# Coherence of higher direct images under proper morphisms

*Written by GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort, October 2026. Self-checked by the writing AI, GPT-6.1 Sol, at Ultra effort. Public domain (CC0).*

A proper morphism need not come with a projective embedding. Nevertheless its coherent sheaves have coherent higher direct images. The passage from projective to proper uses two different reductions. Dévissage says that a property of coherent sheaves can be tested on suitable sheaves of generic rank one on integral closed subschemes. Chow's lemma then constructs such a test sheaf by pushing down a sufficiently positive line bundle from a projective modification. Serre vanishing removes the higher direct images of this modification, allowing Leray to transfer the projective calculation.

We assume [Serre's theorems on projective schemes](serres-theorems-on-projective-schemes.md), including relative finiteness and vanishing. The existing earlier *Morphisms of schemes* unit *Projective morphisms and Chow's lemma*, Theorem 4.1 proves the Noetherian Chow construction. Lemma 4.0 below writes out the integral affine-base case needed by our dévissage and by the later proper comparisons. No finite dimension assumption is imposed on the Noetherian base.

## 1. Noetherian control of coherent sheaves

Throughout the first three sections, \(X\) is a Noetherian scheme. Every quasi-coherent subsheaf of a coherent sheaf is coherent: on an affine open it is a submodule of a finite module over a Noetherian ring. A finite affine cover also proves the ascending chain condition for these subsheaves, since the chains stabilize on every member and the finitely many indices have a maximum. This is [Stacks, Tag 01Y8].

For coherent \(F\), its support is closed. Affine-locally it is \(V(\operatorname{Ann}M)\), where \(M\) is its finite section module. If an ideal sheaf \(J\) cuts out a closed subscheme whose underlying set contains this support, then
\[
J^aF=0\quad\text{for some }a\ge0.
\]
To prove this, on each affine chart every generator of its finite ideal lies in the radical of \(\operatorname{Ann}M\). Powers of those generators kill \(M\), so a sufficiently high power of the ideal does. Take the maximum over a finite cover. Conversely, \(J^aF=0\) forces the support inside \(V(J)\), since \(J\) is the unit ideal off that set. This is the support criterion [Stacks, Tag 01Y9]. The zero sheaf presents no exception.

**Theorem 1.1 (Artin–Rees for coherent sheaves).** Let \(F\) be coherent, \(G\subset F\) quasi-coherent, and \(J\subset\mathcal O_X\) a quasi-coherent ideal. There is an integer \(c\ge0\) such that
\[
J^nF\cap G=J^{n-c}(J^cF\cap G)\qquad(n\ge c).
\]

**Proof.** First work over a Noetherian ring \(A\), with finite module \(M\), submodule \(N\), and ideal \(I\). The Rees algebra
\[
\mathcal R=A\oplus It\oplus I^2t^2\oplus\cdots
\]
is a finitely generated \(A\)-algebra, generated by \(a_it\) for a finite generating set of \(I\); it is Noetherian by the Hilbert basis theorem. The Rees module \(\bigoplus_{n\ge0}I^nM t^n\) is finite over \(\mathcal R\), generated by generators of \(M\) in degree zero. Its graded submodule
\[
\bigoplus_{n\ge0}(I^nM\cap N)t^n
\]
therefore has finitely many homogeneous generators. Let \(c\) bound their degrees. Every homogeneous element in degree \(n\ge c\) is a sum of multiples from degrees \(d\le c\), with coefficient in \(I^{n-d}\). Since
\[
I^{c-d}(I^dM\cap N)\subset I^cM\cap N,
\]
this gives the containment in \(I^{n-c}(I^cM\cap N)\). The reverse containment follows because multiplication preserves membership in \(N\) and raises the \(I\)-power in \(M\). This proves the affine statement.

Apply it on a finite affine cover of \(X\). An index valid on a chart can be increased: if \(c'\ge c\), the equality at \(c'\) and at \(n\ge c'\) gives the same formula with \(c'\). Thus the maximum of the chart indices works everywhere. Localization identifies intersections and products with their sheaf versions, giving the theorem. \(\square\)

This is [Stacks, Tag 01YA], with its underlying algebraic argument [Tag 00IN] written above. Uniformity uses quasi-compactness. On a merely locally Noetherian scheme one obtains this equality locally, without claiming a single global index.

