localisation and polynomial extension of regular rings

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

localisations of regular local rings are regular: Every prime localization $R_{\mathfrak{p}}$ of a regular local ring $R$ is regular, and $\mathrm{edim}{R}_{\mathfrak{p}}=\mathrm{ht}\mathfrak{p}$.

regular noetherian ring: A commutative Noetherian ring $R$ is regular if for every prime ideal $\mathfrak{p}$, the local ring $R_{\mathfrak{p}}$ is regular local. This includes the zero ring vacuously. The maximal-localization test is proved in the localization and polynomial-extension theorem.

flat local ascent of regularity: For a flat local map $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of nonzero Noetherian local rings: if $R$ and $S\mathrm{/}\mathfrak{m}S$ are regular, then $S$ is regular. Conversely, regularity of $S$ implies regularity of $R$; it need not imply regularity of the closed fibre.

polynomial local regularity fibre step: For a prime $\mathfrak{q}\subset R[t]$ with contraction $\mathfrak{p}\subset R$, the closed fibre of $R_{\mathfrak{p}}\to R[t{]}_{\mathfrak{q}}$ is $k(\mathfrak{p})[t]$ localized at a prime. That prime is either zero, giving a field, or generated by an irreducible polynomial, giving a DVR. In both cases the fibre is regular.

If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$: Let $R$ be a Noetherian commutative ring. Then the iterated polynomial ring $R[x_1,\dots ,x_n]$ of def-multivariate-polynomial-ring-by-iteration is Noetherian for every $n\in \mathbb{N}$. The index starts at $0$, where the published definition sets $R[x_1,\dots ,x_0]=R$ and the assertion is the hypothesis itself.

Localisation of modules is exact: If $0⟶M^{\mathrm{\prime }}\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{\mathrm{\prime }\mathrm{\prime }}⟶0$ is a short exact sequence of $R$-modules, then $0⟶S^{-1}{M}^{\mathrm{\prime }}\stackrel{S^{-1}f}{\to }{S}^{-1}M\stackrel{S^{-1}g}{\to }{S}^{-1}{M}^{\mathrm{\prime }\mathrm{\prime }}⟶0$ is a short exact sequence of $S^{-1}R$-modules.

A finite flat module over a Noetherian ring is finite projective: Let $R$ be a Noetherian commutative ring and let $M$ be a finite flat $R$-module. Then $M$ is finite projective.

A module is flat if and only if all prime localizations are flat, equivalently all maximal localizations are flat: Let $R$ be a commutative ring and let $M$ be an $R$-module. The following are equivalent: 1. $M$ is flat over $R$. 2. $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}\subset R$. 3. $M_{\mathfrak{m}}$ is flat over $R_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}\subset R$.

Projective dimension at most n iff the nth syzygy is projective: Let $\mathcal{A}$ be an abelian category with enough projectives, fix a projective resolution $P_{\bullet }\to M$, and let $n\ge 1$. Then $\mathrm{pd}(M)\le n⟺{\mathrm{\Omega }}_P^n(M)\text{ is projective}\mathrm{.}$ In particular, the condition is independent of the chosen projective resolution.

global dimension is detected on cyclic modules: For a unital ring $R$, its left global dimension equals ${\sup }_I{\mathrm{pd}}_R(R\mathrm{/}I)$ over all left ideals $I$, and equals the supremum of the injective dimensions of all left modules. The equalities allow infinity; in the commutative Noetherian case the cyclic modules are finite.

Localizing a Dedekind domain at a nonzero prime gives a DVR: Let $R$ be a Dedekind domain and let $\mathfrak{p}\subset R$ be a nonzero prime ideal. Then $R_{\mathfrak{p}}$ is a discrete valuation ring.

Projective left and right modules are flat over an arbitrary ring: Every projective left or right module over an arbitrary ring is flat on its appropriate side.

auslander buchsbaum serre regularity criterion: For a nonzero Noetherian local ring $(R,\mathfrak{m},k)$ the following are equivalent: $R$ is regular; ${\mathrm{pd}}_{R}k<\mathrm{\infty }$; $\mathrm{gldim}R<\mathrm{\infty }$; and every finite $R$-module has finite projective dimension. When these hold, $\mathrm{gldim}R={\mathrm{pd}}_{R}k=\dim R$. A nonzero finite module over regular local $R$ is maximal Cohen–Macaulay (depth $\dim R$) if and only if it is free.

Every quotient and every localisation of a Noetherian ring is Noetherian: Every localization of a Noetherian commutative ring is Noetherian, including the zero localization.