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An associated point of a coherent sheaf is a point whose corresponding prime is the annihilator of an element of the module on an affine chart. There are finitely many on a Noetherian scheme. A section of a line bundle avoiding them acts injectively on the sheaf, even when the scheme is nonreduced. Over an infinite field a sufficiently ample linear system supplies such a section.

Projection $X\times {\mathbb{P}}^n\to X$ sends closed subsets to closed subsets, with geometric points or schemes understood. On an affine base, choose finitely many homogeneous equations for a closed subset. Its fiber is nonempty exactly when every graded multiplication map for a projective fiber fails to be surjective, by the Projective Nullstellensatz. The image is thus an intersection of closed rank-defect loci. An affine open cover proves the general case. This is the universally closed part of the proper morphism property of projective space.

For a finite morphism between Noetherian schemes, a quasi-coherent sheaf is coherent exactly when its direct image sheaf is coherent. Over an affine target chart, let $B$ be the finite algebra over the target ring $A$, and let $M$ represent the sheaf on $\mathrm{Spec}B$. Finite generation of $M$ over $A$ implies finite generation over $B$, using the same generators; finite generation over $B$ implies finite generation over $A$ because $B$ is finite over $A$.

Coherent cohomology means sheaf cohomology with coefficients in a coherent sheaf. On a separated Noetherian scheme, a finite affine cover computes it using the Čech cochain complex, because intersections of affine opens are affine and positive-degree quasi-coherent sheaf cohomology vanishes there. Short exact sequences give long exact sequences in sheaf cohomology.

An ideal sheaf of a closed subscheme that is a coherent sheaf as a module over the structure sheaf. On a Noetherian scheme, every quasi-coherent ideal sheaf is coherent.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, $X$ is separated, and $\mathcal{F}$ is quasi-coherent on $Z$, thenIndeed, an affine cover of $X$ induces an affine cover of $Z$, and the two associated Čech cochain complexes are termwise identical.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes and $\mathcal{F}$ is a coherent sheaf on $Z$, then $i_{\ast }\mathcal{F}$ is coherent on $X$. Affine-locally this regards a finitely generated $A/I$-module as a finitely generated $A$-module.

For a closed subscheme with ideal sheaf $\mathcal{I}$ in a locally Noetherian scheme, the completion is the formal scheme on $|Y|$ with topological structure sheaf ${\underleftarrow{\lim }}_n({\mathcal{O}}_X/{\mathcal{I}}^{n+1}){|}_{|Y|}$. On a Noetherian affine scheme $\mathrm{Spec}A$, it is $\mathrm{Spf}({\underleftarrow{\lim }}_{n}A/I^{n+1})$. Its adic topology remembers every infinitesimal neighbourhood.

// Target: commutative-algebra.bigb

For an ample line bundle $L$ on a noetherian scheme $X$, every coherent sheaf $F$ becomes a globally generated sheaf after tensoring with $L^{\otimes n}$ for all sufficiently large $n$. Finitely many global sections suffice by finite global generation on a quasi-compact scheme. This is the global-generation characterization of an ample line bundle.^[ref]

For a closed subscheme of projective space, take $L={\iota }^{\ast }\mathcal{O}(1)$. One way to see why twisting helps is to choose finitely many module generators on each standard affine open subscheme. Multiplication by a sufficiently high power of its homogeneous coordinate extends each generator to a global section of a twist. Choose one exponent for all generators. On the original chart that coordinate is invertible, so these extended sections still generate; doing this for every chart proves global generation. Tensoring back by the inverse invertible sheaf gives a surjection from a finite direct sum of sheaves of negative twists onto $F$.

For a coherent sheaf on a Noetherian scheme, its sheaf cohomology vanishes in degrees greater than its support dimension. In particular, a zero-dimensional projective scheme has no positive-degree coherent cohomology.

For a closed point $P$ with residue field $k(P)$, evaluation gives a morphism from the structure sheaf onto the skyscraper sheaf $k_P$. Its kernel is the ideal sheaf of $P$. On a Noetherian scheme it is a coherent ideal sheaf.

If $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf, projection to the last coordinate makes any coherent $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ an extension of a coherent submodule of ${\mathcal{O}}_X^{r-1}$ by an ideal sheaf. The long exact sequence in sheaf cohomology proves the assertion by mathematical induction on $r$.

A locally free resolution of a sheaf of modules $F$ is an exact sequence $\cdots \to G_2\to G_1\to G_0\to F\to 0$ with each $G_i$ a locally free sheaf. If every coherent sheaf is the quotient of a locally free sheaf of finite rank on a noetherian scheme, iteratively taking a kernel sheaf constructs such a resolution with coherent sheaves at every stage. The kernel sheaves need not be locally free sheaves, so a finite resolution requires additional hypotheses. A uniform bound on sheaf cohomology can nevertheless make a finite initial segment sufficient for dimension shifting in sheaf cohomology.

A scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings. It is a Noetherian scheme exactly when it is also a quasi-compact scheme: a finite subcover then supplies the required finite affine open cover. An infinite gluing of spectra of Noetherian rings along open subschemes is still locally Noetherian, but need not be a quasi-compact scheme.

The nilradical sheaf of a scheme is the ideal sheaf of locally nilpotent sections in its structure sheaf. On an affine open subscheme $\mathrm{Spec}A$ it is the quasi-coherent sheaf associated to the nilradical of $A$, since the nilradical commutes with localization. Its quotient defines the reduced induced subscheme. On a Noetherian scheme it has a uniform nilpotence exponent: take nilpotence bounds on a finite affine open cover and use their maximum. On a general scheme, a locally nilpotent section need not have a single nilpotence exponent over a non-quasi-compact open subset.

Write ${\mathbb{P}}^r={\mathbb{P}}_k^r$ over its ground field; the argument also works over any Noetherian affine base. Since $\iota$ is a closed immersion, $X$ is a noetherian scheme and a quasi-compact scheme. LetHere the pullback of a sheaf of modules makes $L$ an invertible sheaf, the restriction of the twisting sheaf on projective space. It is an ample line bundle.

By global generation by sufficiently positive twists, choose $n\ge 0$ such that $F(n)$ is a globally generated sheaf. Finitely many global sections suffice: at each point, finite generation of the stalk of the coherent sheaf $F(n)$ lets us select finitely many generating germs of global sections. Those sections generate on a neighbourhood, because their cokernel sheaf is a coherent sheaf with zero stalk at that point. A finite subcover, using that $X$ is a quasi-compact scheme, supplies finitely many global sections generating everywhere. This is finite global generation on a quasi-compact scheme, and gives a surjective sheaf morphism

Tensor with $(L^{\vee }{)}^{\otimes n}$, where $L^{\vee }$ is the dual of a sheaf. Here tensoring with an invertible sheaf is exact: locally the tensor product of sheaves is tensoring with a free module of rank one. Using the permitted identity $L\otimes L^{\vee }\cong {\mathcal{O}}_X$, we obtainA finite direct sum of sheaves of invertible sheaves is a locally free sheaf of finite rank, and it is a coherent sheaf on the noetherian scheme $X$. This constructs the required $G$. No regularity or smoothness assumption on $X$ is needed.

Fix a coherent sheaf $F$ and an integer $k>M$. Set $K_0=F$. Repeatedly apply a to construct short exact sequences of sheaveswhere each $G_j$ is a coherent sheaf and a locally free sheaf. Each kernel sheaf $K_{j+1}$ is a coherent sheaf because $X$ is a noetherian scheme. Thus the construction can be repeated as many times as needed; it does not require the kernel sheaves themselves to be locally free sheaves.

For every integer $q>M$, the long exact sequence in sheaf cohomology containsBoth outer groups vanish by the hypothesis, since $q$ and $q+1$ exceed $M$. Hence ${\delta }_j$ is an isomorphism. Iterating this dimension shifting in sheaf cohomology gives

Choose $s\ge 0$ with $k+s>N$, where $N$ is the uniform bound from b. The coherent sheaf $K_s$ is a quasi-coherent sheaf, so b makes the last group zero. ThereforeThe argument uses a finite initial segment of a locally free resolution. There is no need to assume that every coherent sheaf has a finite locally free resolution: the uniform bound from b is what terminates the argument.

The immersions and gluing. For $p>0$, the prime ideals of the localization ${\mathbb{Z}}_{(p)}$ are $(0)$ and $(p){\mathbb{Z}}_{(p)}$. Inverting $p$ gives ${\mathbb{Z}}_{(p)}[1/p]=\mathbb{Q}$, sois an open immersion sending the generic point to the generic point. For $p=0$, ${\mathbb{Z}}_{(0)}=\mathbb{Q}$, and take ${\psi }_0$ to be the identity. Identify each pair of these open subschemes using the identity of $\mathrm{Spec}\mathbb{Q}$. The identities compose to the identity on every triple overlap, so gluing of schemes along open subschemes applies. No finiteness of the family is required.

The resulting scheme $Y$ has one common generic point $\eta$ and a distinct closed point $y_p$ for every positive prime number $p$. Its affine open subschemes $U_p=\{\eta ,y_p\}$, for $p>0$, intersect pairwise in $U_0=\{\eta \}$. A subset $W\subseteq Y$ is open precisely when its intersection with every $U_p$ is open. In the two-point topological space $U_p$, the open subsets are $\varnothing$, $\{\eta \}$ and $U_p$. Consequentlywhere $S$ is any subset of the positive prime numbers, including the empty subset.

