Finite twisted resolutions over a regular local base

Given: A regular Noetherian local ring $R$ of finite dimension $d$, the projective space $P={\mathbb{P}}_R^N$ with $S=R[x_0,\dots ,x_N]$, and a coherent sheaf $F$ on $P$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

lem-graded-section-module-finite-projective. Assume the Axiom of Choice as inherited from the cited suppliers (The Axiom of Choice). Let $A$ be a Noetherian commutative ring with $1$ (def-noetherian-ring-and-module), let $n\ge 0$, write $S=A[x_0,\dots ,x_n]$ for the polynomial ring in the total-degree grading (def-polynomial-ring-on-a-family-of-indeterminates), let $X={\mathbb{P}}_A^n$ with twisting sheaves (High-degree section module is finite graded)

lem-extend-sections-from-nonvanishing-open. Assume the Axiom of Choice as inherited from the affine quasi-coherence equivalence and the associated-sheaf construction (The Axiom of Choice). (Extend a quasi-coherent section after multiplying by a power)

thm-localisation-and-polynomial-extension-of-regular-rings. Assume the Axiom of Choice (The Axiom of Choice). Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, $\mathrm{gldim}R=\dim R$, allowing infinity. (localisation and polynomial extension of regular rings)

thm-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. (Projective dimension at most n iff the nth syzygy is projective)

thm-nakayama-lemma. Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I⊴R$ satisfy $I\subseteq J(R)$, and let $M$ be a finitely generated left $R$-module. If $IM=M$, then $M=0$. (Assuming the Axiom of Choice, Nakayama's lemma)

thm-exactness-of-sheaves-stalkwise. Let $\cdots ⟶{\mathcal{F}}_{i-1}\stackrel{d_{i-1}}{\to }{\mathcal{F}}_i\stackrel{d_i}{\to }{\mathcal{F}}_{i+1}⟶\cdots$ be a sequence of sheaves of abelian groups on a topological space $X$. (A sequence of abelian sheaves is exact exactly when it is exact on every stalk)

thm-affine-quasi-coherent-equivalence. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative ring with $1$ and put $X=\mathrm{Spec}A$. Let ${\mathrm{Mod}}_A$ be the category of $A$-modules and $\mathrm{QCoh}(X)$ the full subcategory of ${\mathcal{O}}_X$-modules consisting of the quasi-coherent ones (def-quasi-coherent-module-scheme). (Affine quasi-coherent sheaves are modules)

thm-dimension-of-a-polynomial-ring-over-a-noetherian-ring. Assume the Axiom of Choice. Let $R$ be a Noetherian commutative ring of finite Krull dimension. Then $\dim R[x]=\dim R+1.$ (A Noetherian polynomial ring has dimension one larger)