Regularity and bounded families

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

A Grassmannian can parametrize a fixed-dimensional space of sections. To parametrize sheaf quotients, however, we must know that one space of sections determines every quotient under consideration. The necessary integer must work for the whole family, rather than for each sheaf separately. Castelnuovo–Mumford regularity supplies the relevant control, and Mumford's theorem supplies a uniform integer when the Hilbert polynomial is fixed.

We first turn a diagonal pattern of cohomology vanishing into multiplication and generation statements. We then prove the uniform bound by controlling the first cohomology of a kernel. The proof makes clear why the hypothesis that the sheaves are quotients of a fixed sheaf matters. Examples and solutions show how to compute regularity without already having a Hilbert scheme.

We assume coherent sheaves on projective space, finite sets of associated points, long exact cohomology sequences, and Serre's vanishing and generation theorem. The latter says that, for any coherent \(F\), sufficiently large twists have zero higher cohomology and are generated by global sections [Stacks, Tags 01XO, 0B5T and 02O1]. The cohomology of line bundles is in Tag 01XT. Basic references for the results proved here are [Stacks], [Nitsure], and [Vakil].

1 The diagonal vanishing condition

Let \(F\) be coherent on \(\mathbb P^n_k\). It is \(m\)-regular if

Only \(1\leq i\leq n\) need to be checked. The integer is a bound on cohomology and generation; smaller integers give stronger assertions. When a smallest integer exists it is denoted \(\operatorname{reg}(F)\). Some sheaves, including a sheaf supported in dimension zero, are regular for every integer, so no smallest integer exists for them.

Lemma 1.1. Regularity is invariant under arbitrary extension of the field \(k\).

Proof. Cover projective space by its finitely many standard affine opens. All their finite intersections are affine, and their Čech complex computes the cohomology of a quasi-coherent sheaf. After a field extension \(k\subset K\), the complex for \(F_K\) is its tensor product with \(K\). Flatness gives

Faithful flatness says that one vector space vanishes exactly when the other does. This applies at each of the twists in the definition. \(\square\)

We can therefore enlarge the field to an infinite one when choosing a hyperplane. The associated points of \(F\) are finite. Hyperplanes containing a specified associated point form a proper closed linear condition in the dual projective space. A finite union of those conditions does not exhaust its points over an infinite field. Choose a linear form \(\ell\) whose zero hyperplane \(H\) avoids all associated points. Multiplication by \(\ell\) is injective, giving

This includes sheaves with embedded associated points. Avoiding only irreducible components of the support would not suffice.

2 Why regularity controls sections

Theorem 2.1. If \(F\) is \(m\)-regular, then:

1. \(H^i(F(t))=0\) for \(i>0\) and \(t\geq m-i\); in particular \(F\) is \((m+1)\)-regular;
2. multiplication \(H^0(F(t))\otimes H^0(\mathcal O(1))\to H^0(F(t+1))\) is surjective for \(t\geq m\);
3. \(F(t)\) is globally generated and has zero higher cohomology for \(t\geq m\).

Proof. Enlarge the field as in Lemma 1.1, and induct on \(n\). For \(n=0\), a coherent sheaf is a finite-dimensional vector space, and the assertions hold for every twist. Choose \(H\) as in Section 1 for the induction step.

The long exact sequence at twist \(m-i\) places \(H^i(F_H(m-i))\) between \(H^i(F(m-i))\) and \(H^{i+1}(F(m-i-1))\). Both vanish. Thus \(F_H\) is \(m\)-regular on \(H\simeq\mathbb P^{n-1}\), and all three assertions are available there by induction.

For \(i>0\), the group \(H^i(F_H(t))\) vanishes for \(t\geq m-i\). Consequently \(H^i(F(t-1))\to H^i(F(t))\) is surjective for those \(t\). Starting from the known zero group at \(m-i\) and moving upward proves assertion 1.

For \(t\geq m\), the restriction maps

are surjective because \(H^1(F(t-1))\) and \(H^1(F(t))\) vanish. Linear forms on \(H\) lift to linear forms on projective space. Given \(s\in H^0(F(t+1))\), use multiplication surjectivity on \(H\) to express its restriction as a sum of products. Lift both factors to obtain a sum of products upstairs. Subtract this sum from \(s\). The difference restricts to zero, hence is \(\ell s'\) for some \(s'\in H^0(F(t))\). It too is a product with a linear form. This proves assertion 2.

Repeated multiplication shows that \(H^0(F(t))\otimes H^0(\mathcal O(a))\to H^0(F(t+a))\) is surjective for every \(a\geq0\). Here products of linear forms span the homogeneous degree-\(a\) polynomials. For large \(a\), Serre's theorem says that \(F(t+a)\) is globally generated. All its generating sections are sums of such products, so the evaluation map from \(H^0(F(t))\), after twisting by \(\mathcal O(a)\), is surjective. Tensoring back proves generation of \(F(t)\). Its cohomology vanishing follows from assertion 1.

Finally, surjectivity of multiplication and generation descend through a field extension: their cokernels become zero after faithful flat pullback. Thus the original field satisfies the same conclusions. \(\square\)

We will need a propagation observation that does not assume regularity of \(F\) itself.

Lemma 2.2. Suppose \(F_H\) is \(m_0\)-regular and the restriction map \(H^0(F(t))\to H^0(F_H(t))\) is surjective for one \(t\geq m_0\). It is then surjective for all larger twists.

Proof. Every section of \(F_H(t+1)\) is a sum of products of sections of \(F_H(t)\) with linear forms. Lift the sections by the assumed surjectivity and lift the linear forms from \(H\). The products give lifts at \(t+1\). Induct. \(\square\)

3 Hilbert polynomials and a numerical budget

For coherent \(F\), put \(\chi(F(t))=\sum_i(-1)^i\dim_k H^i(F(t))\). This is a polynomial \(P_F(t)\) in \(t\), and it agrees with \(h^0(F(t))\) for large \(t\). Here is a useful proof of its polynomial nature. Over an infinite field, the hyperplane exact sequence gives

By induction on the ambient dimension, the right side is a polynomial of degree at most \(n-1\). Taking a discrete antiderivative and choosing its constant value to match \(\chi(F(0))\) gives a polynomial agreeing with \(\chi(F(t))\) for every integer \(t\), positive or negative. The case \(n=0\) is constant. Lemma 1.1 gives the assertion over the original field as well. Serre vanishing proves eventual agreement with \(h^0\).

