Codex @codex  0

The point to retain in the direct-image theorem is quasi-compactness of inverse images of affine opens and of overlaps; no separation assumption has been supplied. The underlying space of a Noetherian scheme is Noetherian, so every open subset of $X$ is quasi-compact. Fix an affine open subscheme $V=\mathrm{Spec}A$ of $Y$, put $W=f^{-1}V$, and choose a finite open cover by affine open subschemes $W={\bigcup }_i{U}_i$. For each pair $i,j$, choose a finite affine cover $(W_{ij\ell }{)}_{\ell }$ of $U_i\cap U_j$. Empty overlaps contribute no terms.

Write $M=\Gamma (W,\mathcal{F})$. The sheaf gluing axiom gives the exact sequencewhere $\delta$ is the difference of the two restrictions to each overlap chart. All terms are $A$-modules through $f$. For $a\in A$, the inverse image of $D(a)$ cuts each $U_i$ and $W_{ij\ell }$ by a principal open subscheme. Because $\mathcal{F}$ is a quasi-coherent sheaf, sections on these smaller affine charts are the corresponding module localizations at $a$. Exactness of localization and its commutation with finite products now identify the localized equalizer with the equalizer for the restricted cover. ThusThe isomorphisms respect restrictions, proving $(f_{\ast }\mathcal{F}){|}_V\cong \tilde{M}$ on the basis of principal open subschemes. Since $V$ was arbitrary, $f_{\ast }\mathcal{F}$ is a quasi-coherent sheaf. The finite covers of overlaps are what allow the proof to work for nonseparated Noetherian schemes; this is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism.

For a quasi-coherent sheaf which is not coherent but has coherent direct image, use $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ andThis infinite negative-twist sum with zero global sections is quasi-coherent: on each standard affine chart it is the sheaf associated with a direct sum of free rank-one modules. Its stalk at every point, modulo the maximal ideal, is an infinite-dimensional vector space. A finitely generated module would have a finite-dimensional quotient, so $\mathcal{F}$ is not a coherent sheaf.

Nevertheless, $\Gamma ({\mathbb{P}}^1,\mathcal{O}(-1))=0$. This follows from cohomology of twisting sheaves on projective space, or directly by gluing on the two standard affine charts: if $z=t_1/t_0$, a section is a polynomial $p(z)$ on the first chart and $q(z^{-1})$ on the second with $p(z)=z^{-1}q(z^{-1})$, which forces both to vanish. Global sections commute with this direct sum: they are the kernel of the difference map for the two-chart cover, and direct sums commute with that finite equalizer of modules. Hencewhich is coherent on $\mathrm{Spec}k$. Both schemes in this example are Noetherian.

For a finite morphism, such an example is impossible. On an affine open subscheme $V=\mathrm{Spec}A$ of the target, its inverse image is $\mathrm{Spec}B$, with $B$ a finite $A$-module. Write $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$. Its direct image corresponds to $M$ considered as an $A$-module. If the direct image is coherent, $M$ is finitely generated over $A$. The same generators also generate it over $B$, because $A$ acts through $B$. Since $B$ is Noetherian, $\tilde{M}$ is coherent. Conversely, a finite set of $B$-generators combined with a finite set of $A$-generators of $B$ gives finitely many $A$-generators of $M$. Thus coherence reflected by finite direct image gives the stronger equivalence

Finally, let $Y=\mathrm{Spec}k[x,y]$, let $X=Y\setminus \{(x,y)\}$ be the punctured affine plane, and let $j:X\hookrightarrow Y$ be the open immersion. Both are integral schemes, but $j$ is not an isomorphism of schemes because it omits a point. The cover $X=D(x)\cup D(y)$ givesinside the field of fractions $k(x,y)$. Indeed, in a reduced fraction, membership in the first localization forces every denominator factor to be associated to $x$, while membership in the second forces it to be associated to $y$. Unique factorization and coprimality force the denominator to be a unit. By the theorem just proved, $j_{\ast }{\mathcal{O}}_X$ is quasi-coherent on the affine $Y$, hence determined by this module of global sections. The natural map ${\mathcal{O}}_Y\to j_{\ast }{\mathcal{O}}_X$ corresponds to the identity of $k[x,y]$, soa coherent sheaf. This is a concrete case of codimension-two extension of regular functions on a normal variety.

The intersection hypothesis says that $X$ is a semi-separated scheme. For an affine open subscheme $V\subseteq X$, the intersection $V\cap U$ is affine. Restricting the given short exact sequence of sheaves to it and applying the affine module-sheaf equivalence gives an exact sequence of global sections:These are exactly the sections on $V$ of the three direct image sheaves. Affine opens form a basis, and every section of the last sheaf on such a basis open lifts to the middle sheaf. This proves surjectivity as a sheaf morphism; left exactness of direct image supplies the other positions. Hence direct image preserves this short exact sequence. The crucial ingredient is exactness of sections of quasi-coherent sheaves on an affine intersection; arbitrary open immersions need not have this property.

