Purity of extremal rays of Betti cones
Let $R$ be a standard graded algebra over an infinite field $\mathsf k$, and let $\mathbb{B}_{\mathbb{Q}}(R)$ and $\mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$ denote the rational cones spanned by the Betti tables of all finitely generated $R$-modules and of those with pure resolutions, respectively. We establish several necessary conditions for the equality $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. When $\operatorname{edim}(R)\ge 2$, we prove that $\mathsf k$ has a pure resolution if and only if it has a linear resolution, and consequently, if the extremal rays of $\mathbb{B}_{\mathbb{Q}}(R)$ are pure, then $R$ is Koszul and good (in the sense of Roos). We show that if $R$ has depth zero, it must be Artinian for the equality of the two cones to hold. For rings with linear pairs of exact zerodivisors, we show that the equality of the cones implies that the $h$-polynomial has degree at most $2$, and use it to characterize generic Gorenstein Artin algebras satisfying $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. We also characterize algebras whose extremal rays are exactly the Betti tables of shifts of $R/\mathfrak m^j$ and of pure modules $M$ with $\operatorname{codim}(M)=\operatorname{pdim}(M)$: apart from polynomial rings, these are precisely Cohen--Macaulay algebras of dimension at most one with minimal multiplicity. In addition, we obtain a characterization of Cohen--Macaulay algebras of minimal multiplicity in terms of the extremal rays of the Betti cone of maximal Cohen--Macaulay modules.