flat local ascent of regularity

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

regular local rings are domains and cohen macaulay: A regular local ring $R$ of dimension $d$ is a domain and Cohen–Macaulay. For every regular system $(x_1,\dots ,x_d)$, the tuple is $R$-regular and $R\mathrm{/}(x_1,\dots ,x_c)$ is regular local of dimension $d-c$ for all $0\le c\le d$.

quotient and lifting regularity across a regular element: Let $(R,\mathfrak{m})$ be nonzero Noetherian local. If $x\in \mathfrak{m}$ is a nonzerodivisor and $R\mathrm{/}(x)$ is regular, then $R$ is regular and $x\mathrm{\notin }{\mathfrak{m}}^2$. For every nonzerodivisor $x\in \mathfrak{m}$, $\dim (R\mathrm{/}(x))=\dim R-1$. If $R$ is regular and $0\mathrm{\ne }x\in \mathfrak{m}$, then $R\mathrm{/}(x)$ is regular if and only if $x\mathrm{\notin }{\mathfrak{m}}^2$.

Flat and faithfully flat modules and ring homomorphisms: Let $R$ be a commutative ring and let $M$ be an $R$-module. The module $M$ is flat if the functor $-{\otimes }_{R}M$ preserves exact sequences: whenever $A\to B\to C$ is exact, so is $A{\otimes }_{R}M⟶B{\otimes }_{R}M⟶C{\otimes }_{R}M\mathrm{.}$ Since tensoring is always right exact (thm-right-exactness-of-tensor-products, def-exact-and-short-exact-sequences-of-modules), the definition asks for the remaining left-hand exactness. Its equivalent formulation as preservation of injections is proved separately rather than built into the definition. The module $M$ is faithfully flat if a sequence of $R$-modules is exact exactly when its tensor with $M$ is exact. For a unital ring homomorphism $f:R\to S$ (def-ring-homomorphism) between commutative rings, $S$ is an $R$-module by $r\cdot s=f(r)s$. The map $f$ is flat, respectively faithfully flat, when this $R$-module is flat, respectively faithfully flat.

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.

finite local modules admit minimal free resolutions: Every finite module $M$ over a nonzero Noetherian local ring $(R,\mathfrak{m},k)$ has an augmented resolution $\cdots \to F_1\to F_0\to M\to 0$ by finite-rank free modules, with $d_i(F_i)\subseteq \mathfrak{m}{F}_{i-1}$ for $i>0$. Such a resolution is called minimal; it need not be bounded. This extends the bounded terminology without changing it.

projective dimension from last nonzero betti number: For a nonzero finite module $M$ over a nonzero Noetherian local ring, ${\mathrm{pd}}_{R}M=\sup \{i\ge 0:{\beta }_i^R(M)\mathrm{\ne }0\}$, allowing infinity. For each integer $q\ge 0$, ${\mathrm{pd}}_{R}M\le q$ if and only if ${\mathrm{Tor}}_{q+1}^R(k,M)=0$.