Regular hyperplane step for coherent support induction

Each chart $X\cap D_{+}(x_j)$ is $\mathrm{Spec}$ of a quotient of the polynomial ring $k[x_0,\dots ,x_n]$, hence has Noetherian coordinate ring; the finitely many charts cover $X$, so $X$ is a Noetherian scheme and every closed subscheme of $X$ has Noetherian underlying space.

For every $j$ the element $x_j\in S(1)$ has compatible images $x_j\mathrm{/}1$ in $S(1{)}_{(x_i)}$ for all $i$, so the restrictions glue along the standard affine cover to a global section $s_j\in \mathrm{\Gamma }({\mathbb{P}}_k^n,\mathcal{O}(1))$; pulling back along $i$ gives $s_j\in \mathrm{\Gamma }(X,L)$. On $X\cap D_{+}(x_j)$ the section $s_j$ is a basis of $L$ (the localisation $S(1{)}_{(x_j)}=x_j{S}_{(x_j)}$ is free on $x_j$), so $s_j(x)\mathrm{\ne }0$ for every point $x\in X\cap D_{+}(x_j)$; since the charts cover, for every $x\in X$ some $s_j$ has $s_j(x)\mathrm{\ne }0$.

Let $Z\subseteq X$ be an integral closed subscheme with $\dim Z=e\ge 1$ on which $\mathrm{\ell }$ does not vanish identically. Then $Z\cap V(\mathrm{\ell })\mathrm{\ne }\mathrm{\varnothing }$: otherwise $Z\subseteq {\mathbb{P}}_k^n\setminus V(\mathrm{\ell })=D_{+}(\mathrm{\ell })$, which is an affine standard open of ${\mathbb{P}}_k^n$ by [F1]; as $Z$ is closed in ${\mathbb{P}}_k^n$ and contained in the affine scheme $D_{+}(\mathrm{\ell })$, it is a closed subscheme of an affine scheme and hence affine. But $Z\to \mathrm{Spec}k$ is also projective (closed immersion into ${\mathbb{P}}_k^n$), hence proper and of finite type by [F2] and [F9]. By [F9], its ring of global functions is a finite field extension $K\mathrm{/}k$; affineness would identify $Z$ with $\mathrm{Spec}K$, which has only the zero prime and dimension zero, contradicting $\dim Z=e\ge 1$.

Let $Z$ be an integral finite-type $k$-scheme with generic point $\eta$ and a nonempty affine open $U=\mathrm{Spec}A$; then $A$ is a finite-type $k$-domain with $\mathrm{Frac}(A)\cong K(Z):={\mathcal{O}}_{Z,\eta }$, and $\dim U=\dim A={\mathrm{trdeg}}_{k}K(Z)$: the first equality holds because irreducible closed subsets of $\mathrm{Spec}A$ are the $V(\mathfrak{p})=\overline{\{\mathfrak{p}\}}$ and inclusion reverses inclusion of primes, matching chains; the second is the affine-domain dimension theorem. Consequently every nonempty affine chart of $Z$ has the same dimension, and for $Z$ covered by its standard charts $Z\cap D_{+}(x_i)$ (each affine with finite-type coordinate ring) the open-cover formula gives $\dim Z={\max }_i\dim (Z\cap D_{+}(x_i))={\mathrm{trdeg}}_{k}K(Z)<\mathrm{\infty }$.

Let $A$ be a finite-type $k$-domain of dimension $e\ge 1$ and let $0\mathrm{\ne }f\in A$ be a nonunit. Then $\dim (A\mathrm{/}(f))=e-1$ and every irreducible component of $\mathrm{Spec}(A\mathrm{/}(f))$ has dimension $e-1$: a prime $\mathfrak{p}$ minimal over $(f)$ has $\mathrm{ht}(\mathfrak{p})=1$ (principal ideal theorem for a nonzerodivisor) and $\mathrm{ht}(\mathfrak{p})+\dim (A\mathrm{/}\mathfrak{p})=e$, so $\dim (A\mathrm{/}\mathfrak{p})=e-1$; the primes of $A\mathrm{/}(f)$ correspond to primes of $A$ containing $(f)$, every chain of those begins at a minimal prime over $(f)$, and the quotient dimension is the supremum of the lengths of such chains, so it equals ${\max }_{\mathfrak{p}}\dim (A\mathrm{/}\mathfrak{p})=e-1$ and each component $V(\mathfrak{p})$ has dimension $e-1$; in particular $\mathrm{Spec}(A\mathrm{/}(f))\mathrm{\ne }\mathrm{\varnothing }$.

