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Cohomology of twists on a smooth hypersurface

Statement

Assume the Axiom of Choice, inherited from the projective-space cohomology suppliers (The Axiom of Choice). Let $k$ be a field, $n \geq 2$ , let $f \in S = k [x_{0}, \dots, x_{n}]$ be homogeneous of degree $d \geq 1$ (The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials), and let $X = Z (f) \subseteq P_{k}^{n}$ be the associated hypersurface; assume $X$ is smooth over $k$ of pure dimension $n - 1$ (Relative Jacobian criterion with its presentation hypothesis). Write $I = I_{X}$ for its ideal sheaf and $O_{X} (d)$ for the restriction of the twisting sheaf (Twisting sheaf on Proj). Then:

multiplication by $f$ gives a short exact sequence $0 ⟶ O_{P^{n}} ⟶ O_{P^{n}} (d) ⟶ i_{*} O_{X} (d) ⟶ 0$ of quasi-coherent sheaves (Zero scheme of a line-bundle section, A section of an invertible sheaf has a canonical zero subscheme, Quasi-coherent module on a scheme); in particular $I ≅ O_{P^{n}} (- d)$ and $I / I^{2} ≅ O_{X} (- d)$ ;
$H^{0} (X, O_{X} (d)) ≅ (S / (f))_{d}$ , of dimension $(\binom{n + d}{d}) - 1 = (\binom{n + d}{n}) - 1$ (Nonnegatively graded rings and modules, homogeneous elements, and twists);
$H^{q} (X, O_{X} (d)) = 0$ for every $q > 0$ ;
the normal sheaf of $X$ in $P^{n}$ is $N_{X / P^{n}} = H o m_{O_{X}} (I / I^{2}, O_{X}) ≅ O_{X} (d)$ (Invertible sheaves, Dual of a line bundle is its tensor inverse, Internal Hom of module sheaves).

Facts & Assumptions

Given: a field $k$ , an integer $n \geq 2$ , a homogeneous form $f \in S = k [x_{0}, \dots, x_{n}]$ of degree $d \geq 1$ with $X = Z (f) \subseteq P_{k}^{n}$ smooth over $k$ of pure dimension $n - 1$ , the closed immersion $i : X ↪ P_{k}^{n}$ and its ideal sheaf $I = I_{X}$ ; the Axiom of Choice is assumed as declared in the Statement.

[F1]

For every $t \in Z$ , $H^{q} (P_{k}^{n}, O (t)) = 0$ unless $q = 0$ or $q = n$ ; $H^{0} (P_{k}^{n}, O (t)) ≅ S_{t}$ for $t \geq 0$ and $H^{0} (P_{k}^{n}, O (t)) = 0$ for $t < 0$ ; and $H^{n} (P_{k}^{n}, O (t)) = 0$ for $t > - n - 1$ . In particular $H^{q} (P_{k}^{n}, O) = 0$ for all $q \geq 1$ , and $H^{q} (P_{k}^{n}, O (d)) = 0$ for all $q \geq 1$ because $d \geq 1 > - n - 1$ . (Cohomology of O(d) on projective space, Global sections of projective twists, Relative projective space from standard charts, Nonnegatively graded rings and modules, homogeneous elements, and twists)

[F2]

The zero scheme $Z (f)$ of the global section $f \in Γ (P^{n}, O (d))$ has ideal sheaf $I = Im (c_{f})$ , where $c_{f} : O (- d) \to O$ is multiplication by $f$ , and $O_{P^{n}} / I ≅ i_{*} O_{X}$ ; on an affine chart trivializing $O (d)$ the local equation is the dehomogenization of $f$ . (Zero scheme of a line-bundle section, A section of an invertible sheaf has a canonical zero subscheme, Closed subschemes of projective space and saturated ideals, Twisting sheaf on Proj)

[F3]

For every quasi-coherent $O_{X}$ -module $F$ and every $q \geq 0$ , $H^{q} (X, F) ≅ H^{q} (P_{k}^{n}, i_{*} F)$ . (Closed immersion preserves cohomology and coherent pushforward)

[F4]

A short exact sequence of sheaves of abelian groups on a topological space induces a long exact sequence in cohomology. (Long exact sequence of sheaf cohomology)

[F5]

Twisting by the invertible sheaf $O (t)$ is an exact functor on $O_{P^{n}}$ -modules, $O (- d) \otimes_{O} O (d) ≅ O$ and $H o m_{O} (O (- d), O) ≅ O (d)$ ; for an invertible $O_{X}$ -module $L$ one has $L^{\lor} \otimes_{O_{X}} L ≅ O_{X}$ and $H o m_{O_{X}} (L, O_{X}) ≅ L^{\lor}$ . (Invertible sheaves, Dual of a line bundle is its tensor inverse, Internal Hom of module sheaves, Twisting sheaf on Proj)

[F6]