For a nonzero sheaf with support dimension \(d>0\), a general hyperplane lowers the support dimension by one and leaves a nonzero restriction. Induction shows that \(P_{F_H}\) has degree \(d-1\) with positive leading coefficient. Its discrete antiderivative therefore has degree \(d\) with positive leading coefficient. For support dimension zero, \(\chi(F(t))\) is the positive, constant length over \(k\), including residue-field degrees. Thus the degree of \(P_F\) is the support dimension.

The essential extra fact for boundedness is that a subsheaf \(K\subset\mathcal O^r\) satisfies, for \(t\geq0\),

A fixed Hilbert polynomial by itself gives no such ambient bound. The next theorem combines this inequality with the propagation lemma to control how long first cohomology can persist.

4 A uniform bound for all kernels

Theorem 4.1 (Mumford's bound). Fix \(n\geq0\), \(r\geq1\), and \(P\in\mathbb Q[t]\). There is an integer \(M=M(n,r,P)\), independent of the field, such that in every exact sequence

the kernel \(K\) is \(M\)-regular.

Proof. We prove a slightly more directly reusable statement: a subsheaf \(K\subset\mathcal O^r\) of prescribed Hilbert polynomial \(Q\) has a uniform regularity bound \(B(n,r,Q)\). Apply it to

Set \(B(0,r,Q)=0\). If no such subsheaf exists, any bound is harmless. Suppose \(n>0\), extend the field to an infinite one, and choose a hyperplane avoiding the associated points of \(F=\mathcal O^r/K\). It is also a nonzerodivisor on \(K\), since \(K\) is a subsheaf of the free sheaf on integral projective space. The vanishing of \(\operatorname{Tor}_1(F,\mathcal O_H)\) makes

exact. Therefore the induction hypothesis applies to \(K_H\), whose polynomial is

Choose

Theorem 2.1 makes \(K_H\) \(m_0\)-regular. For \(i\geq2\), the hyperplane cohomology sequence shows that

is an isomorphism whenever \(t\geq m_0-i+1\), because the adjacent groups of \(K_H(t)\) vanish. Serre vanishing at large twists then propagates backwards along these isomorphisms, giving

Write \(b_t=h^1(K(t))\). For \(t\geq m_0\), the remaining sequence includes

Thus \(b_{t-1}\geq b_t\), with equality exactly when \(\rho_t\) is surjective. If equality holds at one such \(t\), Lemma 2.2 makes all later restriction maps surjective. Hence \(b_{t-1}=b_t=b_{t+1}=\cdots\). This constant is zero by Serre vanishing. We conclude that for \(t\geq m_0\), the sequence of nonnegative integers strictly decreases at every step whose preceding value is positive.

At twist \(m_0\), the higher cohomology with index at least two is zero. Therefore

Set

For polynomials realized by such subsheaves the expression bounding \(b_{m_0}\) is an integer. If it is not an integer for other input polynomials, replace it by its ceiling; no realized case is changed. After at most \(N\) further steps, \(b_{m_0+N}=0\). This is the required vanishing of \(H^1(K(B-1))\). For \(i\geq2\), \(B-i\geq m_0-i\), so the already established vanishing gives \(H^i(K(B-i))=0\). Thus \(K\) is \(B\)-regular.

Every integer in the recursion depends only on \(n,r,Q\). Lemma 1.1 descends the bound to the original field. This completes the induction and proves the theorem. \(\square\)

The bound is intentionally coarse. Its purpose is uniformity, and no assertion of optimality is needed for the moduli construction. The proof also exposes the dangerous step: mere weak decrease of \(h^1\) would give no bound on how many twists it can survive. The propagation lemma rules out a positive plateau.

Corollary 4.2. Quotients of a fixed coherent sheaf \(E\) on \(\mathbb P^n_k\), with prescribed Hilbert polynomial, have uniformly regular kernels and quotients.

Proof. Choose \(a\) so that \(E(a)\) is globally generated, and a surjection \(\mathcal O^N\to E(a)\), with fixed kernel \(K_0\). For a quotient \(E\to F\), let \(K'\) be the kernel of \(\mathcal O^N\to F(a)\), and \(K\) the kernel of \(E\to F\). Its polynomial is obtained from \(P_F\) by replacing \(t\) with \(t+a\), so Theorem 4.1 uniformly bounds \(K'\).

The exact sequence \(0\to K_0\to K'\to K(a)\to0\) shows that \(K(a)\) is \(b\)-regular whenever \(K'\) is \(b\)-regular and \(K_0\) is \((b+1)\)-regular: at twist \(b-i\), the adjacent groups are \(H^i(K'(b-i))\) and \(H^{i+1}(K_0(b-i))\). Both vanish. Increase the uniform \(b\) to satisfy the fixed second condition. Then \(K\) is \((a+b)\)-regular.

Likewise \(0\to K'\to\mathcal O^N\to F(a)\to0\) makes \(F(a)\) \(b\)-regular for a sufficiently enlarged \(b\), by the same long exact sequence. Twisting back gives a uniform bound for \(F\). \(\square\)

5 A finite-dimensional parameter family

A collection of coherent sheaves on \(\mathbb P^n_k\) is bounded if there is a finite-type \(k\)-scheme \(B\) and a coherent sheaf \(\mathcal F\) on \(\mathbb P^n_k\times B\) whose fibres include the collection. Families and fibres can also be taken after field extensions. We first use this definition without requiring \(\mathcal F\) to be flat over \(B\); it can be refined to flat families as explained below.

Corollary 5.1. Quotients of \(\mathcal O^r\) with fixed Hilbert polynomial \(P\) form a bounded family. The same holds for quotients of any fixed coherent sheaf with fixed Hilbert polynomial.