The canonical morphism. The ring homomorphisms $\mathbb{Z}\to {\mathbb{Z}}_{(p)}$ give morphisms of schemes $U_p\to \mathrm{Spec}\mathbb{Z}$ agreeing on the overlaps, hence a morphism of schemes $f:Y\to \mathrm{Spec}\mathbb{Z}$. By uniqueness of the structure morphism, this is the required map. Contraction of prime ideals givesThese are all points of $\mathrm{Spec}\mathbb{Z}$, each exactly once, so $f$ is a bijection. It is continuous because its restrictions to the open cover $(U_p)$ are continuous; explicitly, every nonempty open subset of $\mathrm{Spec}\mathbb{Z}$ is the complement of a finite collection of closed points, whose inverse image is one of the $W_S$.

Sections on every open. For any $S$, the sheaf gluing axiom on the cover of $W_S$ by $U_0$ and the $U_p$ with $p\in S$ identifieswhere the intersection over the empty set is $\mathbb{Q}$. Indeed, a section on $U_p$ is an element of ${\mathbb{Z}}_{(p)}$; restrictions to every overlap are its inclusion into $\mathbb{Q}$. Compatibility means that all these elements are the same rational number. In lowest terms $a/b$, this rational number lies in the intersection exactly when no prime number in $S$ divides $b$. Equivalently, this is the integer localization at a set of primes obtained by inverting all positive prime numbers outside $S$.

Now let $V\subseteq \mathrm{Spec}\mathbb{Z}$ be nonempty and open. Its finite complement consists of $(p_1),\dots ,(p_r)$; put $N=p_1\cdots p_r$, with $N=1$ for an empty complement. Then $V=D(N)$ andThe equality follows by the denominator criterion above. The map $f^{\#}$ is this same equality of subrings of $\mathbb{Q}$, so it is a ring isomorphism. Over the empty open subset both rings of sections are the zero ring. These identifications respect every restriction map. Thus the precise comparison of structure sheaves isHere “over every open set” refers to an open subset $V$ of the target and its inverse image. The preceding intersection formula additionally describes the sections on every open subset of $Y$, including ones whose images are not open in the target.

Topology and finiteness. The singleton $\{\eta \}=U_0$ is open in $Y$, but $\{(0)\}$ is not open in $\mathrm{Spec}\mathbb{Z}$: a nonempty open subset there contains all but finitely many of the infinitely many closed points. Therefore $f$ is not a homeomorphism, despite being a continuous bijection and giving the indicated isomorphism of structure sheaves after direct image.

The open cover $(U_p{)}_{p>0}$ of $Y$ has no finite subcover, because $y_q$ belongs only to $U_q$. Thus $Y$ is not a quasi-compact scheme, and part (a) rules out an affine scheme. Each ${\mathbb{Z}}_{(p)}$ is a Noetherian ring, being a localization of the Noetherian ring $\mathbb{Z}$. Hence the $U_p$ form an affine open cover by spectra of Noetherian rings, proving that $Y$ is a locally Noetherian scheme. A Noetherian scheme is a quasi-compact scheme, so

Failure in general. Let $f:{\mathbb{A}}_k^1=\mathrm{Spec}k[t]\to \mathrm{Spec}k$ be the structure morphism and take the trivial line bundle $\mathcal{L}={\mathcal{O}}_{{\mathbb{A}}_k^1}$. The polynomial ring $k[t]$ is a Noetherian ring, so this is a coherent sheaf. Its direct image on the one-point target has sectionsA coherent sheaf on $\mathrm{Spec}k$ corresponds to a finite-dimensional vector space over $k$, whereas $1,t,t^2,\dots$ are linearly independent. Thus

Quasi-coherence for the stated morphism. Now let $f$ satisfy (a), and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Over any affine open subscheme $V=\mathrm{Spec}A$ of $Y$, write $f^{-1}V=\mathrm{Spec}B$ and $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$ for a $B$-module $M$. Regard $M$ as an $A$-module through the ring homomorphism $A\to B$. For $a\in A$,The last two localizations are identical: both invert the action of the same element on $M$. These identifications commute with restrictions, so on the basis of principal open subsets they identify $(f_{\ast }\mathcal{F}){|}_V$ with the affine module sheaf $\widetilde{{}_{A}M}$. Hence this direct image is a quasi-coherent sheaf on each $V$, and therefore on $Y$. This part uses only that $f$ is an affine morphism.

Coherence under the Noetherian hypotheses. If $X,Y$ are Noetherian schemes and $\mathcal{F}$ is a coherent sheaf, then $M$ is a finitely generated module over $B$. Choose generators $m_1,\dots ,m_s$ of $M$ over $B$, and generators $b_1,\dots ,b_r$ of $B$ over $A$, supplied by (a). Then the finite familygenerates $M$ as an $A$-module: expand each coefficient of an expression $m={\sum }_{\ell }{c}_{\ell }{m}_{\ell }$ in the $b_j$. Since $A$ is a Noetherian ring, the affine module sheaf $\widetilde{{}_{A}M}$ is a coherent sheaf. This holds on an affine open cover of $Y$, proving