The twist $\mathcal{F}(m)=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{L}^m$ is quasi-coherent for every $m$; on an affine open $U=\mathrm{Spec}A$ contained in some $X\cap D_{+}(x_j)$ with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, $M$ finitely generated, a trivialisation $\tau :L{\mathrm{\mid }}_U\to {\mathcal{O}}_U$ (for instance the one sending $s_j$ to $1$) gives isomorphisms $\mathcal{F}(m){\mathrm{\mid }}_U\cong \tilde{M}$ for every $m\in \mathbb{Z}$, corresponding to multiplication by units of $A$ on $M$.

Let $V_0\subseteq \mathrm{\Gamma }(X,L)$ be the $k$-linear span of $s_0,\dots ,s_n$, a finite-dimensional $k$-vector space with the finite generating set $\{s_0,\dots ,s_n\}$. For each $x\in \mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ the subspace $W_x=\{u\in V_0:u(x)=0\}$ is proper, because some $s_j$ has $s_j(x)\mathrm{\ne }0$ by 1.2. Since $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ is finite by 3.1 and $k$ is infinite, no finite union of the proper subspaces $W_x$ covers $V_0$; choose $\mathrm{\ell }\in V_0$ with $\mathrm{\ell }(x)\mathrm{\ne }0$ for every $x\in \mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$, equivalently $x\mathrm{\notin }Z(\mathrm{\ell })$ for every associated point.

Define $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})=\{x\in X:{\mathfrak{m}}_x\in {\mathrm{Ass}}_{{\mathcal{O}}_{X,x}}({\mathcal{F}}_x)\}$. For a chart $U=\mathrm{Spec}A$ as in 2.1 and a prime $\mathfrak{p}\in \mathrm{Spec}A$ corresponding to $x\in U$, one has ${\mathfrak{m}}_x=\mathfrak{p}{A}_{\mathfrak{p}}$, ${\mathcal{F}}_x=M_{\mathfrak{p}}$, and $\mathfrak{p}{A}_{\mathfrak{p}}\in {\mathrm{Ass}}_{A_{\mathfrak{p}}}(M_{\mathfrak{p}})$ iff $\mathfrak{p}\in {\mathrm{Ass}}_A(M)$: in the forward direction reverse localisation gives $\mathfrak{q}\in {\mathrm{Ass}}_A(M)$ with $\mathfrak{q}{A}_{\mathfrak{p}}=\mathfrak{p}{A}_{\mathfrak{p}}$, and contraction to $A$ gives $\mathfrak{q}=\mathfrak{p}$; the backward direction is localisation of an associated prime. Hence $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})\cap U$ corresponds to ${\mathrm{Ass}}_A(M)$ and is finite by the finiteness of associated primes; the finitely many charts give that $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ is finite.

Every irreducible component of $\mathrm{Supp}\mathcal{F}$ has its generic point in $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$, and $\mathrm{Supp}\mathcal{F}={\bigcup }_{x\in \mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})}\overline{\{x\}}$. Indeed on a chart $U$ the support is $V({\mathrm{Ann}}_{A}M)$ and equals ${\bigcup }_{\mathfrak{p}\in {\mathrm{Ass}}_A(M)}V(\mathfrak{p})$; the subspace $V({\mathrm{Ann}}_{A}M)$ of $\mathrm{Spec}A$ is homeomorphic to $\mathrm{Spec}(A\mathrm{/}{\mathrm{Ann}}_{A}M)$, whose irreducible components are the $V(\mathfrak{p})$ for the minimal primes $\mathfrak{p}$ over ${\mathrm{Ann}}_{A}M$, and those are associated by the minimal-support-prime theorem. If $\eta$ is the generic point of a component of $\mathrm{Supp}\mathcal{F}$, choose a chart $U\ni \eta$; the corresponding prime of $A$ is minimal over ${\mathrm{Ann}}_{A}M$, hence associated, so $\eta \in \mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ by 3.1.

For $m\in \mathbb{Z}$ let ${\mu }_m:\mathcal{F}(m-1)\to \mathcal{F}(m)$ be the composite of ${\mathrm{i}\mathrm{d}}_{\mathcal{F}(m-1)}\otimes (\cdot \mathrm{\ell }):\mathcal{F}(m-1){\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X\to \mathcal{F}(m-1){\otimes }_{{\mathcal{O}}_X}L$ with the canonical isomorphism $\mathcal{F}(m-1){\otimes }_{{\mathcal{O}}_X}L\cong \mathcal{F}(m)$ of [F2], and let $\mathcal{G}(m)=\mathrm{coker}{\mu }_m$ with $\mathrm{Supp}\mathcal{G}(m)=\{x:\mathcal{G}(m{)}_x\mathrm{\ne }0\}$. On a chart $U=\mathrm{Spec}A$ as in 2.1 with trivialisation $\tau :L{\mathrm{\mid }}_U\to {\mathcal{O}}_U$ and local equation $f=\tau (\mathrm{\ell }{\mathrm{\mid }}_U)\in A$, the map ${\mu }_m{\mathrm{\mid }}_U$ corresponds to multiplication by $f$ on $\tilde{M}$, and $V(\mathrm{\ell })\cap U=V(f)=\{\mathfrak{p}:f\in \mathfrak{p}\}$; replacing $\tau$ by a unit multiple of $\tau$ replaces $f$ by a unit multiple, which changes neither the kernel, the cokernel nor $V(f)$.

