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Let $X_0\hookrightarrow X$ be a nilpotent thickening defined by ${\mathcal{I}}^m=0$. If $X$ is an affine scheme, its closed subscheme $X_0$ is an affine scheme. Conversely, suppose $X_0$ is an affine scheme. The underlying topological spaces are identical, so $X$ is a quasi-compact scheme. For any quasi-coherent sheaf $\mathcal{F}$, each subquotient ${\mathcal{I}}^j\mathcal{F}/{\mathcal{I}}^{j+1}\mathcal{F}$ is naturally a quasi-coherent sheaf on $X_0$. By sheaf cohomology under a closed inclusion and vanishing of quasi-coherent cohomology on an affine scheme, its first sheaf cohomology on $X$ vanishes. The long exact sequence in sheaf cohomology and descending mathematical induction through this finite filtration prove $H^1(X,\mathcal{F})=0$. The Serre criterion for affineness now makes $X$ an affine scheme.

An affine open cover of a scheme is an open cover by affine open subsets. Every scheme has such a cover by definition. A quasi-compact scheme has a finite affine open cover. Whether the intersections are affine is a further condition, supplied in particular by separatedness over an affine base.

A line bundle $L$ is ample when some positive tensor power $L^{\otimes m}$ is very ample.

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.

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.

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

For a quasi-compact scheme $X$, the Serre criterion for affineness gives the equivalencesIndeed, an affine scheme has vanishing of quasi-coherent cohomology on an affine scheme. Conversely, vanishing of the first sheaf cohomology for every quasi-coherent sheaf includes every quasi-coherent sheaf of ideals, which suffices for the criterion. No separated scheme hypothesis is needed in this formulation. Nor may one test only the structure sheaf.^[ref]

Suppose, to the contrary, that $H^1(X,\mathcal{F})=0$ for every quasi-coherent sheaf $\mathcal{F}$. Since $X$ is a quasi-compact scheme, part (a) makes it an affine scheme.

Every affine scheme $X=\mathrm{Spec}A$ is a separated scheme: its diagonal morphism over $\mathrm{Spec}\mathbb{Z}$ is induced by the surjective ring homomorphism $A{\otimes }_{\mathbb{Z}}A\to A$, $a\otimes b\mapsto ab$, so it is a closed immersion. This contradicts the assumption that $X$ is not a separated scheme. ThereforeIn fact, the version of the Serre criterion for affineness using ideals shows that $\mathcal{F}$ may be chosen to be a quasi-coherent sheaf of ideals. This argument does not use a comparison with Čech cohomology on the nonseparated scheme.

Suppose $X=U_1\cup \cdots \cup U_n$ with each $U_i$ an affine open subscheme. Every affine scheme is a quasi-compact topological space: a cover by principal open subsets $D(f_{\lambda })$ means that the $f_{\lambda }$ generate the unit ideal; an expression $1={\sum }_{j=1}^r{a}_j{f}_{{\lambda }_j}$ gives a finite subcover. Every open cover has a refinement by principal open subsets, so this proves the general covering property.

Restrict an arbitrary open cover of $X$ to each $U_i$ and choose a finite subcover there. The union of these finitely many finite families covers $X$. Thus $X$ is a quasi-compact scheme.

Now assume $X$ is a separated scheme. By (c), every finite intersection of the $U_i$ is an affine scheme, and hence has vanishing positive sheaf cohomology for the structure sheaf. This affine open cover therefore computes sheaf cohomology using its Čech cochain complex. Its degree $p$ term isThere are no strictly increasing $(p+1)$-tuples when $p\ge n$, so those Čech cochain groups vanish. ConsequentlyThe same argument works for every quasi-coherent sheaf. The relevant comparison requires the acyclicity of the finite intersections; the bare assertion that arbitrary Čech cohomology equals sheaf cohomology would be too broad.