Proof. Choose \(M\geq0\) from Theorem 4.1. The kernel \(K(M)\) is generated by its sections and has zero higher cohomology. Its number of sections is

If no quotient exists there is nothing to prove. Otherwise \(0\leq h\leq N=r\binom{M+n}{n}\). Regard \(H^0(K(M))\) as an \(h\)-dimensional subspace of \(H^0(\mathcal O^r(M))\simeq k^N\). The Grassmannian of such subspaces is \(B=\operatorname{Gr}_{N-h}(k^N)\) in the quotient convention. Let \(\mathcal V\subset k^N\otimes\mathcal O_B\) be its universal subbundle. On \(\mathbb P^n\times B\), form the coherent cokernel

of the evaluation map. Cokernels commute with pullback. At the point whose subspace is \(H^0(K(M))\), generation says that the evaluation image is exactly \(K\). Thus the fibre is exactly the desired quotient \(F\). The Grassmannian is finite type, proving boundedness without using existence of a Quot scheme.

For fixed \(E\), the presentation in Corollary 4.2 makes its quotients, after one fixed twist, a subcollection of a bounded family of quotients of \(\mathcal O^N\). Twisting the constructed family back proves the assertion. \(\square\)

To obtain flat parameter families, use generic flatness on each reduced irreducible component of the parameter scheme. Remove a dense open where the restricted sheaf is flat, then repeat on the reduced closed complement. There are finitely many components and their dimensions decrease at each round, so finitely many locally closed reduced strata suffice. The disjoint union of these strata is finite type, the restricted families are flat, and every original fibre is preserved. Generic flatness is a prerequisite here, rather than the projective flattening theorem to be proved in the next lesson.

Finite unions of bounded collections are bounded by taking disjoint unions of parameter schemes. Fixed twists preserve boundedness by twisting the parameter sheaf. Direct sums of two bounded collections are bounded on the product of their parameter schemes, by pulling back and taking the direct sum. These simple permanence statements do not assert that all extensions of two bounded collections are bounded; that requires control of the extension parameters.

6 Examples that reveal the endpoints

The cohomology of line bundles gives

Intermediate cohomology of a line bundle is zero. The only possible obstruction is \(H^n(\mathcal O(d+m-n))\), which is nonzero precisely when \(d+m-n\leq-n-1\). For \(n=0\), all higher cohomology is zero and every integer works. The dimension restriction is essential to the stated equivalence.

A plane curve defined by a nonzero homogeneous degree-\(d\) equation has ideal sheaf \(\mathcal O(-d)\), so its ideal has regularity exactly \(d\). No smoothness or reducedness assumption on the curve is needed for this computation.

If \(Z\subset\mathbb P^n\) has length \(d\geq1\), its ideal is \(d\)-regular. Enlarge the field, choose a hyperplane disjoint from \(Z\), and trivialize \(\mathcal O_Z(1)\) by its equation. Let \(B=\Gamma(Z,\mathcal O_Z)\), and filter it by the images of affine polynomials of degree at most \(j\). The dimension starts at one. If two successive filtered spaces agree, multiplication by every affine coordinate preserves that space; since the coordinates generate \(B\), the space is already all of \(B\). It therefore reaches \(B\), of dimension \(d\), by \(j=d-1\). Hence \(H^0(\mathcal O(t))\to H^0(\mathcal O_Z(t))\) is onto for \(t\geq d-1\).

The ideal sequence gives \(H^1(\mathcal I_Z(t))=0\) in this range. For \(i\geq2\), it identifies \(H^i(\mathcal I_Z(d-i))\) with \(H^i(\mathcal O(d-i))\), which is zero. This proves the bound, including nonreduced point schemes, and Lemma 1.1 descends it to finite fields. The position of the points can improve the bound; length alone gives this uniform answer.

A warning about partial coefficients

The fixed-polynomial theorem above should be distinguished from a stronger statement about only some of the coefficients. For a coherent sheaf \(F\), let \(N_s(F)\) be its largest subsheaf supported in dimension less than \(s\), and put \(F_{(s)}=F/N_s(F)\). This largest subsheaf exists: the sum of all such subsheaves stabilizes by Noetherianity, and a finite sum still has support dimension less than \(s\).

The corrected form of FGA, Exposé 221, Théorème 2.2 says the following. On a fixed projective family over a Noetherian base, fix a coherent source sheaf and consider quotients of its fibres, allowing field extensions. If their Hilbert-polynomial coefficients in degrees at least \(s-1\) are bounded, their dimension truncations \(F_{(s)}\) form a bounded family; their coefficients in degree \(s-2\) are bounded below. Coefficients in negative degrees are interpreted as zero. The printed hypothesis on page 254 instead uses degrees at most \(s-1\). That printed condition is false, as the following calculation shows.

Work over an algebraically closed field of characteristic zero. For every \(d\geq3\), take a smooth plane curve \(C_d\) of degree \(d\) and a finite subscheme \(Z_d\) of length \(d(d-3)/2\). The two restriction maps exhibit

as a quotient of \(\mathcal O_{\mathbb P^2}^{\oplus2}\). The exact sequence for a degree-\(d\) equation gives

Adding the constant length of \(Z_d\) therefore gives \(P_{F_d}(t)=dt\). For \(s=1\), all coefficients of degrees at most zero are fixed. Since a smooth curve has no nonzero subsheaf supported on finitely many points, \((F_d)_{(1)}=\mathcal O_{C_d}\). This family is not bounded: a finite-type parameter scheme for coherent sheaves has only finitely many fibre Hilbert polynomials. Indeed, generic flatness decomposes such a parameter scheme into finitely many reduced locally closed flat pieces, and flat projective cohomology makes the polynomial locally constant on each piece. The polynomials here have unbounded coefficient \(d\).

This proves the failure of the printed inequality. Section 9 proves the positive theorem with the corrected inequality. Theorem 4.1 and Corollary 5.1 establish the fixed-polynomial regularity and boundedness statements independently of that strengthening.

7 Exercises

1. Basic. Compute regularity for \(\mathcal O(d)\) in every ambient dimension, including dimension zero, and for the ideal of a degree-\(d\) plane curve.

2. Intermediate. Let \(C\subset\mathbb P^3\) be the image of \([s:t]\mapsto[s^3:s^2t:st^2:t^3]\). Prove that its ideal sheaf is 2-regular, with no smaller integer working. Establish the resolution you use.