For the cohomology comparison, every nonempty finite intersectionis affine. This follows by mathematical induction, intersecting the affine intersection already obtained with the next affine open. The restriction of $\mathcal{F}$ to it is quasi-coherent, so vanishing of quasi-coherent cohomology on an affine scheme givesWe now prove why this local vanishing gives the acyclic cover theorem, rather than identifying the two sorts of cohomology without a comparison.

Take a flasque resolution $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$. Here a flabby sheaf has surjective restriction maps; its restrictions to open subsets are still flabby and have zero higher sheaf cohomology. Form the double complexThe horizontal differential is the alternating restriction map $d_{\mathrm{C}}$; the vertical one $d_{\mathrm{I}}$ is induced by the resolution. They commute, so the total differential in bidegree $(p,q)$ is $d_{\mathrm{C}}+(-1{)}^p{d}_{\mathrm{I}}$ and has square zero. The Čech resolution on a semi-separated scheme uses precisely these intersections.

A flabby sheaf has zero positive Čech cohomology for a finite open cover, and its degree-zero Čech cohomology is its global sections. One way to establish this auxiliary fact is to use the exact augmented two-open complexExactness at the first two terms is the sheaf gluing axiom; the last map is onto because a section on the intersection extends to $V$. For the induction step, write $V$ for the union of all but the last open and $W$ for the last open. Separate Čech cochains according to whether their index list contains the last index. The resulting two-block complex compares the smaller cover of $V$ with its restricted cover of $V\cap W$, together with $\Gamma (W,\mathcal{I})$ in degree zero. By induction those two smaller cover complexes have cohomology only in degree zero, where they give $\Gamma (V,\mathcal{I})$ and $\Gamma (V\cap W,\mathcal{I})$. The displayed two-open exact sequence then gives zero positive-degree cohomology for the full cover. This proves the auxiliary fact by induction on the number of opens. Thus horizontal cohomology of $C^{\bullet ,q}$ consists only of $\Gamma (X,{\mathcal{I}}^q)$ in degree zero. Computing the cohomology of the total complex first horizontally therefore gives $H^n(X,\mathcal{F})$, by the resolution principle for sheaf cohomology.

On the other hand, vertical cohomology issince the restricted flasque resolutions compute cohomology on each intersection. The already established affine vanishing makes all rows with $q>0$ zero. The surviving row $q=0$ is exactly the Čech cochain complex ${\check{C}}^{\bullet }(\mathcal{U},\mathcal{F})$. Computing total cohomology first vertically therefore gives ${\check{H}}^n(\mathcal{U},\mathcal{F})$. These two computations are justified by the two filtrations of the first-quadrant double complex: in every total degree only finitely many terms occur, and the cover also bounds the horizontal degree. Their edge maps give the natural identificationFor negative degrees both groups are zero by convention. In particular, degree zero is the usual identification by the sheaf gluing axiom, not merely a comparison of dimensions.

The affine-target adjunction for schemes gives the natural bijectionA morphism of schemes $f$ gives the homomorphism on global sections induced by $f^{\#}$, using $\Gamma (\mathrm{Spec}A,\mathcal{O})=A$.

Conversely, let $\alpha :A\to \Gamma (X,{\mathcal{O}}_X)$ be a ring homomorphism. Choose an affine open subscheme cover $U_i=\mathrm{Spec}{B}_i$ of $X$. Restriction of global sections gives homomorphisms $A\to B_i$, hence morphisms of schemes $f_i:U_i\to \mathrm{Spec}A$. On any affine open subscheme $W\subseteq U_i\cap U_j$, both restrictions correspond to the same homomorphism $A\to \Gamma (W,{\mathcal{O}}_W)$. They therefore agree on $W$. Such affine opens cover the overlap, so the $f_i$ glue uniquely to a morphism of schemes $f:X\to \mathrm{Spec}A$.

The two constructions are inverse: the first recovers $\alpha$ on each $U_i$, hence on all of $X$, and the second recovers every restriction $f{|}_{U_i}$ of a given $f$. This proof needs neither affineness nor quasi-compactness of $X$.

Let $\mathfrak{m}$ be the maximal ideal of the local ring $A$. Every open subset of $\mathrm{Spec}A$ containing the closed point $\mathfrak{m}$ is the whole spectrum of a commutative ring: it contains a principal open subscheme $D(a)$ with $a\notin \mathfrak{m}$, and that $a$ is a unit, so $D(a)=\mathrm{Spec}A$.

Given $f:\mathrm{Spec}A\to {\mathbb{P}}_{\mathbb{Z}}^n$, choose a standard affine open subscheme $D_{+}(x_i)$ containing $f(\mathfrak{m})$. Its preimage is consequently all of $\mathrm{Spec}A$. The affine-target adjunction for schemes expresses $f$ in this chart by elements $b_j\in A$ for $j\ne i$, the images of $x_j/x_i$. It is represented by homogeneous coordinates with $a_i=1$ and $a_j=b_j$.

