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Compensating h0 and h1 jumps with constant Euler characteristic

Statement

Let $k$ be a field (Field), let $S = Spec k [a]$ with origin $(a)$ and generic point $η$ , and let $E$ be the rank-two $S$ -flat vector bundle on $X = P_{S}^{1}$ of An upper jump of h0 in a flat projective family, glued from the frames $(u_{0}, v_{0})$ over $U_{0}$ and $(u_{1}, v_{1})$ over $U_{1}$ by $u_{1} = z^{- 2} u_{0}, v_{1} = v_{0} + a z^{- 1} u_{0},$ with $0 \to O_{X} (- 2) \to E \to O_{X} \to 0$ . For a point $u \in S$ let $X_{u}$ be the fibre, $E_{u}$ the pullback of $E$ and $h^{q} (E_{u}) = \dim_{κ (u)} H^{q} (X_{u}, E_{u}) (q = 0, 1),$ with cohomology as in Sheaf cohomology as right derived global sections and Euler characteristic $χ (E_{u}) = h^{0} (E_{u}) - h^{1} (E_{u})$ (Euler characteristic of a coherent sheaf). Then

at the origin, $(h^{0}, h^{1}) (E_{(a)}) = (1, 1)$ ;
at every other point $u \neq (a)$ of $S$ , including every closed point and the generic point $η$ , $(h^{0}, h^{1}) (E_{u}) = (0, 0)$ ;
$χ (E_{u}) = 0$ on every fibre, so the Euler characteristic is constant although $h^{0}$ and $h^{1}$ jump; this agrees with the local constancy of Euler characteristic in a proper flat family is locally constant and shows that the corollary cannot be strengthened to local constancy of the individual $h^{q}$ .

The field $k$ is arbitrary, including $k = F_{2}$ .

Facts & Assumptions

Given: The Axiom of Choice (and the Axiom of Dependent Choice through the Euler-characteristic corollary), a field $k$ , the base $S = Spec k [a]$ and the glued rank-two $S$ -flat bundle $E$ on $P_{S}^{1}$ of the companion example.

[F1]

The bundle and its fibre sequences: $E$ is locally free of rank two, finitely presented and flat over $S$ , and sits in an exact sequence $0 \to O_{X} (- 2) \to E \to O_{X} \to 0$ with extension cocycle $a x_{0}^{- 1} x_{1}^{- 1}$ , so that the fibre over a point $u \in S$ with residue field $κ (u)$ and scalar $λ = a (u)$ sits in $0 \to O (- 2) \to E_{u} \to O \to 0$ with extension class $λ \cdot [1 / (x_{0} x_{1})]$ , a class that vanishes at the origin and is nonzero at every other point. Moreover $X \to S$ is proper of finite presentation. (An upper jump of h0 in a flat projective family, Flat and faithfully flat modules and ring homomorphisms, Locally finite presentation morphisms, The residue field at a point of an affine scheme)

[F2]

Cohomology of the twisting sheaves on $P^{1}$ : for every field $κ$ and every $d \in Z$ one has $h^{0} (O (d)) = \max (d + 1, 0)$ and $h^{1} (O (d)) = \max (- d - 1, 0)$ ; in particular $H^{0} (O) ≅ κ$ , $H^{1} (O) = 0$ and $H^{0} (O (- 2)) = 0$ , $H^{1} (O (- 2)) ≅ κ$ . All higher cohomology vanishes. (All twists on the projective line, Sheaf cohomology as right derived global sections)

[F3]

The long exact sequence of the fibre sequence: $0 \to H^{0} (O (- 2)) \to H^{0} (E_{u}) \to H^{0} (O) \overset{δ_{u}}{\to} H^{1} (O (- 2)) \to H^{1} (E_{u}) \to H^{1} (O) \to 0$ is exact, with $H^{0} (O)$ and $H^{1} (O (- 2))$ one-dimensional $κ (u)$ -vector spaces by [F2], and the connecting map $δ_{u}$ is multiplication by the extension class $λ \cdot [1 / (x_{0} x_{1})]$ , hence the zero map when $λ = 0$ and an isomorphism when $λ \neq 0$ . The companion An upper jump of h0 in a flat projective family computes the connecting map directly: lifting $1$ by $v_{0}$ and $v_{1}$ on the standard affine charts gives the Čech coboundary $v_{1} - v_{0} = λ z^{- 1} u_{0}$ and hence $δ_{u} (1) = λ [1 / (x_{0} x_{1})]$ . (Long exact sequence of sheaf cohomology, Exact sequences of sheaves, [F1], [F2])

[F4]