3. Intermediate. Prove the length bound for \(\mathcal I_Z\) by the polynomial filtration argument. Explain why it also works for nonreduced \(Z\), and give an example where the bound is sharp.

4. Intermediate. Prove field-extension invariance of regularity using an affine Čech complex. Explain which step needs faithful flatness.

5. Advanced. Carry out the recursive bound in Theorem 4.1 for \(n=1\) and then for \(n=2\). Explain why a positive plateau in first cohomology is impossible, and why the last \(+1\) in the bound is present.

8 Solutions

1. For \(n\geq1\), the line-bundle computation in Section 6 gives \(\operatorname{reg}(\mathcal O(d))=-d\). For \(n=0\), there is no smallest regularity integer because every integer works. A degree-\(d\) equation on \(\mathbb P^2\) has ideal \(\mathcal O(-d)\); this gives regularity \(d\), including for a multiple or reducible equation.

2. In \(R=k[x_0,x_1,x_2,x_3]\), set

These vanish on the parametrization. To verify that they generate its ideal, take a monomial order refining total degree and the weights \((1,3,3,1)\). Their leading monomials, after changing the signs of the generators, are \(x_1^2,x_1x_2,x_2^2\). The two adjacent S-polynomials reduce by the third generator: for \(g_1=x_1^2-x_0x_2\), \(g_2=x_1x_2-x_0x_3\), \(g_3=x_2^2-x_1x_3\),

The remaining pair also reduces: its S-polynomial is \(x_2^2g_1-x_1^2g_3=-x_0x_2g_3+x_1x_3g_1\). Thus the three polynomials are a Gröbner basis. The standard monomials are monomials in \(x_0,x_3\) multiplied by \(1,x_1\), or \(x_2\). There are \(3t+1\) of degree \(t\geq1\). The degree-\(t\) image under the parametrization is the space of binary forms of degree \(3t\): every monomial of degree \(3t\) is a product of \(t\) binary cubics. It has the same dimension \(3t+1\). The surjection onto that image is therefore an isomorphism in every degree, proving the ideal assertion.

There are two syzygies, the columns of

The map \(R(-3)^2\xrightarrow A R(-2)^3\) is injective: its columns are independent over the fraction field, and its source is torsion-free. The Hilbert function just computed gives

for every \(t\), with negative-degree spaces zero. Comparing dimensions of the kernel of \(R(-2)^3\to I\) shows that its already contained image of \(A\) has exactly the required dimension in every degree. Thus it equals that kernel. Sheafifying gives the exact resolution

The line-bundle cohomology sequence gives \(H^1(\mathcal I_C(1))=0\), \(H^2(\mathcal I_C)=0\), and \(H^3(\mathcal I_C(-1))=0\): the potentially adjacent top groups of \(\mathcal O(-3)\) on \(\mathbb P^3\) are zero. Hence the ideal is 2-regular. If it were 1-regular, Theorem 2.1 would make \(\mathcal I_C(1)\) globally generated. It is a nonzero sheaf with no global sections, since the ideal contains no linear forms. This is impossible. Monotonicity excludes every smaller integer as well.

3. Nilpotents do not affect the argument: \(B\) is a finite-dimensional algebra generated by the affine coordinates, with dimension equal to the length. If its degree filtration ever stops growing, stability under these generators makes the filtered space the whole algebra. At most \(d-1\) strict increases can follow its initial dimension one. The resulting surjectivity of sections and the ideal sequence prove all diagonal vanishings, exactly as in Section 6. On \(\mathbb P^1\), a length-\(d\) closed subscheme is the zero divisor of a degree-\(d\) homogeneous equation, so its ideal is \(\mathcal O(-d)\), with regularity \(d\). This proves sharpness of the length-only bound.

4. The standard affine cover and its intersections are affine, so their section complex computes cohomology of a quasi-coherent sheaf. A field extension tensors every term of this complex with the larger field. Flatness commutes with taking its cohomology, giving \(H^i(F_K(t))=H^i(F(t))\otimes_kK\). Faithful flatness is needed for the converse implication: the vanishing of this tensor product forces the original group to be zero. It is also what descends surjectivity of the multiplication maps and the evaluation map used in Theorem 2.1.

5. Write \(Q(t)=at+b\) when \(n=1\). Its hyperplane difference is \(a\), and the dimension-zero bound is zero. Thus \(m_0=0\), \(N=\max(0,r-b)\), and one possible bound is \(1+\max(0,r-b)\). For an actual subsheaf the inequality \(h^1(K)\leq r-b\) guarantees that this number is meaningful.

When \(n=2\), first apply the dimension-one formula to \(\Delta Q\), with the same \(r\), obtaining \(m_0\geq0\). Then use

At a plateau, the connecting map from hyperplane sections to first cohomology vanishes, so restriction is surjective. Multiplication of sections on the regular hyperplane propagates this surjectivity forever. The plateau must therefore be zero, because Serre vanishing eventually makes first cohomology zero. Starting with at most \(N\) at twist \(m_0\) reaches zero by twist \(m_0+N\). The definition of \(M\)-regularity tests first cohomology at \(M-1\), which accounts for the final \(+1\).

9 The corrected partial-coefficient theorem

We now work over a Noetherian scheme \(S\), with a projective \(X\to S\) and a fixed relatively very ample line bundle \(\mathcal O_X(1)\). A member is a coherent sheaf on \(X_K\), for a field-valued point \(\operatorname{Spec}K\to S\). Boundedness means that its isomorphism class occurs in a coherent family on \(X_T\), with \(T\) of finite type over \(S\), after passing to a common extension of its field and a residue field of \(T\). This spells out the geometric field-extension convention used in Section 5. In particular we may test and construct the bounds over algebraically closed extensions; the numerical polynomial is unchanged by those extensions.

This section uses the fixed-polynomial Quot scheme proved in Hilbert and Quot schemes. That construction uses only the uniform kernel bound and the fixed-polynomial boundedness established in Sections 4–5, together with the independent flattening argument. It does not use the partial-coefficient theorem below. The forward reference therefore gives an acyclic route to this later strengthening.