**Lemma 1.2 (extend a morphism across a closed complement).** Let \(U=X\setminus V(J)\), and let \(F,G\) be coherent. Any map \(G|_U\to F|_U\) extends to a map \(J^aG\to F\) for some \(a\ge0\). Two such maps that agree on \(U\) agree after restricting to a sufficiently higher power. Equivalently,
\[
\underset{a}{\operatorname{colim}}\operatorname{Hom}_X(J^aG,F)
\simeq\operatorname{Hom}_U(G|_U,F|_U).
\]

**Proof.** Let \(j:U\hookrightarrow X\). A Noetherian open is quasi-compact, and this open immersion is quasi-separated. Its pushforward preserves quasi-coherence by the earlier direct-image theorem. Let \(H\subset G\oplus F\) be the inverse image of the graph of the given map on \(U\). More precisely it is the kernel of
\[
G\oplus F\longrightarrow j_*((G\oplus F)|_U/\operatorname{graph}).
\]
It is quasi-coherent and hence coherent, and restricts to the graph. Let \(K\) and \(M\subset G\) be the kernel and image of its first projection. Then \(K|_U=0\) and \((G/M)|_U=0\). The support criterion supplies powers killing these two coherent sheaves, so \(J^bG\subset M\) for some \(b\), and \(J^dK=0\) for some \(d\).

Artin–Rees applied to \(K\subset H\) gives \(J^cH\cap K=0\) for a sufficiently large \(c\): take an Artin–Rees index plus \(d\). Projection consequently identifies \(J^cH\) with \(J^cM\). This image contains \(J^{b+c}G\). Invert that projection on this subsheaf and compose with the second projection to \(F\); the resulting map extends the original one on \(U\). If two maps agree on \(U\), their difference, after restricting to a common source power, has coherent image supported on \(V(J)\). A further power kills that image, so the restrictions agree. This proves both assertions. \(\square\)

This is the coherent case of [Stacks, Tag 01YB]. The ideal power modifies the source, rather than asserting that a morphism on an open always extends to the unchanged sheaf. That distinction is what permits control of a generic isomorphism.

## 2. Integral supports and coherent filtrations

An integral closed subscheme means a closed subscheme that is reduced and irreducible. Its generic local ring is its function field. A nonzero coherent ideal on such a scheme has generic rank one and embeds into that field on every nonempty affine chart.

**Lemma 2.1 (a generic lattice).** Let \(i:Z\hookrightarrow X\) be integral, with generic point \(\eta\). If \(F\) is coherent and \(\mathfrak m_\eta F_\eta=0\), there is a coherent ideal \(I\subset\mathcal O_Z\) and an injection
\[
i_*(I^{\oplus r})\hookrightarrow F,
\qquad r=\dim_{\kappa(\eta)}F_\eta,
\]
that is an isomorphism near \(\eta\). For \(r>0\), the ideal can be chosen nonzero.

**Proof.** Let \(J\) define \(Z\), and replace \(F\) by its subsheaf annihilated by \(J\). This is a coherent subsheaf, and its stalk at \(\eta\) equals \(F_\eta\): the ideal has finitely many generators, so annihilation commutes with localization, and \(J_\eta=\mathfrak m_\eta\). The quotient has zero generic stalk, hence vanishes on some neighborhood of \(\eta\). A sheaf annihilated by \(J\) is the pushforward of a coherent sheaf \(E\) on \(Z\), by the affine module correspondence for a quotient ring.

On an affine neighborhood of \(\eta\), choose representatives of a basis of \(E_\eta\), clearing their finitely many denominators. They define a map \(\mathcal O_Z^{\oplus r}\to E\) after shrinking. Its coherent kernel and cokernel have zero generic stalk; shrink again to remove their supports, making the map an isomorphism on a nonempty open \(V\). Let \(A\subset\mathcal O_Z\) define its closed complement. Lemma 1.2 extends this map to \((A^a)^{\oplus r}\to E\). Its kernel is zero: on an affine integral chart its source embeds into a finite sum of copies of the domain, and a vector that becomes zero in the fraction field was already zero. Thus \(I=A^a\) works. It is the unit ideal on \(V\), so is nonzero when needed; the extended map is still an isomorphism there. Composing with the inclusion back into \(F\) proves the statement. \(\square\)

The case \(r=0\) means that \(F\) itself is zero near \(\eta\), and the zero source suffices. This is [Stacks, Tag 01YE], including the torsion-free injectivity that a generic isomorphism alone would not supply for an arbitrary source sheaf.

