Relative derived duality on projective space over a regular local ring

Given: A regular Noetherian local ring $R$ of finite dimension, the projective space $P={\mathbb{P}}_R^N$ with $W={\mathcal{O}}_P(-N-1)$, the residue $t$ on the all-minus-one Laurent monomial, and $K\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(P)$.

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-relative-projective-space-regular-local-base-twisted-resolution. Assume AC and DC. Let $R$ be a regular Noetherian local ring of finite dimension $d$, $P={\mathbb{P}}_R^N$, and $F$ a coherent sheaf on $P$. There is a finite resolution of $F$ by finite direct sums of twists ${\mathcal{O}}_P(m)$, of length at most $d+N+1$. Every bounded coherent complex on $P$ is perfect and belongs to the triangulated subcategory generated by these twists. (Finite twisted resolutions over a regular local base)

thm-cohomology-projective-space-twisting-sheaves. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative ring with $1$ (def-commutative-ring), let $n\ge 0$, let $d\in \mathbb{Z}$ and let $X={\mathbb{P}}_A^n\cong \mathrm{Proj}A[x_0,\dots ,x_n]$ be relative projective space (def-relative-projective-space-standard-charts, def-polynomial-ring-on-a-family-of-indeterminates), with twisting sheaf (Cohomology of O(d) on projective space)

def-cup-product-sheaf-cohomology. Assume the Axiom of Choice (The Axiom of Choice) and the standing smallness or supplied cofinal-denominator hypothesis of def-derived-category-of-an-abelian-category. (Cup product in sheaf cohomology)

thm-cech-computes-qc-cohomology-separated-scheme-affine-cover. Assume the Axiom of Choice, inherited from sheaf cohomology. Let $X$ be a quasi-compact separated scheme (def-separated-morphism-schemes), let $U_0,\dots ,U_r$ be a finite affine open cover of $X$ and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module (def-quasi-coherent-module-scheme). (Cech cohomology computes quasi-coherent cohomology on a separated scheme)

thm-hom-tensor-adjunction-for-modules. Let $R$ be a commutative ring and let $M,N,P$ be $R$-modules. There is a natural $R$-module isomorphism ${\mathrm{Hom}}_R(M{\otimes }_{R}N,P)\cong {\mathrm{Hom}}_R(M,{\mathrm{Hom}}_R(N,P))\mathrm{.}$ It sends $F$ to the map $m\mapsto [n\mapsto F(m\otimes n)]$ and sends $u:M\to {\mathrm{Hom}}_R(N,P)$ to the homomorphism determined by $m\otimes n\mapsto u(m)(n)$. (Hom-tensor adjunction: ${\mathrm{Hom}}_R(M{\otimes }_{R}N,P)\cong {\mathrm{Hom}}_R(M,{\mathrm{Hom}}_R(N,P))$)

thm-five-lemma-for-a-morphism-of-long-exact-sequences. Let $\cdots \to A_{n-1}\to A_n\to A_{n+1}\to A_{n+2}\to A_{n+3}\to \cdots$ and $\cdots \to B_{n-1}\to B_n\to B_{n+1}\to B_{n+2}\to B_{n+3}\to \cdots$ be long exact sequences in an abelian category, together with a morphism of these sequences. (Five lemma for a morphism of long exact sequences)