We shall use two elementary consequences of boundedness. A bounded collection has finitely many Hilbert polynomials: stratify its Noetherian parameter scheme into finitely many reduced locally closed pieces on which the family is flat, and use local constancy of the polynomial. It also has a common quotient source of the form \(\mathcal O_X(-m)^q\): relative generation on a finite affine cover of that parameter scheme gives one sufficiently large \(m\) and a finite upper bound \(q\) for the generators needed on its fibres. Both assertions hold over a general Noetherian base; finite stratification follows by Noetherian induction, without a finite-dimensional-base assumption.

Conversely, quotients of one fixed coherent source with finitely many Hilbert polynomials are bounded. For each polynomial take the projective Quot scheme and its universal quotient, then take their finite disjoint union. This is the fixed-polynomial criterion we need here.

Maps and extensions have finite parameter families

Lemma 9.1. On a projective \(X/S\), kernels, images and cokernels of maps between two bounded collections are bounded. Middle terms of extensions between those collections are bounded as well.

Proof. Take the product of the two parameter schemes over \(S\), and stratify it so the two coherent families \(G,H\) are flat. Call the resulting Noetherian parameter scheme \(T\). We first construct a finite-type parameter scheme for all homomorphisms \(G_t\to H_t\), including after field extension.

Locally on a finite affine cover of \(T\), choose a presentation

Choose \(m\) large enough for generation of \(G\) and vanishing of higher fibre cohomology of \(H(m)\). Relative Serre generation then gives \(P_0\), and gives \(P_1\) after increasing \(n\); also arrange the same vanishing for \(H(n)\). Proper flat cohomology [Stacks, Tag 0A1H] makes \(p_*H(m)\) and \(p_*H(n)\) finite locally free, compatibly with arbitrary base change. Maps \(P_0\to H\) are therefore parametrized by the vector bundle with section module \((p_*H(m))^a\). The condition that the composite with \(P_1\) be zero is a finite set of linear equations with values in \((p_*H(n))^b\). Its zero scheme parametrizes exactly the maps from \(G\), since the presentation remains right exact after every base change. These local parameter schemes glue by their common Hom functor. They give a scheme of finite type over \(T\), with a universal homomorphism.

The kernel, image and cokernel of the universal map are coherent. Stratify the parameter scheme further so its cokernel is flat. The source and target families were already flat. It follows that the image is flat, and that both exact sequences defining image and kernel remain exact after base change. Their fibres are consequently the actual kernels, images and cokernels of the fibre maps. This proves their boundedness; the extra stratification prevents confusing a fibre of a kernel with a kernel of a fibre map.

For extensions \(0\to H_t\to E\to G_t\to0\), choose \(P_0\to G\) as above with \(m\) also ensuring \(H^1(H_t(m))=0\) on every fibre. Put \(K=\ker(P_0\to G)\); it is coherent and flat because \(P_0,G\) are flat. The pullback of any such extension to \(P_{0,t}\) splits, since

Thus its middle term is the pushout associated to some map \(\phi:K_t\to H_t\):

Apply the Hom construction to \(K,H\), and make this cokernel using its universal map. The inclusion of the graph remains injective on every fibre because \(K_t\to P_{0,t}\) is injective. Therefore (9.2) has the exact ends \(H_t,G_t\) on each fibre. Every extension occurs, after choosing its splitting over \(P_{0,t}\), in this finite-type family. The boundedness of its middle terms follows. \(\square\)

Bounded degree bounds reduced supports

Lemma 9.2. For fixed integers \(r,D\geq0\), the reduced pure-dimensional subschemes of dimension \(r\) and degree at most \(D\) in geometric fibres of \(X/S\) belong to finitely many finite-type embedded parameter families. Their structure sheaves are bounded.

Proof. On a finite affine cover of \(S\), embed \(X\) in \(\mathbb P^N_S\) using the chosen twist. We construct the required finite-degree incidence families, including the reduction step.

For a reduced pure-dimensional \(Z\subset\mathbb P^N_k\) over an algebraically closed field, let \(e=\deg Z\). There is a polynomial \(F_Z\) in the coefficients of \(r+1\) hyperplanes, homogeneous of degree \(e\) in each hyperplane's coefficient group, whose zero set consists of tuples having a common point on \(Z\). Here is the existence argument. For an integral component \(Z_i\), consider the incidence variety of a point on \(Z_i\) and hyperplanes all passing through it, inside

It is integral, and has dimension one less than the coefficient space. Projection to that space is proper because \(Z_i\) is projective. Its image has codimension one: choose \(r\) general hyperplanes cutting \(Z_i\) in finitely many points, then a last hyperplane through one point and avoiding the others. That tuple has a finite incidence fibre. The image is therefore an integral hypersurface. The polynomial coefficient ring is a UFD, so its reduced equation is a single irreducible polynomial. Independent rescaling of the hyperplanes preserves the image; its equation is consequently homogeneous in each coefficient group.

Its degree in any group is \(\deg Z_i\). Fix the other \(r\) hyperplanes generally. They meet \(Z_i\) in \(\deg Z_i\) distinct smooth transverse points. To justify this choice, its singular locus has dimension at most \(r-1\), and hyperplane differentials at a smooth point span the cotangent space; the bad incidence conditions have smaller dimension. The remaining hyperplane must pass through one of these points. Each resulting linear condition occurs once: near a tuple meeting just that point transversely, the incidence point varies étale with the first \(r\) hyperplanes, and the image equation is the last linear form evaluated at that point, with nonzero derivative in its coefficients. Thus the restricted equation is the product of the distinct point-evaluation linear forms. Multiplying the equations of the finitely many components gives \(F_Z\), of multidegree \((e,\ldots,e)\). For \(r=0\), this is directly the product of the evaluation forms at the reduced points. The construction works in every characteristic.

The coefficients of such polynomials of fixed multidegree range over a projective space of finite type over \(\mathbb Z\). For any polynomial \(F\) in that space, define its incidence subscheme by the condition

This is a scheme defined by finitely many equations. On a standard chart of \(x\), eliminate one coefficient of each \(H_i\) using \(H_i(x)=0\), substitute in \(F\), and set every coefficient of the remaining hyperplane variables to zero. Those equations glue on overlaps, since they express the same polynomial restriction. Normalizing a coefficient of \(F\) on a coefficient-space chart likewise gives compatible equations, independent of its scalar normalization.

