Positive canonical twists on projective Cohen–Macaulay curves

Given: A projective pure Cohen--Macaulay curve $E$ over a field $\kappa$ with $H^0(E,{\mathcal{O}}_E)=\kappa$ and canonical module ${\omega }_E$, a globally generated nontrivial invertible sheaf $L$ on $E$, and in the generation part a very ample $L$ with ${\mathrm{deg}}_{\kappa }L\ge 2$.

cor-cohen-macaulay-modules-have-no-embedded-associated-primes. Under the hypotheses of lem-associated-primes-of-cohen-macaulay-module-have-full-dimension, every associated prime of $M$ is minimal in ${\mathrm{Supp}}_R(M)$. Thus $M$ has no embedded associated primes. (Cohen--Macaulay modules have no embedded associated primes)

cor-flat-local-depth-additivity. Assume the Axiom of Choice. For a flat local homomorphism $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, $\mathrm{depth}(S)=\mathrm{depth}(R)+\mathrm{depth}(S\mathrm{/}\mathfrak{m}S)\mathrm{.}$ (Depth is additive for a flat local homomorphism)

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-degree-invertible-sheaf-proper-dimension-one. Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field (def-field) and let $C$ be a proper $k$-scheme (def-proper-morphism) whose underlying topological space is Noetherian of dimension at most one (def-dimension-noetherian-topological-space, def-locally-noetherian-and-noetherian-scheme). (Degree of an invertible sheaf on a proper one-dimensional scheme)

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-finite-closed-immersion-derived-coinduction-adjunction. Assume AC. For a finite homomorphism $A\to B$ of Noetherian rings and $G\in D^{+}(A)$, the complex $f^{!}G=R{\mathrm{Hom}}_A(B,G)$ has its natural $B$-action and is right adjoint to restriction of scalars. (Derived adjunction for finite rings and closed immersions)

lem-projective-pure-cm-dualizing-complex-concentration. Assume AC. If $X$ is projective over a field, Cohen–Macaulay and pure of dimension $d$, its normalized dualizing complex has the canonical concentration $D_X\cong {\omega }_X[d],{\omega }_X={\mathcal{H}}^{-d}(D_X)\mathrm{.}$ For $i:X\hookrightarrow {\mathbb{P}}_k^N$, $i_{\ast }{\omega }_X=\mathcal{E}x{t}_P^{N-d}(i_{\ast }{\mathcal{O}}_X,{\omega }_P)\mathrm{.}$ The sheaf ${\omega }_X$ is coherent, has support $X$, and is CM. (Concentration of the projective dualizing complex on a pure CM scheme)

lem-surface-flat-base-change-coherent-cohomology-by-cech. Assume AC and DC. For a quasi-compact separated scheme $X$ over a ring $A$, a quasi-coherent sheaf $F$ and a flat $A$-algebra $C$, the canonical maps $H^q(X,F){\otimes }_{A}C\to H^q(X_C,F_C)$ are isomorphisms for every $q$. No flatness of $F$ over $A$ is required. (Flat base change for quasi-coherent surface cohomology by Čech)

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)

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-serre-duality-for-coherent-sheaves-on-projective-cm-scheme. Assume AC. Let $k$ be a field and let $X$ be a projective, pure $d$-dimensional Cohen–Macaulay $k$-scheme. Let $D_X={\omega }_X[d]$ be its normalized dualizing complex, and let $t_X:H^d(X,{\omega }_X)\to k$ be its trace. (Serre duality for coherent sheaves on a projective Cohen–Macaulay scheme)