Euler characteristics: for a point $u$ the Euler characteristic $χ (E_{u}) = h^{0} (E_{u}) - h^{1} (E_{u})$ is a finite alternating sum, and for a proper morphism of finite presentation with a finitely presented module flat over the base the function $s \mapsto χ (X_{s}, F_{s})$ is locally constant on $S$ (Euler characteristic in a proper flat family is locally constant, Euler characteristic of a coherent sheaf); the corollary inherits the Axiom of Choice and the Axiom of Dependent Choice (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct: read the two low-degree cohomology groups of each fibre off the long exact sequence of $0\to\mathcal O(-2)\to\mathcal E_u\to\mathcal O\to0$, whose outer terms are known explicitly on $\mathbb P^1$, distinguishing only whether the connecting map (multiplication by the scalar $a(u)$ between one-dimensional spaces) is zero or an isomorphism, and compare the resulting alternating sums with the locally constant Euler characteristic

1.1F1F2F3

The sequence and its outer terms. Fix $u \in S$ and write $λ = a (u) \in κ (u)$ . By [F1] the fibre sequence $0 \to O (- 2) \to E_{u} \to O \to 0$ is exact, and by [F3] its long exact cohomology sequence begins and ends as $0 \to H^{0} (O (- 2)) \to H^{0} (E_{u}) \to H^{0} (O) \overset{δ_{u}}{\to} H^{1} (O (- 2)) \to H^{1} (E_{u}) \to H^{1} (O) \to 0,$ with $H^{0} (O (- 2)) = 0$ , $H^{0} (O) ≅ κ (u)$ and $H^{1} (O (- 2)) ≅ κ (u)$ by [F2] applied with $d = 0$ and $d = - 2$ , and $H^{1} (O) = 0$ .

1.2F1F2F3

The connecting map. By [F3] the map $δ_{u} : H^{0} (O) \to H^{1} (O (- 2))$ sends $1$ to the extension class $λ \cdot [1 / (x_{0} x_{1})]$ ; under the one-dimensional identifications of [F2], it is multiplication by $λ$ . Hence $δ_{u} = 0$ when $λ = 0$ , and $δ_{u}$ is an isomorphism when $λ \neq 0$ .

1.3F21.11.2

The special fibre. At the origin $u = (a)$ one has $λ = 0$ , so $δ_{u} = 0$ ; exactness of the sequence of 1.1 gives $H^{0} (E_{(a)}) ≅ H^{0} (O) ≅ κ (u)$ and $H^{1} (E_{(a)}) ≅ H^{1} (O (- 2)) ≅ κ (u)$ , the maps $H^{0} (O (- 2)) \to H^{0} (E_{u})$ and $H^{1} (E_{u}) \to H^{1} (O)$ having zero source and target respectively. Therefore $(h^{0}, h^{1}) (E_{(a)}) = (1, 1)$ .

1.4F21.11.2

The other fibres. If $u \neq (a)$ then $λ \neq 0$ : a prime of $k [a]$ containing $a$ contains the maximal ideal $(a)$ and hence equals $(a)$ , so this includes all closed points of arbitrary residue degree and the generic point — so by 1.2 the map $δ_{u}$ is an isomorphism between one-dimensional spaces. Exactness of 1.1 then gives $H^{0} (E_{u}) = \ker δ_{u} = 0$ and $H^{1} (E_{u}) = coker δ_{u} = 0$ , that is, $(h^{0}, h^{1}) (E_{u}) = (0, 0)$ .

1.5F1F41.31.4

The Euler characteristic. By 1.3, $χ (E_{(a)}) = 1 - 1 = 0$ ; by 1.4, $χ (E_{u}) = 0 - 0 = 0$ for every $u \neq (a)$ . Hence $χ (E_{u}) = 0$ for every $u \in S$ , a constant, in agreement with the local constancy of [F4] applied to the proper morphism of finite presentation $X \to S$ and the finitely presented module $E$ flat over $S$ .

2.1F3F41.11.31.41.5∎

Boundaries and consistency. The field $k$ is arbitrary, including $k = F_{2}$ ; the special fibre is the origin $(a)$ with residue field $k$ and the generic fibre is computed over $k (a)$ ; every other point, including closed points of higher residue degree, falls under 1.4. The example shows that the Euler characteristic can be locally constant — here constant $0$ — while $h^{0}$ and $h^{1}$ both jump from $0$ to $1$ at the origin; their contributions to the alternating sum have opposite signs, so it is unchanged; the local constancy in [F4] therefore cannot be improved to local constancy of the individual $h^{q}$ . The Axiom of Choice and the Axiom of Dependent Choice are inherited through [F4] and the long exact sequence of [F3], and the connecting-map identification is proved by the companion example's two-chart lift calculation; 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)