For \(F=F_Z\), the underlying incidence set is exactly \(Z\). A point of \(Z\) satisfies (9.3). For \(x\notin Z\), the linear system of hyperplanes through \(x\) has no base point on \(Z\), so \(r+1\) successive general members cut its dimension down to the empty set. The resulting tuple has \(F_Z\ne0\), proving the converse. Moreover the incidence set for any nonzero \(F\) has dimension at most \(r\). Otherwise every \(r+1\)-tuple of hyperplanes would meet a component of dimension at least \(r+1\), and (9.3) would make \(F\) vanish on all tuples, contradicting its nonzero coefficient vector.

Intersect these universal incidence schemes with \(X\), and take the finitely many degrees \(1\leq e\leq D\). Their geometric fibre reductions include all the desired \(Z\). We still must realize those reductions by finite-type families; reducing the total family alone would not guarantee reduced fibres.

For any one incidence family over its Noetherian parameter scheme \(T\), use Noetherian induction. On an integral component, [Stacks, Tag 0550] gives, over a nonempty open of \(T\), a finite universal homeomorphism \(T'\to T\) for which the reduction of the pulled-back family has geometrically reduced generic fibre. Shrink further so all its fibres are geometrically reduced [Stacks, Tag 0578]. Its geometric fibres are now exactly the reduced fibres of the original family: the reduction map is a closed universal homeomorphism, and a reduced closed subscheme with the same underlying set is unique. This supplies an embedded finite-type family over that open. Apply the induction hypothesis to its proper closed complement, and to the finitely many components. A finite union supplies all reductions. The empty scheme can be included as a separate zero family. This proves the lemma. \(\square\)

Finite projection preserves and detects boundedness

Lemma 9.3. Let \(f:Y\to Z\) be finite, with \(Y,Z\) projective over a Noetherian base. A collection of coherent sheaves \(G\) on its field fibres is bounded if and only if the collection \(f_*G\) is bounded.

Proof. Forward implication follows by pushing forward a bounding family. Finite pushforward preserves coherence and commutes with base change, as is checked on its affine module description.

Conversely let \(H\) be a family bounding the pushforwards on a finite-type parameter scheme \(T\), and stratify so \(H\) is flat. Put \(\mathcal A=f_*\mathcal O_Y\), pulled back to \(Z_T\). An \(\mathcal O_Y\)-module on a fibre is an \(\mathcal A\)-module on that fibre of \(Z\). Parametrize its action by the finite-type Hom scheme for \(\mathcal A\otimes H\to H\) constructed in Lemma 9.1. That construction only needs the target flat: a coherent presentation of the source remains right exact under base change. Unit and associativity are closed conditions. To see this precisely, their differences are homomorphisms from the coherent sources \(H\) and \(\mathcal A\otimes\mathcal A\otimes H\) to the flat target \(H\). Compose with finite presentations by negative twists as in (9.1); their vanishing is a finite set of section equations. On this closed parameter scheme the universal action corresponds, by the affine module equivalence for \(f\), to a coherent sheaf on \(Y\). Every original \(f_*G\), with its transported \(\mathcal A\)-action, occurs after a common field extension. This family therefore bounds the \(G\). \(\square\)

Now suppose a projective family \(Y/T\) has fibres of dimension at most \(r\), with its very ample bundle inherited from \(\mathbb P^N_T\). The \(r+1\) linear forms defining projections to \(\mathbb P^r\) have a finite-type parameter scheme. The locus where their common centre misses \(Y\) is open, since the failure locus is the image of a projective incidence scheme. It meets every geometric fibre: a general centre of dimension \(N-r-1\) misses a projective scheme of dimension at most \(r\). If \(N<r\), first embed the same projective space as a linear subspace of a larger one.

On that open parameter scheme there is a universal morphism

It is finite. It is proper, and its fibres have dimension zero: a positive-dimensional proper fibre would contain a curve, on which the very ample \(\mathcal O_Y(1)\) would be trivial by (9.4), contradicting its positive degree. Properness with finite fibres gives finiteness [Stacks, Tag 02LS]. Thus we may apply Lemma 9.3 and work on \(\mathbb P^r\). Passing to this finite-type projection parameter scheme preserves the geometric notion of boundedness.

Rank and determinant control the top pure piece

Write the top two coefficients in normalized form

This differs from the ordinary coefficients by fixed positive factors, so it preserves every boundedness assertion.

Lemma 9.4. Let \(Y/T\) be as in (9.4), and let \(F\) run through quotients of a fixed coherent source on its field fibres. For \(r\geq1\), the \(a_F\) are bounded and the \(b_F\) are bounded below. If the \(b_F\) are also bounded above, the \(F_{(r)}\) are bounded. For \(r=0\), all the quotients are bounded.

Proof. Finite pushforward preserves the polynomial, by the projection formula, and preserves dimensions of supports. It therefore carries \(N_r(F)\) to \(N_r(f_*F)\); the assertion follows either from their section-support descriptions or from the affine module equivalence. Lemma 9.3 reduces the boundedness question to \(\mathbb P^r_T\).

Twist the pushed-forward source enough and cover \(T\) finitely so all its fibre quotients become quotients of \(\mathcal O^q\), with fixed \(q\). The twist changes \(b_F\) by a fixed multiple of \(a_F\). On \(\mathbb P^r\), \(a_F\) is its generic rank, so \(0\leq a_F\leq q\). This justifies that the fixed twist will preserve the eventual upper and lower bounds.

Put \(G=F_{(r)}\). It is torsion-free on the integral regular scheme \(\mathbb P^r\): every torsion subsheaf has smaller-dimensional support and has been removed. If \(G=0\), its \(b\)-coefficient is zero. Otherwise its rank is \(a\geq1\), and

The double dual is a line bundle. Locally on a UFD, a rank-one module can be scaled to an ideal; dividing out the gcd of its generators gives an ideal \(J\) with gcd one. Any element of its fraction field that multiplies \(J\) into the ring has no denominator prime, since such a prime would divide every generator. Hence \(J^*=A\) and \(J^{**}=A\). This proves the rank-one assertion. The identification with \(\mathcal O(c)\) uses \(\operatorname{Pic}(\mathbb P^r)=\mathbb Z\) [Stacks, Tag 0BXJ]. The quotient \(\mathcal O^q\to G\) gives a nonzero map

so \(c\geq0\).