**Theorem 2.2 (filtration by ideals on integral subschemes).** Every coherent \(F\) on \(X\) has a finite filtration whose nonzero successive quotients are
\[
i_*I,
\]
where \(i:Z\hookrightarrow X\) is integral closed and \(I\subset\mathcal O_Z\) is a nonzero coherent ideal.

**Proof.** Use Noetherian induction on the closed support, proving the assertion for all sheaves with smaller support before handling the given support. This is induction under strict inclusion of closed subsets; it does not assume finite Krull dimension.

If the support is a union \(Z_1\cup Z_2\) of two strictly smaller closed subsets, let \(J\) define the reduced structure on \(Z_1\). The quotient \(F/J^aF\) is supported in \(Z_1\) for every \(a\). On \(X\setminus Z_2\), the support of \(F\) lies in \(Z_1\), so the support criterion kills \(J^aF\) there for some \(a\). This uses that this open is itself Noetherian. Hence \(J^aF\) is supported in \(Z_2\). By induction both ends of the resulting short exact sequence have the desired filtrations; taking inverse images of the quotient filtration joins them into one for \(F\).

If the support is irreducible, give it its reduced structure \(Z\), with ideal \(J\). Some power \(J^a\) kills \(F\). The finite filtration by the powers \(J^bF\) has quotients annihilated by \(J\). It suffices to handle such a quotient. It is the pushforward of a coherent sheaf on integral \(Z\). If its generic stalk is zero, its support is smaller and induction applies. Otherwise Lemma 2.1 embeds \(i_*(I^{\oplus r})\) with quotient supported in a strictly smaller closed subset. The quotient has the required filtration by induction, and the source is filtered by its \(r\) copies of \(i_*I\). Joining these filtrations completes the proof. The empty support is the zero sheaf and gives the starting case. \(\square\)

This is [Stacks, Tag 01YF]. The ideals cannot generally all be replaced by structure sheaves using only extension closure: comparing an ideal with its structure sheaf introduces a quotient that must also be controlled.

## 3. The property form of dévissage

**Theorem 3.1 (one generic-rank-one witness per integral support).** Suppose a property \(\mathcal P\) of coherent sheaves is invariant under isomorphism, holds for zero, and satisfies two out of three in short exact sequences. Assume that for every integral closed \(Z\subset X\), with generic point \(\eta\), there is a coherent sheaf \(G_Z\) such that
\[
\operatorname{Supp}G_Z=Z,\qquad
\mathfrak m_\eta(G_Z)_\eta=0,\qquad
\dim_{\kappa(\eta)}(G_Z)_\eta=1,
\]
and \(\mathcal P(G_Z)\) holds. Then \(\mathcal P\) holds for every coherent sheaf on \(X\). The result remains true if closure under direct summands is included as an additional assumption.

**Proof.** By Theorem 2.2 and extension closure it suffices to prove \(\mathcal P(i_*I)\) for every nonzero coherent ideal on every integral closed subscheme. Suppose there is a failure and choose a minimal integral closed support \(Z\) with one. Every coherent sheaf supported on a proper closed subset of \(Z\) satisfies \(\mathcal P\): its filtration factors are on strictly smaller integral closed subschemes.

Apply Lemma 2.1 to the witness \(G_Z\). Its generic rank is one, giving
\[
0\to i_*I_0\to G_Z\to Q\to0
\]
with \(Q\) supported on a proper closed subset. Thus \(\mathcal P(Q)\) and \(\mathcal P(G_Z)\) imply \(\mathcal P(i_*I_0)\). For any other nonzero coherent ideal \(I\subset\mathcal O_Z\), consider \(I\cap I_0\). At the generic point all three ideals are the function field, so the two quotients \(I_0/(I\cap I_0)\) and \(I/(I\cap I_0)\) have proper closed support. Push the two short exact sequences to \(X\); this is exact for a closed immersion. Two out of three first proves the property for the intersection, then for \(i_*I\). This contradicts the choice of \(Z\), proving the theorem. \(\square\)

This is [Stacks, Tag 01YI]. A useful stronger hypothesis is to know the property for every \(i_*\mathcal O_Z\): these sheaves themselves are rank-one witnesses. Merely saying that the property holds for a sheaf with nonzero generic stalk, of an unspecified rank, would require a different dévissage lemma. Here the witness has exactly rank one.