$S = ⨁_{m \geq 0} S_{m}$ has $S_{0} = k$ and is a domain: the leading monomials in a lexicographic order multiply with nonzero product coefficient. The degree- $d$ monomials form a basis; their exponent tuples sum to $d$ and are counted by placing $n$ separators among $n + d$ positions, giving $\dim_{k} S_{d} = (\binom{n + d}{n})$ . If $g \neq 0$ is homogeneous and $g f \in S_{d}$ , then $\deg g = 0$ , so $g \in k$ ; the zero multiplier is also in $k$ . (Nonnegatively graded rings and modules, homogeneous elements, and twists, The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials)

Proof

technique · identify the ideal of the hypersurface with the image of the section $f$ of $\mathcal O(d)$, twist the resulting section sequence, and read off the cohomology from the projective-space computation

1.1F2given

Since $X = Z (f)$ has pure dimension $n - 1 < n$ , the form $f$ is nonzero; fix an affine chart $D_{+} (x_{i})$ . In the chart ring the local equation of the section $f$ is the dehomogenization $f / x_{i}^{d}$ , regarded as an element of the localization $S_{x_{i}}$ ; localization $S ↪ S_{x_{i}}$ is injective because $S$ is a domain, and $f / x_{i}^{d} \neq 0$ because $f \neq 0$ , so the local equation is nonzero, hence a nonzerodivisor in the polynomial chart ring. By [F2] the ideal sheaf $I$ equals $Im (c_{f})$ for the contraction $c_{f} : O (- d) \to O$ that is multiplication by $f$ on local trivializations, and $O / I ≅ i_{*} O_{X}$ ; thus $0 \to O_{P^{n}} (- d) \overset{\cdot f}{\to} O_{P^{n}} \to i_{*} O_{X} \to 0$ is a short exact sequence of quasi-coherent sheaves, with $I ≅ O (- d)$ because $c_{f}$ is injective.

2.1F2F5step 1.1algebra

Twist the sequence of step 1.1 by the invertible sheaf $O (d)$ and use exactness of twisting ([F5]); this gives the sequence of claim (1), $0 \to O_{P^{n}} \to O_{P^{n}} (d) \to i_{*} O_{X} (d) \to 0$ . Because the ideal $I ≅ O (- d)$ is invertible, its square $I^{2}$ is as well and the quotient $I / I^{2}$ is the pullback $i^{*} I ≅ i^{*} O (- d) = O_{X} (- d)$ ; this proves the remaining assertions of (1).

3.1F1F3F4step 2.1

By [F1], $H^{q} (P^{n}, O) = 0$ and $H^{q} (P^{n}, O (d)) = 0$ for every $q \geq 1$ . Apply [F4] to the sequence of step 2.1 and use the identification $H^{q} (X, O_{X} (d)) ≅ H^{q} (P^{n}, i_{*} O_{X} (d))$ of [F3]: for every $q \geq 1$ the group $H^{q} (X, O_{X} (d))$ is sandwiched between $H^{q} (P^{n}, O (d)) = 0$ and $H^{q + 1} (P^{n}, O) = 0$ , hence vanishes. This proves (3).

4.1F1F3F4F6step 2.1algebra

Still in the long exact sequence of step 3.1, the start reads $0 \to H^{0} (P^{n}, O) \to H^{0} (P^{n}, O (d)) \to H^{0} (X, O_{X} (d)) \to H^{1} (P^{n}, O) = 0$ , so $H^{0} (X, O_{X} (d))$ is the cokernel of the multiplication map $k = S_{0} \to S_{d}$ , $1 \mapsto f$ , namely $S_{d} / k \cdot f$ . By [F6], a homogeneous multiple of $f$ lying in degree $d$ has multiplier of degree $0$ , hence lies in $k \cdot f$ ; therefore $k \cdot f = (f) \cap S_{d}$ and $H^{0} (X, O_{X} (d)) ≅ S_{d} / k \cdot f ≅ (S / (f))_{d}$ , of dimension $\dim_{k} S_{d} - 1 = (\binom{n + d}{n}) - 1$ by [F6]. This proves (2).

5.1F5step 2.1algebra∎

By step 2.1 the conormal sheaf $I / I^{2} ≅ O_{X} (- d)$ is invertible. Taking its dual and using [F5], $N_{X / P^{n}} = H o m_{O_{X}} (I / I^{2}, O_{X}) ≅ H o m_{O_{X}} (O_{X} (- d), O_{X}) ≅ O_{X} (d)$ , which proves (4). Together with steps 1.1, 2.1, 3.1 and 4.1 this proves all four claims; the Axiom of Choice is inherited from the cited cohomology, zero-scheme and closed-immersion suppliers.

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Sources

The Stacks Project, Cohomology of Schemes, complete chapter (Chapter 30) (standard reference, not scraped)
Robin Hartshorne, Lectures on Deformation Theory (Berkeley Math 274 draft, 2004/2005) (standard reference, not scraped)