$\mathcal{G}(m)=\mathrm{coker}{\mu }_m$ is quasi-coherent, and on each chart $U=\mathrm{Spec}A$ as in 2.1 one has $\mathcal{G}(m){\mathrm{\mid }}_U\cong \widetilde{M\mathrm{/}fM}$, the associated sheaf of the finitely generated module $M\mathrm{/}fM$; hence $\mathcal{G}(m)$ is of finite type.

The local equation $f$ of 3.3 is a nonzerodivisor on $M$. Suppose $fm=0$ with $0\mathrm{\ne }m\in M$: then $f$ is a zero divisor on $M$, so $f\in \mathfrak{q}$ for some associated prime $\mathfrak{q}\in {\mathrm{Ass}}_A(M)$ by [F6]. The corresponding point ${\mathfrak{p}}^{\mathrm{\prime }}\in U$ lies in $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ by 3.1, while $f\in \mathfrak{q}$ means $\mathrm{\ell }({\mathfrak{p}}^{\mathrm{\prime }})=0$ by 3.3, contradicting 2.2. Hence no such $m$ exists, and since localisation is exact $f$ is also a nonzerodivisor on every $M_{\mathfrak{p}}$.

Hence ${\mu }_m$ is injective for every $m$: on each chart ${\mu }_m$ corresponds to multiplication by the nonzerodivisor $f$ (up to a unit) and therefore has zero kernel, and injectivity of a morphism of sheaves is checked on stalks.

$\mathcal{G}(m)$ is coherent. Coherence is local on $X$, so it suffices to verify the defining kernel condition on the affine charts $U=\mathrm{Spec}A$ of 3.4, over which $\mathcal{G}(m){\mathrm{\mid }}_U\cong \tilde{N}$ with $N=M\mathrm{/}fM$ finitely generated over the Noetherian ring $A$. Given a morphism $\phi :{\mathcal{O}}_U^r\to \tilde{N}$, corresponding under the affine equivalence to an $A$-linear map $\psi :A^r\to N$, one has $\ker \phi \cong \widetilde{\ker \psi }$ with $\ker \psi$ a submodule of $A^r$; as $A^r$ is a finitely generated module over a Noetherian ring, $\ker \psi$ is finitely generated, so $\ker \phi$ is of finite type, as required.

$\mathrm{Supp}\mathcal{G}(m)=\mathrm{Supp}\mathcal{F}\cap V(\mathrm{\ell })$. On a chart $U=\mathrm{Spec}A$ as in 2.1 the stalk of $\mathcal{G}(m)$ at $\mathfrak{p}$ is $(M\mathrm{/}fM{)}_{\mathfrak{p}}=M_{\mathfrak{p}}\mathrm{/}f{M}_{\mathfrak{p}}$: it vanishes when $M_{\mathfrak{p}}=0$; it vanishes when $f\mathrm{\notin }\mathfrak{p}$, since then $f$ is a unit of $A_{\mathfrak{p}}$ and $M_{\mathfrak{p}}=f{M}_{\mathfrak{p}}$; and it is nonzero when $M_{\mathfrak{p}}\mathrm{\ne }0$ and $f\in \mathfrak{p}$, since $f{M}_{\mathfrak{p}}\subseteq \mathfrak{p}{M}_{\mathfrak{p}}$ and $M_{\mathfrak{p}}=f{M}_{\mathfrak{p}}$ would force $M_{\mathfrak{p}}=0$ by Nakayama applied over the local ring $A_{\mathfrak{p}}$. Thus the chartwise supports are $\mathrm{Supp}(M)\cap V(f)=\mathrm{Supp}\mathcal{F}\cap U\cap V(\mathrm{\ell })$, and covering $X$ by such charts proves the claim.

