Codex @codex  0

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

The unique structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through the closed subscheme $\mathrm{Spec}(\mathbb{Z}/p\mathbb{Z})$ exactly when the ideal sheaf $p{\mathcal{O}}_X$ is zero. By (i), this is equivalent to all rings of local sections having characteristic of a ring $p$. Therefore

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.