For the cohomology property used below, closure under direct summands is automatic. Applying any additive cohomology functor to a decomposition gives the corresponding decomposition of its values, and a direct summand of a finite module is finite. The proof nevertheless uses the precise rank-one witness, so it does not need to extract a rank-one quotient from a higher-rank sheaf without control.

## 4. A projective modification supplies the witness

**Lemma 4.0 (the integral Chow construction).** Let \(Z\) be an integral separated scheme of finite type over a Noetherian ring \(A\). There is an integral quasi-projective \(A\)-scheme \(Y\) and a projective surjection \(\pi:Y\to Z\), an isomorphism over a nonempty open. If \(Z\) is proper over \(A\), then \(Y\) is projective over \(A\).

**Proof.** Choose a finite affine cover \(U_i\) of \(Z\). Its coordinate rings are finite-type \(A\)-algebras, so finite generating sets give closed embeddings \(U_i\hookrightarrow\mathbf A_A^{n_i}\). Let \(Z_i\subset\mathbf P_A^{n_i}\) be their reduced projective closures. Their intersections with the affine charts are precisely \(U_i\). The common open \(U=\bigcap_iU_i\) is nonempty because \(Z\) is irreducible. In \(Z\times_A P\), \(P=\prod_iZ_i\), take the reduced closure \(Y\) of the graph of \(U\to P\). It is integral. The projection \(\pi\) is projective, since \(P\) is projective (the Segre monomials give its projective embedding). It is an isomorphism above \(U\), because the graph is closed over \(U\). Its image is closed and contains this dense open, hence is all of \(Z\).

Write \(h_i:Y\to Z_i\) and \(h:Y\to P\). The graph of \(U_i\to Z_i\) is closed in \(U_i\times Z_i\); the graph of \(U_i\hookrightarrow Z\) is closed in \(Z\times U_i\), by separatedness of \(Z\). Taking the closure of the common graph in these two open sets gives
\[
h_i^{-1}(U_i)=\pi^{-1}(U_i),\qquad
\pi=j_i^{-1}h_i\quad\hbox{there},
\]
where \(j_i:U_i\hookrightarrow Z_i\). The equalities hold scheme-theoretically because all restrictions are reduced and the maps agree on the dense open of the integral scheme \(Y\). On \(W_i=U_i\times\prod_{j\ne i}Z_j\subset P\), the \(i\)-th coordinate therefore recovers \(\pi\). The closed subscheme \(Y\cap(Z\times W_i)\) lies in the closed graph of this map \(W_i\to Z\); hence \(h^{-1}W_i\to W_i\) is a closed immersion. The \(W_i\) cover (h(Y)), since the \(U_i\) cover \(Z\). Thus \(h\) embeds \(Y\) as a closed subscheme of the open \(\bigcup_iW_i\subset P\), proving quasi-projectivity. If \(Z\) is proper, then \(Y\) is proper, and \(h\) is proper by the closed-graph factorization. Its image is closed; add its open complement to the \(W_i\). On this open cover \(h\) is a closed immersion, so it is a closed immersion into \(P\) globally. This proves projectivity. \(\square\)

We recall the relative Leray spectral sequence for composable morphisms of ringed spaces:
\[
E_2^{p,q}=R^pg_*(R^q\pi_*H)
\Longrightarrow R^{p+q}(g\pi)_*H.
\]
It follows from the same composition-of-derived-functors construction as the first lesson's Leray sequence: injective module sheaves are flasque; their pushforwards are flasque and hence acyclic for the next pushforward. These facts verify the acyclicity hypothesis of the composition spectral sequence. The earlier programme proof is *Derived pullback and pushforward*, Proposition 3.1, which proves pushforward composition and its Leray filtration; the finite-filtration construction in the Čech lesson also applies sheafwise. [Stacks, Tags 01F6 and 015N] are supplementary locators. In particular, if \(R^q\pi_*H=0\) for \(q>0\), the edge maps give
\[
R^pg_*(\pi_*H)\simeq R^p(g\pi)_*H
\]
in every degree. This collapse does not require \(\pi\) to be affine.

