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h0 differs from the Euler characteristic before vanishing

Statement refuted

The Hilbert function of a coherent sheaf need not agree with its Euler-characteristic function away from large twists; the equality asserted for $m ≫ 0$ cannot be extended to all $m$ . Let $k$ be a field (Field) and let $X = P_{k}^{1}$ carry the fixed embedding $i = id : X ↪ P_{k}^{1}$ of the convention of Hilbert function and Euler characteristic on a projective scheme, so that $O_{X} (1) = O_{X} (1)$ is the twisting sheaf (Relative projective space from standard charts, Twisting sheaf on Proj) and for a coherent $G$ and $m \in Z$ the twist is $G (m) = G \otimes_{O_{X}} O_{X} (1)^{\otimes m}$ (Twists of a quasi-coherent sheaf, Tensor product of sheaves of modules). Take $F = O_{X} (- 2),$ a coherent invertible $O_{X}$ -module (Invertible sheaves, Coherent module sheaves). Then, with $h_{F} (m) = \dim_{k} H^{0} (X, F (m))$ and $P_{F} (m) = χ (X, F (m))$ (Sheaf cohomology as right derived global sections, Euler characteristic of a coherent sheaf):

$h_{F} (0) = \dim_{k} H^{0} (X, O_{X} (- 2)) = 0$ , while $P_{F} (0) = χ (X, O_{X} (- 2)) = - 1$ , because $h^{1} = \dim_{k} H^{1} (X, O_{X} (- 2)) = 1$ ;
more generally $h_{F} (m) = \max (m - 1, 0)$ and $P_{F} (m) = m - 1$ for every $m \in Z$ , so the two functions differ precisely at the twists $m \leq 0$ , while the polynomial $q (t) = t - 1$ agrees with $P_{F}$ at every integer.

Thus the Hilbert function need not equal the Hilbert polynomial at negative twists. The field $k$ is arbitrary, including $k = F_{2}$ ; $F$ is nonzero.

Facts & Assumptions

Given: The Axiom of Choice as inherited, a field $k$ , the projective line $X = P_{k}^{1}$ with its standard embedding, and the sheaf $F = O_{X} (- 2)$ .

[F1]

Conventions: with the fixed embedding of the statement, every twist $G (m)$ of a coherent $G$ is coherent, and $h_{G} (m) = \dim_{k} H^{0} (X, G (m)), P_{G} (m) = χ (X, G (m)) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (X, G (m))$ are defined for every $m \in Z$ ; a polynomial $q \in Q [t]$ with $q (m) = P_{G} (m)$ for all $m$ is a Hilbert polynomial of $G$ . (Hilbert function and Euler characteristic on a projective scheme, Sheaf cohomology as right derived global sections, Euler characteristic of a coherent sheaf, Coherent module sheaves)

[F2]

Twisting sheaves multiply: on the projective line with its twisting sheaves $O_{X} (d)$ one has $O_{X} (a) \otimes_{O_{X}} O_{X} (b) ≅ O_{X} (a + b)$ for all $a, b \in Z$ , and each $O_{X} (d)$ is invertible, so the twist of $F = O_{X} (- 2)$ is $F (m) ≅ O_{X} (m - 2)$ . (Invertible twists for degree-one generated rings, Twisting sheaf on Proj, Invertible sheaves, Tensor product of sheaves of modules, Twists of a quasi-coherent sheaf)

[F3]

Cohomology of twists on $P^{1}$ : for every field $k$ and every $d \in Z$ , writing $h^{q} = \dim_{k} H^{q} (P_{k}^{1}, O_{X} (d))$ , one has $h^{0} = \max (d + 1, 0)$ , $h^{1} = \max (- d - 1, 0)$ and $χ (P_{k}^{1}, O_{X} (d)) = d + 1$ , all higher cohomology vanishing. (All twists on the projective line, Euler characteristic of a coherent sheaf, Sheaf cohomology as right derived global sections)

[F4]

The Axiom of Choice is the choice principle named in the statement, inherited from the cohomology and finiteness suppliers cited in [F1] and [F3]. (The Axiom of Choice)

[F5]

The two standard affine charts of $P_{k}^{1}$ have coordinate rings $k [t]$ and $k [u]$ (Relative projective space from standard charts). Each is Noetherian: for a nonzero ideal, choose a nonzero polynomial of least degree; cancel the leading term of any other member by a multiple of that polynomial, and repeat until the remainder has smaller degree, hence is zero. Thus every ideal is principal. On this locally Noetherian scheme, an invertible sheaf is locally free of rank one and therefore coherent by the local kernel criterion of Coherent module sheaves (Invertible sheaves).

Counterexample

technique · direct: insert $d=m-2$ into the explicit two-dimensional cohomology computation on $\mathbb P^1$, using the multiplicativity of twisting sheaves to identify the twist of $\mathcal O_X(-2)$ with $\mathcal O_X(m-2)$, and compare the two functions at the twist $m=0$ and at negative twists

1.1F1F2F5

Identification of the twists. The projective line is locally Noetherian by [F5], and $F = O_{X} (- 2)$ is invertible by [F2], so [F5] establishes that this concrete $F$ is coherent. By [F2] its twist is $F (m) ≅ O_{X} (- 2) \otimes O_{X} (m) ≅ O_{X} (m - 2)$ for every $m \in Z$ ; the twist remains coherent by [F1].

1.2F1F31.1

The two functions. Fix $m \in Z$ and apply [F3] with $d = m - 2$ , using the identification of 1.1: $h_{F} (m) = \dim_{k} H^{0} (X, O_{X} (m - 2)) = \max (m - 1, 0),$ $P_{F} (m) = χ (X, O_{X} (m - 2)) = (m - 2) + 1 = m - 1.$ In particular, at $m = 0$ one gets $h_{F} (0) = \max (- 1, 0) = 0$ and $P_{F} (0) = - 1$ , the latter because $h^{1} = \max (1, 0) = 1$ in [F3].

1.3F11.2algebra

The polynomial and the comparison. The polynomial $q (t) = t - 1 \in Q [t]$ satisfies $q (m) = m - 1 = P_{F} (m)$ for every $m \in Z$ by 1.2, so it is an Euler-characteristic polynomial of $F$ in the sense of [F1], and $q (0) = - 1$ . Since $h_{F} (m) = \max (m - 1, 0)$ while $q (m) = m - 1$ , the two functions agree exactly for $m \geq 1$ and differ for every $m \leq 0$ ; no polynomial in $Q [t]$ can agree with $h_{F}$ at all integers, because such a polynomial would have to agree with $q$ at the infinitely many $m \geq 1$ and hence equal $q$ , contradicting $h_{F} (0) \neq q (0)$ .

2.1F1F2F3F41.3∎

Boundaries and choice. The field $k$ is arbitrary, including $k = F_{2}$ where $1 / (x_{0} x_{1})$ and the signs of the alternating sum still make sense; the sheaf $F = O_{X} (- 2)$ is nonzero of support $X$ , so neither the empty scheme nor the zero sheaf is involved. The endpoint $m = 0$ is the exhibited disagreement, the endpoints $m \leq 0$ are all covered by 1.3, and the twist conventions for negative powers of an invertible sheaf are those of [F1] and [F2]. The Axiom of Choice is inherited through [F4] and the suppliers of [F1] and [F3]; no further selection is made.

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Sources

The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2 (standard reference, not scraped)