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arXiv · 2306.14833

A connectedness theorem for spaces of valuation rings

William Heinzer·K. Alan Loper·Bruce Olberding·Matt Toeniskoetter
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Abstract

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$ of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally closed in $F$ if and only if there is a closed and connected subspace $Z$ of Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian.

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William Heinzer, K. Alan Loper, Bruce Olberding, Matt Toeniskoetter. 2023-06-26. A connectedness theorem for spaces of valuation rings. https://arxiv.org/abs/2306.14833

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