**Theorem 4.1 (Grothendieck's coherence theorem).** Let \(S\) be locally Noetherian, \(f:X\to S\) proper, and \(F\) coherent on \(X\). Then every \(R^qf_*F\), \(q\ge0\), is coherent on \(S\).

**Proof.** Restrict to an affine open \(\operatorname{Spec}R\subset S\). The ring is Noetherian, and its inverse image in \(X\) is Noetherian because \(f\) is of finite type. Coherence and higher direct images are local on the base, so it suffices to prove the assertion in this situation.

Define \(\mathcal P(F)\) by coherence of all its higher direct images. They are already quasi-coherent by the earlier quasi-compact quasi-separated direct-image theorem. The long exact sequence proves two out of three: each unknown term lies between two known coherent terms and is an extension of a quotient of one by a submodule of the other. On the Noetherian base these submodules and quotients remain coherent. The zero sheaf satisfies the property. We can therefore apply Theorem 3.1 once rank-one witnesses are constructed.

Fix an integral closed \(i:Z\hookrightarrow X\), with generic point \(\eta\), and put \(g=f|_Z\). This morphism is proper. Chow's lemma gives a proper surjective \(\pi:Z'\to Z\), an isomorphism over a dense open \(U\subset Z\), and an immersion \(h:Z'\to\mathbf P_R^m\). Since \(Z'\) is proper over \(R\), this immersion is closed. Moreover
\[
(h,\pi):Z'\hookrightarrow\mathbf P_Z^m
\]
is closed: it is the graph of \(\pi\) inside \(Z'\times_R Z\), followed by the base change of the closed immersion \(h\); the graph is closed because \(Z\) is separated over \(R\). Consequently both \(g'=g\pi\) and \(\pi\) are projective. The line bundle \(L=h^*\mathcal O(1)\) is relatively ample for both, by these embeddings.

Choose \(a\) large enough for the preceding lesson's relative Serre vanishing to hold simultaneously:
\[
R^q\pi_*L^a=0,
\qquad R^qg'_*L^a=0\qquad(q>0).
\]
The sheaf \(G=\pi_*L^a\) is coherent on \(Z\) by the already proved projective finiteness theorem. On \(U\), it is a line bundle because \(\pi\) is an isomorphism there. Thus \(G_\eta\) is one-dimensional over \(\kappa(\eta)\). Relative Leray and the first vanishing give
\[
R^pg_*G\simeq R^pg'_*L^a.
\]
The right side is zero for \(p>0\), and its degree-zero sheaf is coherent by projective finiteness. Hence \(i_*G\) is the required witness on \(X\): its support is exactly \(Z\), its generic stalk is annihilated by \(\mathfrak m_\eta\), and its higher direct images under \(f\) have the stated property. Exactness and absence of higher direct images for the closed immersion \(i\) identify them with those under \(g\). Theorem 3.1 now proves the assertion for every coherent \(F\). \(\square\)

This is [Stacks, Tag 02O5]. The projective finiteness theorem is used for \(\pi\) before any proper finiteness claim: that ordering prevents a circular argument. One sufficiently positive line bundle controls both maps; pulling back the arbitrary sheaf \(F\) and assuming its pushforward approximates \(F\) would not provide the same immediate Leray collapse.

## 5. Finite cohomology and global functions

**Corollary 5.1.** If \(X\) is proper over a Noetherian ring \(R\) and \(F\) is coherent, every \(H^q(X,F)\) is a finite \(R\)-module.

**Proof.** Over the affine base, the earlier localization theorem identifies
\[
R^qf_*F\simeq\widetilde{H^q(X,F)}.
\]
By Theorem 4.1 this sheaf is coherent. The affine module correspondence over the Noetherian ring says precisely that its section module is finite. This proves the claim, [Stacks, Tag 02O6]. \(\square\)

In particular \(B=H^0(X,\mathcal O_X)\) is a finite algebra over a field \(k\). It is Artinian, since any descending chain of ideals is a descending chain of subspaces in its finite-dimensional vector space. If \(X\) is reduced, so is \(B\): a nilpotent global section vanishes on every affine open. If \(X\) is connected, \(B\) has no nontrivial idempotent, since an idempotent section splits \(X\) into the disjoint opens where it equals zero and one. These facts show that for a nonempty connected reduced proper scheme, \(B\) is a finite field extension of \(k\); they do not by themselves force that extension to be \(k\).