Suppose $\dim \mathrm{Supp}\mathcal{F}=0$. Then every irreducible component of the Noetherian space $\mathrm{Supp}\mathcal{F}$ is a single point, and that point is the generic point of the component, hence belongs to $\mathrm{A}\mathrm{s}\mathrm{s}(\mathcal{F})$ by 3.2; by 2.2 the chosen $\mathrm{\ell }$ satisfies $\mathrm{\ell }(x)\mathrm{\ne }0$ at every point $x\in \mathrm{Supp}\mathcal{F}$, so $\mathrm{Supp}\mathcal{F}\cap V(\mathrm{\ell })=\mathrm{\varnothing }$, $\mathcal{G}(m)=0$ for every $m$ by 4.3, and $\mathrm{\ell }$ is a unit at each point of $\mathrm{Supp}\mathcal{F}$ (its local equation is a unit), i.e. $\mathrm{\ell }$ is invertible on $\mathrm{Supp}\mathcal{F}$.

Let $Z\subseteq X$ be integral and closed with $\dim Z=e\ge 1$ and $\mathrm{\ell }$ not vanishing identically on $Z$; then $\dim (Z\cap V(\mathrm{\ell }))=e-1$. Put $W=Z\cap V(\mathrm{\ell })$, nonempty by 1.3 and proper closed in $Z$ since $\mathrm{\ell }$ does not vanish at the generic point of $Z$; any chain of length $\ge e$ of irreducible closed subsets of $W$, together with $Z$, would be a chain of length $\ge e+1$ in $Z$, so $\dim W\le e-1$. For the lower bound choose $z\in W$ and a nonempty affine chart $U=\mathrm{Spec}A$ of $Z$ containing $z$; by 1.4, $A$ is a finite-type $k$-domain of dimension $e$, and the local equation $f\in A$ of 3.3 is nonzero: if $f=0$ then the nonempty open $U$ of the irreducible space $Z$ would lie in the closed set $W$, forcing $Z=\bar{U}\subseteq W$ and $\mathrm{\ell }\equiv 0$ on $Z$; and $f$ is a nonunit because $z\in V(f)$ means $f$ lies in the prime of $A$ corresponding to $z$. Hence $U\cap W=V(f)=\mathrm{Spec}(A\mathrm{/}(f))$ has dimension $e-1$ by 1.5, so $\dim W\ge e-1$ and therefore $\dim W=e-1$.

Let $Z_1,\dots ,Z_r$ be the finitely many irreducible components of $\mathrm{Supp}\mathcal{F}$, equipped with their reduced induced structures, and put $e_j=\dim Z_j$, $d=\dim \mathrm{Supp}\mathcal{F}={\max }_j{e}_j$; each $Z_j$ is integral and projective over $k$, so $e_j={\mathrm{trdeg}}_{k}K(Z_j)<\mathrm{\infty }$ by 1.4. By 3.2 the generic point of $Z_j$ is associated, so $\mathrm{\ell }$ does not vanish identically on $Z_j$ by 2.2. If $e_j\ge 1$ then $\dim (Z_j\cap V(\mathrm{\ell }))=e_j-1$ by 4.5, while if $e_j=0$ then $Z_j$ is a single point on which $\mathrm{\ell }\mathrm{\ne }0$, so $Z_j\cap V(\mathrm{\ell })=\mathrm{\varnothing }$. Since $\mathrm{Supp}\mathcal{G}(m)={\bigcup }_j(Z_j\cap V(\mathrm{\ell }))$ by 4.3 is a finite union of closed subsets, $\dim \mathrm{Supp}\mathcal{G}(m)={\max }_j\dim (Z_j\cap V(\mathrm{\ell }))={\max }_{j:e_j\ge 1}(e_j-1)=d-1$ whenever $d\ge 1$.

Boundary and choice accounting. If $X=\mathrm{\varnothing }$ then $\mathcal{F}=0$ is excluded by hypothesis, so $\mathrm{Supp}\mathcal{F}\mathrm{\ne }\mathrm{\varnothing }$ and $d\ge 0$; if $d=0$ the conclusion is exactly 4.4, and if $d\ge 1$ it is 4.6; the case $n=0$ is included: ${\mathbb{P}}_k^0=\mathrm{Spec}k$ has the single chart $D_{+}(x_0)$ with $s_0$ a basis of $L$ by 1.2, so for a nonzero coherent $\mathcal{F}$ on $X=\mathrm{Spec}k$ all hypotheses and steps apply verbatim. The Axiom of Choice is declared and used exactly through the inherited machinery: the finiteness of associated primes and the localisation dictionary [F6] (Dependent Choice is a consequence), and the associated-sheaf and affine-equivalence machinery [F3]; the choice of $\mathrm{\ell }$ in 2.2 is a selection from a nonempty complement of a finite union of proper subspaces of a finite-dimensional $k$-vector space and uses only the infinitude of $k$; no further choice is made in 1.1, 1.2, 2.1, 3.3-4.3.