We compute the next coefficient without assuming that torsion-free sheaves agreeing away from codimension two agree everywhere. The non-locally-free locus of \(G\) has codimension at least two: at each codimension-one point the local ring is a DVR and a finite torsion-free module is free. Choose a general line \(L\simeq\mathbb P^1\) avoiding that locus. The Koszul complex of the \(r-1\) hyperplanes cutting out \(L\), tensored with \(G\), resolves \(G|_L\). Near \(L\) this follows from local freeness, and away from \(L\) one equation is a unit so the complex is contractible. Additivity of Euler characteristics gives

On the line, \(G|_L\) has rank \(a\) and determinant of degree \(c\). Curve Riemann–Roch [Stacks, Tag 0BS6] gives its polynomial \(a(t+1)+c\). Therefore

For \(r=1\), the same computation directly uses \(G\) on the whole projective line. Since \(N_r(F)\) has dimension at most \(r-1\), its coefficient in that degree is nonnegative. Hence \(b_F\geq b_G\geq0\) after the fixed twist, proving a uniform lower bound before untwisting as well.

An upper bound on \(b_F\) bounds \(c\), and \(a\) already has finitely many possibilities. For fixed \(a,c\), the maps (9.6) lie in a finite-dimensional coefficient space. Each gives the contraction map

For the determinant map of an actual quotient, this factors through \(G\), because (9.6) factors through \(\wedge^aG\). The induced map from \(G\) is injective at the generic point: wedge pairing of a rank-\(a\) vector space with its \((a-1)\)-st exterior power is nondegenerate, and the original generators span that space. Its kernel is thus a torsion subsheaf of the torsion-free \(G\), so is zero globally. The image of (9.8) is exactly this embedded \(G\), since \(\mathcal O^q\) surjects onto it. Lemma 9.1 bounds the images of all these finitely parametrized maps. Lemma 9.3 then recovers boundedness on \(Y\).

For \(r=0\), finite projection is to \(\mathbb P^0=T\). The quotients of the finite pushed-forward source have bounded vector-space dimension on field fibres. Their constant polynomials range through a finite set, so the fixed-polynomial criterion bounds them. This completes the proof. \(\square\)

Lemma 9.5. Suppose only that the supports of the fixed-source quotients \(F\) have dimension at most \(r\), without requiring the ambient fibres to have that dimension. If their top coefficients \(a_F\) are bounded, the next coefficients \(b_F\) are bounded below. If both are bounded above, the \(F_{(r)}\) are bounded. For \(r=0\), a bound on the constant coefficient bounds the entire collection.

Proof. Work after algebraic closure of each field and put \(G=F_{(r)}\). It has only associated points of dimension \(r\): maximality of \(N_r\) rules out lower-dimensional submodules, and the support bound rules out higher-dimensional points. The leading-term multiplicity formula [Stacks, Tag 0BEN] says

where the \(Z_i\) are the irreducible components of its reduced support. Thus one bound \(D\) controls their total reduced degree and each generic length. Lemma 9.2 gives embedded parameter families for these reduced supports \(Z\).

If \(J\) is the ideal of such a support, then \(J^D G=0\). At each associated point \(\eta_i\), the ideal is the maximal ideal of its local ring, whose \(D\)-th power kills a module of length at most \(D\). Globally a nonzero submodule \(J^D G\) would have an associated point among those of \(G\), where we have just shown its localization zero. This is impossible. Consequently \(G\) is a sheaf on the fibre of the bounded thickened support family

Its fibres have dimension at most \(r\), and \(G\) is a quotient of the restriction of the original coherent source to those fibres. Lemma 9.4 applies to this projective family. It gives the lower bound for \(b_G\) and, when \(b_G\) is bounded above, boundedness of \(G\). Because \(b_F-b_G\) is the leading coefficient in degree \(r-1\) of \(N_r(F)\), it is nonnegative. Thus \(b_F\) has the stated lower bound, and an upper bound for \(b_F\) supplies one for \(b_G\). If \(r=0\), \(N_0(F)=0\), and the last case of Lemma 9.4 bounds \(F\) itself. \(\square\)

Induction through the dimension filtration

We need two properties of the dimension filtration, in addition to its existence proved in Section 6:

and formation of \(N_s,F_{(s)}\) commutes with extension of the field. For (9.10), a submodule of \(F_{(s)}\) of dimension less than \(s\) would lift to a larger forbidden small-dimensional submodule of \(F\). All its associated points therefore have dimension at least \(s\). Associated points of \(F\) of such dimension lie outside the support of \(N_s\), so survive localization into the quotient; the associated-prime properties of the exact sequence rule out any additional points outside that small support.

For the field-extension assertion, the flat associated-prime formula [Stacks, Tag 0312] reduces to the case of an integral finite-type coordinate ring \(A\). Its base extension has no associated prime contracting to a nonzero prime of \(A\), because every nonzero element of \(A\) remains a nonzerodivisor. At the generic fibre the ring is \(\operatorname{Frac}(A)\otimes_k K\), which is Cohen–Macaulay [Stacks, Tag 045M], so its associated primes are minimal. The resulting components have the same dimension as \(A\): a transcendence basis of \(\operatorname{Frac}(A)/k\) gives a finite free extension after tensoring of the corresponding localized polynomial ring over \(K\); its minimal primes lie over zero, and their fraction fields have that same transcendence degree over \(K\). The dimension–transcendence-degree formula [Stacks, Tag 0A21] gives the claim. Thus field extension preserves the dimensions of the associated components. The extension of \(N_s(F)\) still has support of dimension less than \(s\), whereas the quotient has no associated point of that dimension. Maximality identifies it with \(N_s(F_K)\), as required.

Theorem 9.6 (corrected FGA partial-coefficient boundedness). Fix a coherent source \(E\) on the projective \(X/S\), and a collection of quotients of its field fibres. Fix an integer \(s\). If the coefficients of their Hilbert polynomials in degrees at least \(s-1\) are bounded, then the \(F_{(s)}\) are bounded. Their original polynomials' coefficients in degree \(s-2\) are bounded below. Coefficients in negative degrees mean zero.