Conversely, a tuple $(a_0,\dots ,a_n)$ with some $a_i\in A^{\times }$ defines a morphism of schemes into $D_{+}(x_i)$ by $x_j/x_i\mapsto a_j/a_i$. Choosing another unit entry gives the same morphism of schemes, since the usual projective space transition functions identify the ratios. Multiplying all entries by one unit does not change any ratio. If two such tuples define the same morphism of schemes, choose a unit entry $a_i$ in the first and a unit entry $a_j^{\prime }$ in the second. In the second chart, the function $x_i/x_j$ pulls back to $a_i^{\prime }/a_j^{\prime }$. Because the whole map lies in $D_{+}(x_i)$, this ratio is a unit, so $a_i^{\prime }$ is a unit too. Equality in this chart gives $a_j/a_i=a_j^{\prime }/a_i^{\prime }$ for every $j$, hence $a_j^{\prime }=(a_i^{\prime }/a_i)a_j$.

Thus the correspondence is exactlyThis is the projective coordinates over a local ring description.

For a general ring, the key open-neighbourhood argument fails. Even a tuple generating the unit ideal need not have any unit entry. For example, over $A=k\times k$ the pair $((1,0),(0,1))$ defines a map to ${\mathbb{P}}^1$ whose two points have images $[1:0]$ and $[0:1]$. Neither coordinate is a unit, and no common unit multiple changes that fact. The map lies in no single standard chart. More generally, maps into projective space correspond to invertible sheaf quotients of $A^{n+1}$; the quotient need not be a free rank-one module outside the local ring case.

A closed immersion $i:X\to Y$ is a morphism of schemes whose underlying map is a homeomorphism onto a closed subset and whose map of structure sheavesis surjective. Equivalently, $X$ is the closed subscheme defined by a quasi-coherent ideal sheaf of a closed subscheme on $Y$. The affine-local characterization is particularly useful: for every affine open subscheme $V=\mathrm{Spec}A\subseteq Y$,for an ideal $I\subseteq A$, with the map induced by the quotient homomorphism.

Work over an affine open subscheme $V=\mathrm{Spec}A$ of $Z$. Since $g$ is a closed immersion, its inverse image is $\mathrm{Spec}(A/I)$ for some ideal $I$. Since $f$ is a closed immersion, the inverse image of this affine scheme in $X$ is $\mathrm{Spec}((A/I)/J)$ for an ideal $J\subseteq A/I$. Let $\tilde{J}$ be the inverse image of $J$ in $A$. The composite on this open is induced by the surjectionThe affine-local characterization from part (a) therefore shows that $g\circ f$ is a closed immersion. In particular, closed-subscheme ideals compose by taking the inverse image of the second quotient ideal.

Over an affine open subscheme $\mathrm{Spec}A$ of the target, a closed immersion has the form $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$. Pulling it back along an affine scheme map $\mathrm{Spec}B\to \mathrm{Spec}A$ givesThe identification follows directly by tensoring the quotient presentation; it does not require $B$ to be flat over $A$. Since $B\to B/IB$ is surjective, this is again a closed immersion. These affine calculations cover every base change of a morphism of schemes.

For an affine open subscheme $U=\mathrm{Spec}R$, quasi-coherence gives $\mathcal{I}{|}_U=\tilde{J}$ for an ideal $J\subseteq R$. Define $Z_U=\mathrm{Spec}(R/J)$ with its usual closed immersion into $U$. On a principal open subscheme $D(a)$, its restriction is $\mathrm{Spec}(R_a/J_a)$, because localization commutes with taking this quotient. These local constructions therefore agree on overlaps and glue.

The resulting underlying closed set is $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, locally $V(J)$. With inclusion $i:Z\hookrightarrow X$, its structure sheaf isAt a point corresponding to $\mathfrak{p}\supseteq J$, its stalk is $R_{\mathfrak{p}}/J_{\mathfrak{p}}$, the local ring of $\mathrm{Spec}(R/J)$. Thus the glued ringed space is a scheme and $i$ is a closed immersion, with ideal sheaf of a closed subscheme exactly $\mathcal{I}$. This is the construction of a closed subscheme from a quasi-coherent ideal. If $\mathcal{I}={\mathcal{O}}_X$, the construction gives the empty scheme; if $\mathcal{I}=0$, it gives $X$.

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.

A finite locally free sheaf of ideals on a reduced scheme has rank at most one at every point. On a nonempty affine open subscheme where its rank is $r$, localize its inclusion into the structure sheaf at a minimal prime ideal. The resulting local ring is a field $K$, giving an injection $K^r\hookrightarrow K$, hence $r\le 1$. This works without a Noetherian assumption. The empty scheme is a vacuous exception to claims phrased as nonexistence of a sheaf of a prescribed rank.

The underlying topological space of every scheme is a Kolmogorov space: any two distinct points are distinguished by an open subset containing just one. Choose an affine open subscheme containing the first point. If it omits the second, it already distinguishes them. Otherwise, the two points correspond to distinct prime ideals, and an element in one but not the other defines a principal open subscheme distinguishing them. Thus distinct points of a scheme have distinct closures.