To determine when this finite extension equals \(k\), one must distinguish geometric connectedness from geometric reducedness. For a field extension \(K/k\), the affine-cover complex gives
\[
B\otimes_k K\simeq H^0(X_K,\mathcal O_{X_K}),
\]
because tensoring with \(K\) is exact and the cover complex base-changes term by term. Take an algebraic closure \(\bar k\). A finite field extension \(B/k\) with separable degree greater than one makes \(B\otimes_k\bar k\) have a nontrivial idempotent, contradicting connectedness of \(X_{\bar k}\). Thus \(B/k\) is purely inseparable. Geometric connectedness alone does not eliminate this possibility: \(\operatorname{Spec}B\) for a finite purely inseparable extension is a reduced, proper, geometrically connected counterexample to \(B=k\). The correct sufficient hypothesis is that \(X\) is **geometrically reduced and geometrically connected**, or that \(k\) is perfect in addition to reducedness and geometric connectedness. Under geometric reducedness, \(B\otimes_k\bar k\) is reduced, excluding a nontrivial purely inseparable extension, and so \(B=k\).

For contrast, the affine line over \(k\) has \(H^0(\mathcal O)=k[t]\), an infinite-dimensional vector space. The morphism is separated and of finite type, but is not proper. Higher cohomology is already zero there; properness in the theorem controls the finiteness of degree zero as well.

## 6. Exercises with solutions

**Exercise 6.1 (easy: an omitted hypothesis).** Show explicitly that the conclusion on coherent pushforward fails for \(\mathbf A_k^1\to\operatorname{Spec}k\), despite vanishing of every positive higher direct image of \(\mathcal O\).

**Solution.** The source is affine, so its positive quasi-coherent cohomology groups are zero. Its zeroth group is \(k[t]\), whose monomials \(1,t,t^2,\ldots\) are linearly independent. A coherent sheaf on \(\operatorname{Spec}k\) is a finite-dimensional vector space, so this pushforward is not coherent. The missing hypothesis is properness; separated finite type alone does not supply the finiteness assertion.

**Exercise 6.2 (easy: recover the module).** For a proper morphism to \(\operatorname{Spec}R\), with \(R\) Noetherian, identify the section module of \(R^qf_*F\) and its restriction to \(D(a)\).

**Solution.** Quasi-coherence of higher direct images gives the sheaf associated to \(M=H^q(X,F)\). Its sections over the full base are \(M\), and over \(D(a)\) they are \(M_a\). The latter also equal \(H^q(f^{-1}D(a),F)\), by the localization theorem. Proper coherence proves \(M\) finite, so these are coherent restrictions. The identification comes from the natural restriction map and localization, not merely from knowing abstractly that some finite module represents the direct image.

**Exercise 6.3 (medium: decompose the global-function algebra).** If \(X\) is proper over a field, prove that \(H^0(X,\mathcal O_X)\) is a finite product of Artinian local algebras. Determine what reducedness and connectedness add.

**Solution.** The algebra \(B\) is finite-dimensional and hence Artinian. It has only finitely many maximal ideals: arbitrarily many distinct ones would, by the Chinese remainder theorem, produce quotients of arbitrarily large vector-space dimension. Their intersection \(J\) has a power equal to zero. To see this, the descending sequence \(J^a\) stabilizes; the stabilized finite module \(M\) satisfies \(JM=M\), so Nakayama's lemma for the Jacobson radical gives \(M=0\). Distinct maximal ideals are comaximal, as are their powers; with a power \(a\) killing \(J\), the product of these powered ideals is zero. Chinese remainders therefore give \(B\simeq\prod B/\mathfrak m_i^a\), with each factor Artinian local. Reducedness kills the nilpotent radicals and makes the factors fields. Connectedness excludes more than one nonzero factor by their idempotents. For nonempty connected reduced \(X\), one obtains a finite field extension, which can differ from the original field as explained above.

**Exercise 6.4 (medium: maps to affine targets).** Let \(X\) be a nonempty \(k\)-scheme with \(H^0(X,\mathcal O_X)=k\). Prove that every \(k\)-morphism \(X\to Y\) to an affine \(k\)-scheme factors through a \(k\)-rational point of \(Y\).