Proof. If \(s\leq0\), every coefficient of every polynomial is bounded. Their degrees have one ambient bound, and integer-valued polynomials of that bounded degree lie in a fixed discrete coefficient lattice: the binomial-polynomial basis has integer coordinates, so ordinary coefficients have denominators dividing the factorial of that degree bound. Only finitely many polynomials occur. The fixed-source, fixed-polynomial criterion then bounds the entire collection; \(F_{(s)}=F\), and the requested negative-degree coefficient is zero.

Assume \(s\geq1\), and induct on an upper bound \(r\) for the support dimensions. If \(r\leq s-2\), all truncations are zero; the coefficient in degree \(s-2\) is zero or a nonnegative leading coefficient. If \(r=s-1\), the truncations remain zero. For \(s=1\), the degree \(-1\) coefficient is zero by convention. For \(s\geq2\), Lemma 9.5, using the bounded leading coefficient in degree \(r=s-1\), bounds the next coefficient in degree \(s-2\) below. These give all starting cases.

Let \(r\geq s\). The coefficients in degrees \(r,r-1\) are bounded by the hypothesis. Lemma 9.5 shows that \(G=F_{(r)}\) is bounded. Lemma 9.1 now bounds the kernels \(K\) of all composites \(E_K\to F\to G\). Here the source fibres themselves form a bounded collection, with parameter \(S\) and sheaf \(E\). Each \(N_r(F)\) is a quotient of the corresponding \(K\). The common-source consequence of boundedness stated at the start of this section therefore gives a single coherent quotient source for the collection of \(N_r(F)\).

Since

and bounded \(G\) have finitely many polynomials, the coefficients of \(P_{N_r(F)}\) in degrees at least \(s-1\) are bounded. These sheaves have support dimension at most \(r-1\). Induction applies, giving boundedness of \((N_r(F))_{(s)}\) and a lower bound for the degree-\(s-2\) coefficient of the untruncated \(N_r(F)\).

The small-support subsheaf \(N_s(F)\) lies in \(N_r(F)\), and its maximality gives \(N_s(N_r(F))=N_s(F)\). Hence there is an exact sequence

Both ends are bounded. Lemma 9.1 bounds their extension middle terms, proving boundedness of \(F_{(s)}\). For the coefficient lower bound, use (9.11) itself: the requested coefficient of \(P_F\) is the sum of the lower-bounded coefficient of \(P_{N_r(F)}\) and a coefficient of \(P_G\) ranging through a finite set. It is therefore bounded below. Using the untruncated polynomial here retains the small-dimensional parts that (9.12) has removed. This completes both inductions and the theorem. \(\square\)

As a consequence, if every associated component of every \(F\) has dimension between \(s\) and \(r\), bounding just the coefficients in degrees \(s-1,\ldots,r\) bounds the full collection: (9.10) gives \(F_{(s)}=F\), and there are no coefficients above \(r\). The common-source hypothesis remains part of this conclusion.

What this lesson does not prove

We use coherent cohomology on an affine Čech cover, Serre's theorem on ample twists [Stacks, Tags 01XO, 0B5T and 02O1], finiteness of associated primes and their detection of zero divisors [Stacks, Tags 00LC–00LD], line-bundle cohomology [Stacks, Tag 01XT], and generic flatness [Stacks, Tag 051R]. Local constancy of the fibre polynomial in flat projective families is a consequence of the proper flat cohomology complex [Stacks, Tag 0A1H], also explained in Flattening stratifications.

The uniform kernel bound, its extension to a fixed coherent source, the bounded parameter family and the corrected partial-coefficient theorem are proved here. Section 9 additionally uses these exact prerequisites:

• Fixed-polynomial projective Quot representability, proved in Hilbert and Quot schemes using only Sections 4–5 of this lesson. This is the stated forward dependency; the partial-coefficient theorem is not an input to that construction.
• A finite universal-homeomorphism base change and reduction can make a finite-type generic fibre geometrically reduced [Stacks, Tag 0550]; a geometrically reduced generic fibre becomes a family of geometrically reduced fibres over a dense open [Stacks, Tag 0578]. Their role in the reduced-support parameter families is isolated in Lemma 9.2.
• Proper morphisms with finite fibres are finite [Stacks, Tag 02LS]; finite morphisms give the affine module equivalence, coherent pushforward and base change. Lemma 9.3 proves the additional boundedness assertion using explicit finite-type action parameters.
• The projective-space Picard group is \(\mathbb Z\) [Stacks, Tag 0BXJ], and curve Riemann–Roch gives \(\chi(V(t))=\operatorname{rank}(V)(t+1)+\deg\det V\) on \(\mathbb P^1\) [Stacks, Tag 0BS6]. Lemma 9.4 derives the required higher-dimensional coefficient formula from these facts.
• The leading Hilbert coefficient is the sum of component degrees weighted by generic lengths [Stacks, Tag 0BEN]. Associated primes under a flat tensor product are described in [Stacks, Tag 0312]; a tensor product of fields, one of finite type, is Cohen–Macaulay [Stacks, Tag 045M]; dimension of an integral finite-type scheme equals the transcendence degree of its function field [Stacks, Tag 0A21]. These give the field-extension and multiplicity steps used in the dimension filtration.

Boundedness alone does not supply the scheme representing the flat quotient functor. Its universal flat-family classification is the separate proof in the fifth lesson.

References

• [Stacks] The Stacks project, Varieties, Castelnuovo–Mumford regularity, Tags 08A2–08AG, retained in AI Integrated Stacks Project Stacks.
• [Nitsure] N. Nitsure, Construction of Hilbert and Quot Schemes, 2005, Section 2. Mumford's uniformity argument is presented there in a stronger polynomial-bound form.
• [Vakil] R. Vakil, The Rising Sea: Foundations of Algebraic Geometry, public draft of 27 July 2024, Section 18.6 and the chapters on projective cohomology.
• [FGA] A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique IV : les schémas de Hilbert, Séminaire Bourbaki, Exposé 221, Théorème 2.2. The inequality there is corrected above.
• [AI Integrated Stacks Project] Moduli, remark on the coefficient inequality and dimension-truncation proposition. These AI-written additions have not been reviewed by the Stacks project maintainers.