Serre vanishing for coherent sheaves and ample twists

Given: The Axiom of Choice as inherited, a Noetherian commutative ring $A$, a scheme $X$ projective over $A$ in the H-projective convention, an ample invertible module $L$ on $X$, and a coherent module $\mathcal{F}$ on $X$.

$X$ is proper of finite type over $\mathrm{Spec}A$ (Projective morphisms before Proj, Projective morphisms are proper), and for $A$ Noetherian the projective space ${\mathbb{P}}_A^N$ is locally Noetherian, since its standard charts are spectra of the polynomial rings $A[x_{\mathrm{\ell }}^{(i)}]$, which are Noetherian. (Relative projective space from standard charts, If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$, Locally Noetherian and Noetherian schemes)

For $A$ Noetherian, $f:X\to \mathrm{Spec}A$ proper of finite type and $L$ ample there exist $d\ge 1$ and a closed immersion $i:X\hookrightarrow {\mathbb{P}}_A^N$ over $\mathrm{Spec}A$ with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes d}$; that is, $L^{\otimes d}$ is closed H-very ample relative to the affine base. (High powers of an ample line bundle embed a proper scheme, Relative very ampleness in the finite projective-space convention)

Tensor powers of the invertible module $L$ are invertible, and for a coherent $\mathcal{F}$ every twist $\mathcal{F}\otimes L^{\otimes r}$ is coherent: coherence is local on $X$, and on an open set where $L$ is trivial the twist is isomorphic to $\mathcal{F}$. (Invertible sheaves, Coherent module sheaves, Tensor product of sheaves of modules)

For a closed immersion $j:Y\to Z$ of schemes and a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{G}$ one has $H^q(Y,\mathcal{G})\cong H^q(Z,j_{\ast }\mathcal{G})$ for every $q\ge 0$; if in addition $Z$ is locally Noetherian and $\mathcal{G}$ is coherent, then $j_{\ast }\mathcal{G}$ is coherent. (Closed immersion preserves cohomology and coherent pushforward)

For a closed immersion $j:Y\to Z$ and any ${\mathcal{O}}_Y$-module $\mathcal{G}$ one has $j_{\ast }\left(\mathcal{G}{\otimes }_{{\mathcal{O}}_Y}{j}^{\ast }\mathcal{F}\right)\cong (j_{\ast }\mathcal{G}){\otimes }_{{\mathcal{O}}_Z}\mathcal{F}$ for every ${\mathcal{O}}_Z$-module $\mathcal{F}$: at a point $z=j(y)$ both sides have stalk ${\mathcal{G}}_y{\otimes }_{{\mathcal{O}}_{Z,z}}{\mathcal{F}}_z$, by associativity of tensor products and the stalk formula for pullback; outside the closed image both stalks are zero. The natural projection morphism is therefore an isomorphism on all stalks. Consequently, when $\mathcal{G}$ and $\mathcal{F}$ are quasi-coherent, $H^q(Z,(j_{\ast }\mathcal{G})\otimes \mathcal{F})\cong H^q(Y,\mathcal{G}\otimes j^{\ast }\mathcal{F})$ by [F4]. (Closed immersions are affine quotients and survive base change, Direct image of a sheaf along a continuous map, Tensor product of sheaves of modules)

For a coherent module $\mathcal{H}$ on ${\mathbb{P}}_A^N$ with $A$ Noetherian there is $n_0$ such that $H^q({\mathbb{P}}_A^N,\mathcal{H}(n))=0$ for every $q>0$ and every $n\ge n_0$. Indeed Projective coherent finiteness and large twist vanishing gives a threshold for each $q=1,\dots ,N$. Take the maximum of these finitely many thresholds and $0$; for $q>N$ every twist vanishes by Projective n-space has quasi-coherent cohomological dimension at most n. This also covers $N=0$.

Pullback of modules is monoidal: $j^{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Z}\mathcal{G})\cong j^{\ast }\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{j}^{\ast }\mathcal{G}$ for every morphism $j:Y\to Z$ of schemes and ${\mathcal{O}}_Z$-modules $\mathcal{F},\mathcal{G}$. Indeed at a point $y$ both sides have stalk ${\mathcal{F}}_{j(y)}{\otimes }_{{\mathcal{O}}_{Z,j(y)}}{\mathcal{O}}_{Y,y}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{G}}_{j(y)}{\otimes }_{{\mathcal{O}}_{Z,j(y)}}{\mathcal{O}}_{Y,y}$ by the stalk formulas for pullback and for the tensor product, and a morphism of sheaves is an isomorphism once it is one on every stalk. Hence for an invertible sheaf and $n\ge 0$ one has $j^{\ast }({\mathcal{L}}^{\otimes n})\cong (j^{\ast }\mathcal{L}{)}^{\otimes n}$. (Pullback of a module along a morphism of ringed spaces, The stalk of an inverse image sheaf is the stalk over the image point, The stalk of a tensor product sheaf is the tensor product of the stalks, A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk, Invertible sheaves)

Arithmetic of the residue classes: for integers $d\ge 1$ and $n_r\ge 0$ indexed by $r\in \{0,\dots ,d-1\}$, put $m_0=d\cdot {\max }_r{n}_r+(d-1)$. Every integer $m\ge m_0$ has a unique presentation $m=dn+r$ with $n\ge 0$ and $0\le r<d$, and then $n=\lfloor m\mathrm{/}d\rfloor \ge \lfloor m_0\mathrm{/}d\rfloor ={\max }_r{n}_r\ge n_r$. [algebra]