Interpretation of the removed locus. Over $\mathbb{Z}$, the notation $[1:0:0]$ specifies a section $\mathrm{Spec}\mathbb{Z}\to {\mathbb{P}}_{\mathbb{Z}}^2$, with image $V_{+}(x_1,x_2)\cong \mathrm{Spec}\mathbb{Z}$. We remove this entire closed subscheme. This is the interpretation forced by the asserted open cover $X=U_1\cup U_2$. Removing one closed point in a particular fiber would not give that open cover. The native PDF uses the same notation as the transcription.

Put $a=x_0/x_1$, $b=x_2/x_1$ on $U_1$. On $U_2$ put $c=x_0/x_2$, $d=x_1/x_2$. On their intersection $c=a/b$ and $d=b^{-1}$, giving coordinate ringsThe Čech cochain complex for the structure sheaf, using increasing distinct indices, isThus ${\check{H}}^0$ is the intersection of the two subrings of $L$, while ${\check{H}}^1$ is the quotient module of $L$ by their additive sum.

The Laurent polynomial ring $L$ is a free abelian group on the monomials $a^i{b}^j$ with $i\ge 0$, $j\in \mathbb{Z}$. The first subring has the basis $j\ge 0$. The second has the basisso its allowed exponents satisfy $j\le -i$. A monomial belongs to both precisely when $i=j=0$. The monomials belonging to neither satisfy $i\ge 2$ and $-i<j<0$. Since the subgroups are spanned by subsets of the same basis, their sum introduces no other relations. ThereforeHere the square brackets denote cohomology classes, and the direct sums are as abelian groups. In particular, $[a^2{b}^{-1}]=[x_0^2/(x_1{x}_2)]$ is a nonzero cohomology class. This scheme is a separated scheme, being an open subscheme of projective space over $\mathbb{Z}$, and is a quasi-compact scheme by (d). Both charts and their intersection are affine schemes, so this Čech cohomology equals sheaf cohomology. The nonzero first sheaf cohomology of the quasi-coherent structure sheaf rules out an affine scheme:

Let $\mathcal{F}$ be any quasi-coherent sheaf on $X$, and choose $m$ from (i). Filter it by the powers of the nilradical sheaf:Each subquotientis a quasi-coherent sheaf annihilated by $\mathcal{N}$. Apply (ii) to the quasi-coherent sheaf ${\mathcal{N}}^j\mathcal{F}$. Since $X_{\mathrm{red}}$ is an affine scheme, vanishing of quasi-coherent cohomology on an affine scheme givesThe short exact sequencesyield the segments of the long exact sequence in sheaf cohomologyStarting from ${\mathcal{N}}^m\mathcal{F}=0$, descending mathematical induction makes every middle group zero, finally giving $H^1(X,\mathcal{F})=0$.

This holds for every quasi-coherent sheaf $\mathcal{F}$. Since a Noetherian scheme is a quasi-compact scheme, the Serre criterion for affineness in (a) applies and provesThis is an instance of the general fact that affineness is invariant under nilpotent thickening.

A Noetherian scheme is a quasi-compact scheme, so choose a finite affine open cover $X={\bigcup }_{i=1}^s\mathrm{Spec}{A}_i$ with all $A_i$ Noetherian rings. The nilradical sheaf restricts on this chart to the quasi-coherent sheaf associated to $\mathrm{Nil}(A_i)$: the nilradical commutes with localization.

By the allowed fact, choose $d_i\ge 1$ such that $\mathrm{Nil}(A_i{)}^{d_i}=0$ and put $m={\max }_i{d}_i$. On every chart, the ideal sheaf ${\mathcal{N}}^m$ is zero, and hence it is zero globally. Higher powers are also zero. ThusIf $X$ is empty, take $m=1$. The finite affine open cover is essential for choosing one bound valid everywhere.

A morphism of schemes is quasi-compact if the inverse image of every quasi-compact open subset is quasi-compact. It suffices to test affine open subschemes of the target. This is a relative condition: the source need not be a quasi-compact scheme when the target is not one.