率构建的$k<!--FUNCTION\; APPLICATION-->$。当$右$被幺半群代数取代，用于任何子群的正圆锥${ℝ}^n$满足温和的密度条件。