**Solution.** Write \(Y=\operatorname{Spec}A\). The universal property of an affine target identifies the morphism with a \(k\)-algebra map \(A\to H^0(X,\mathcal O_X)=k\). This map defines a \(k\)-point \(\operatorname{Spec}k\to Y\), and the original morphism is its composite with \(X\to\operatorname{Spec}k\). Nonemptiness ensures that the unital map is consistent with a nonzero function ring. Properness, connectedness and reducedness can help establish the assumed equality, but the factorization itself only uses that equality and the affine universal property.

**Exercise 6.5 (medium: use structure sheaves as witnesses).** Suppose \(\mathcal P\) has the closure properties of Theorem 3.1 and holds for \(i_*\mathcal O_Z\) for every integral closed \(Z\). Prove that it holds for every coherent sheaf, explaining why extension closure alone would be insufficient for this argument.

**Solution.** The structure sheaf of an integral \(Z\) has support \(Z\), generic stalk its function field, and generic rank one. Thus it is a witness in Theorem 3.1, which proves the assertion. More concretely, under induction on support, the quotient \(\mathcal O_Z/I\) of any nonzero coherent ideal has smaller support. From \(0\to I\to\mathcal O_Z\to\mathcal O_Z/I\to0\), two out of three gives the property for \(I\); the filtration theorem then gives it for all coherent sheaves. This step deduces a subobject's property from those of the middle and quotient terms. Extension closure by itself only deduces the middle term's property from the two ends, so it would not justify this deduction. If invertible twists of \(\mathcal O_Z\) are used as witnesses instead, their generic rank is still one and the same theorem applies.

**Exercise 6.6 (challenging: inseparable constants).** In characteristic \(p>0\), let \(k=\mathbf F_p(u)\), \(K=k(v)\) with \(v^p=u\), and \(X=\operatorname{Spec}K\). Verify the hypotheses that do hold and explain why geometrically connected plus reduced does not imply \(H^0(X,\mathcal O_X)=k\).

**Solution.** The polynomial \(T^p-u\) is irreducible over \(k\), since \(u\) is not a \(p\)-th power in the rational function field. Thus \(K\) is a field of degree \(p\), so \(X\) is reduced and connected, and finite over \(k\). Finite morphisms are proper, so it is proper as well. After any field extension \(k'/k\), the polynomial either remains purely inseparably irreducible or becomes a \(p\)-th power of a linear factor when its root lies in \(k'\); its quotient algebra has a single prime in either case. Hence the base-changed scheme is connected. Over an algebraic closure it is \(\bar k[\epsilon]/(\epsilon^p)\), so it is not geometrically reduced. Its global-function algebra is \(K\ne k\). The additional geometric reducedness, or perfectness of the base field with the other stated hypotheses, is exactly what removes this counterexample.

## References and the projective modification

- **[Stacks]** The Stacks project authors, *The Stacks project*, in its AI Integrated Stacks Project edition. Coherent ascending chains: [Tag 01Y8](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-acc-coherent); support and ideal powers: [Tag 01Y9](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-power-ideal-kills-sheaf); sheaf Artin–Rees: [Tag 01YA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-Artin-Rees); extension of morphisms: [Tag 01YB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-homs-over-open). Sections 1–2 supply the arguments used here, including generic injectivity and extension uniqueness.
- Generic lattices: [Tag 01YE](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-prepare-filter-irreducible); coherent filtrations: [Tag 01YF](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-coherent-filter); property dévissage: [Tag 01YI](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-property); proper coherence: [Tag 02O5](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-proposition-proper-pushforward-coherent); finite cohomology: [Tag 02O6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-proper-over-affine-cohomology-finite).
- Lemma 4.0 gives the integral Chow construction used by Theorem 4.1. The existing earlier *Projective morphisms and Chow’s lemma*, Theorem 4.1, proves the full Noetherian version; the actual reader is linked at the start of this lesson. Relative Leray and pushforward composition are proved in the earlier *Derived pullback and pushforward*, Proposition 3.1. The free Stacks locators for parallel reading are [Tag 0200](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-chow-Noetherian), [Tag 01F6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html#cohomology-lemma-relative-Leray) and [Tag 015N](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-grothendieck-spectral-sequence).
- The linked reference treatments retain the GNU Free Documentation License 1.2. The present exposition, proofs, examples and solutions are independently written CC0 content. AI Integrated Stacks Project contains AI-proposed corrections and additions and is not reviewed by maintainers of the [official Stacks project](https://stacks.math.columbia.edu/).