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A Hilbert polynomial bounds regularity independently of ambient dimension

Statement

Assume AC and DC. For every numerical polynomial $P$ there is an integer $b (P) \geq 0$ such that, over every field and for every $n, p \geq 0$ , the kernel and quotient of any coherent quotient $O_{P^{n}}^{p} ↠ F$ with Hilbert polynomial $P$ are $b (P)$ -regular. The bound is independent of both $n$ and $p$ . For realizable nonzero $P$ it can be defined recursively by $a = b (Δ P)$ , $b (P) = a + \max (0, P (a)) + 1$ , where $Δ P (t) = P (t) - P (t - 1)$ and $b (0) = 0$ . For unrealizable polynomials any bound suffices.

Facts & Assumptions

Given: AC and DC and the hypotheses of the statement.

[F1]

Regularity propagation and section multiplication are supplied in Regularity gives generation, multiplication, and vanishing. For fixed $n, p$ and the polynomial of $I$ , Uniform regularity for all quotients with a fixed Hilbert polynomial supplies some regularity bound; propagation therefore gives eventual vanishing of every positive cohomology group of $I$ . Associated points are finite and detect zero divisors (Finite modules over Noetherian rings have finitely many associated primes, Zero divisors on a module over a Noetherian ring are the union of its associated primes).

[F2]

A nonzero coherent sheaf on projective space has Hilbert polynomial of degree equal to its support dimension and nonzero leading coefficient (Degree of the coherent Hilbert polynomial). In particular zero polynomial means the zero sheaf.

[F3]

On projective space, coherent cohomology commutes with every field extension (Flat field extension commutes with coherent cohomology). Twists commute with pullback, as is seen from their standard-chart transition functions; hence the Hilbert polynomial and regularity are preserved. Vanishing descends because a vector space whose scalar extension is zero is itself zero. Flatness also preserves the quotient sequence and its kernel.

[F4]

The Hilbert polynomial equals the Euler characteristic at every integral twist (Euler characteristic is a Hilbert polynomial), and Euler characteristic is additive in coherent exact sequences (Euler characteristic is additive in short exact sequences). The cohomology of twists gives $h^{0} (O (t)) = (\binom{n + t}{n})$ for $t \geq 0$ and shows that $O$ is $0$ -regular (Cohomology of O(d) on projective space).

Proof

1.1F1F2F3F4construct

Induct on the degree of a realizable nonzero $P$ , adjoining the zero polynomial as the initial case. If $P = 0$ , [F2] gives $F = 0$ and the kernel is $O^{p}$ , which is $0$ -regular. If $n = 0$ , all coherent sheaves are regular in every degree, so the proposed bound also works. Otherwise extend to the infinite field $k (u)$ using [F3], and choose a hyperplane $H$ avoiding the associated points of $F$ and of its kernel $I$ . Tor exactness gives $0 \to I_{H} \to O_{H}^{p} \to F_{H} \to 0$ , and the restriction sequence and [F4] give $P_{F_{H}} = Δ P$ . For degree-zero $P$ the hyperplane misses the finite support, so $F_{H} = 0$ ; in all other cases [F2] reduces the induction degree. Thus $I_{H}$ is $a$ -regular with $a = b (Δ P)$ , independent of $n, p$ .

2.1F1F3F4step 1.1algebra∎

The exact hyperplane sequence and [F1] give $H^{i} (I (t)) = 0$ for $i \geq 2$ and $t \geq a - i$ : compare successive twists using vanishing of the two adjacent groups of $I_{H}$ , then iterate to an eventual-vanishing twist supplied by [F1]. They also show that $h^{1} (I (t))$ decreases strictly after twist $a$ until it becomes zero, because equality $h^{1} (I (t)) = h^{1} (I (t + 1))$ for $t \geq a$ makes $H^{0} (I (t + 1)) \to H^{0} (I_{H} (t + 1))$ onto; multiplication for $I_{H}$ propagates that surjectivity, so equality would persist to the eventual zero value. At twist $a \geq 0$ , the vanishings and [F4] therefore give $h^{1} (I (a)) = h^{0} (I (a)) - χ (I (a)) \leq p (\binom{n + a}{n}) - (p (\binom{n + a}{n}) - P (a)) = P (a) .$ The entire ambient term cancels. After at most $\max (0, P (a))$ decreases, $H^{1} (I (b (P) - 1)) = 0$ ; the other regularity positions vanish by the sharper range above. Thus $I$ is $b (P)$ -regular and the exact sequence with $O^{p}$ makes $F$ regular as well. Field descent in [F3] proves the result over all fields.

Depends on

Used by

Global Hilbert strata in a coherent projective bundle over a locally Noetherian base Lemma

Dependency tree · two levels

121 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nitsure, Construction of Hilbert and Quot Schemes, Sections 2–5 (standard reference, not scraped)
Grothendieck, Les schémas de Hilbert, Bourbaki 221, Section 3 (standard reference, not scraped)