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Pushforward and vanishing for an affine point blowup

Statement

Assume the Axiom of Choice, inherited from the Proj construction and from the cohomology suppliers (The Axiom of Choice). Let $A$ be a commutative ring with $1$ , let $I = (x, y) \subseteq A$ be an ideal generated by a regular sequence $(x, y)$ (for example $A$ a regular local ring of dimension two with parameters $x, y$ ), let $π ⁣ : {Bl}_{I} Spec A \to Spec A$ be the blowup of Blowup of a scheme along an ideal sheaf and let $E$ be its exceptional subscheme. Then the two standard charts $X_{0} = D_{+} (x t)$ and $X_{1} = D_{+} (y t)$ cover ${Bl}_{I} Spec A$ , the ordered Čech complex of this two-element affine cover computes the cohomology of $O_{{Bl}_{I} Spec A}$ (Cech cohomology computes quasi-coherent cohomology on a separated scheme), and $H^{0} ({Bl}_{I} Spec A, O) = A, H^{1} ({Bl}_{I} Spec A, O) = 0, H^{q} ({Bl}_{I} Spec A, O) = 0 (q \geq 2) .$ Consequently $π_{*} O_{{Bl}_{I} Spec A} = O_{Spec A}$ and $R^{q} π_{*} O_{{Bl}_{I} Spec A} = 0$ for all $q > 0$ .

Facts & Assumptions

Given: A commutative ring $A$ , an ideal $I = (x, y) \subseteq A$ such that $x$ is a nonzerodivisor of $A$ and $y$ is a nonzerodivisor of $A / (x)$ , the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup $π ⁣ : {Bl}_{I} Spec A = Proj R (I) \to Spec A$ (Blowup of a scheme along an ideal sheaf), and the relative projective line $P_{A}^{1} = Proj A [u, v]$ with twisting sheaves $O (d)$ (Twisting sheaf on Proj).

[F1]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ and $B_{i} = A [I / f_{i}]$ the standard opens $D_{+} (f_{i} t) = Spec B_{i}$ cover ${Bl}_{I} Spec A$ , with overlap identifications given by the ratios $(f_{j} t) / (f_{i} t)$ .

[F2]

Closed subschemes of projective space and saturated ideals: A homogeneous ideal $a \subseteq A [u, v]$ determines a closed subscheme $V_{+} (a) ↪ P_{A}^{1}$ , equal to $Proj (A [u, v] / a)$ under the canonical closed immersion, and $V_{+} (a) \cap D_{+} (u) = Spec (A [u, v]_{(u)} / a_{(u)})$ .

[F3]

Hypersurface cohomology sequence: For $f \in A [u, v]$ homogeneous of degree $d > 0$ with every dehomogenisation $f / u^{d}$ , $f / v^{d}$ a nonzerodivisor of the corresponding chart ring, the multiplication map $O_{P_{A}^{1}} (- d) \overset{\cdot f}{\to} O_{P_{A}^{1}}$ and the structure map of the closed immersion $i ⁣ : V_{+} (f) ↪ P_{A}^{1}$ form a short exact sequence $0 \to O_{P_{A}^{1}} (- d) \overset{\cdot f}{\to} O_{P_{A}^{1}} \overset{i^{♯}}{\to} i_{*} O_{V_{+} (f)} \to 0,$ and the induced long exact sequence of sheaf cohomology computes the cohomology of $V_{+} (f)$ from that of the twists $O_{P_{A}^{1}} (d)$ .

[F4]

Cohomology of O(d) on projective space: On $P_{A}^{1}$ one has $H^{0} (O) = A$ , $H^{1} (O) = 0$ , $H^{q} (O) = 0$ for $q \geq 2$ , and $H^{0} (O (- 1)) = 0$ , $H^{1} (O (- 1)) = 0$ ; indeed $H^{q} (P_{A}^{1}, O (d)) = 0$ unless $q = 0$ or $q = 1$ , with $H^{0}$ the degree- $d$ part of $A [u, v]$ for $d \geq 0$ and $H^{1}$ the free $A$ -module on the Laurent monomials $u^{e_{0}} v^{e_{1}}$ with $e_{0}, e_{1} < 0$ and $e_{0} + e_{1} = d$ , which is zero for $d = - 1$ .

[F5]

Cech cohomology computes quasi-coherent cohomology on a separated scheme: For a quasi-compact separated scheme $X$ , a finite affine open cover and a quasi-coherent $O_{X}$ -module $F$ , the canonical comparison map ${\overset{ˇ}{H}}^{q} (U, F) \to H^{q} (X, F)$ is an isomorphism for every $q \geq 0$ .

[F6]

Čech complex for a two-open cover: For a cover by two open sets $U_{0}, U_{1}$ , the ordered Čech complex has $C^{0} = F (U_{0}) \oplus F (U_{1})$ , $C^{1} = F (U_{0} \cap U_{1})$ and $C^{p} = 0$ for $p \geq 2$ , with $δ^{0} (s_{0}, s_{1}) = s_{1} ∣_{U_{0} \cap U_{1}} - s_{0} ∣_{U_{0} \cap U_{1}}$ ; hence ${\overset{ˇ}{H}}^{0} = \ker δ^{0}$ , ${\overset{ˇ}{H}}^{1} = coker δ^{0}$ and ${\overset{ˇ}{H}}^{p} = 0$ for $p \geq 2$ .

[F7]

Higher direct images localize over an affine base: For a quasi-compact separated morphism $f ⁣ : X \to S$ and a quasi-coherent $O_{X}$ -module $F$ , each $R^{q} f_{*} F$ is quasi-coherent (Higher direct image of a sheaf), and for every affine open $V = Spec A \subseteq S$ there is a canonical isomorphism $(R^{q} f_{*} F) ∣_{V} ≅ \tilde{H^{q} (f^{- 1} V, F)}$ .

[F8]

Closed immersions of schemes, Quasi-compact and quasi-separated schemes, Separated morphism of schemes: A closed subscheme of a quasi-compact scheme is quasi-compact, and a closed subscheme of a separated scheme is separated; the relative projective line $P_{A}^{1}$ is quasi-compact and separated over $Spec A$ .

Proof

1.1F1

The graded $A$ -algebra homomorphism $A [u, v] \to R (I)$ with $u \mapsto x t$ , $v \mapsto y t$ is surjective because $I^{n} = (x, y)^{n}$ is generated by the monomials $x^{n - i} y^{i}$ , and its kernel is $(x v - y u)$ : for a homogeneous $F = \sum_{i = 0}^{n} c_{i} u^{n - i} v^{i}$ with $F (x, y) = 0$ , reduction modulo $x$ gives $c_{n} y^{n} \equiv 0 (m o d x)$ , so ${\overset{ˉ}{c}}_{n} = 0$ because $y$ is a nonzerodivisor modulo $x$ , say $c_{n} = x d$ ; then $F - d v^{n - 1} (x v - y u)$ has zero $v^{n}$ -coefficient, hence equals $u G$ for a homogeneous $G$ of degree $n - 1$ , and evaluating at $(x, y)$ gives $x G (x, y) = 0$ , so $G (x, y) = 0$ because $x$ is a nonzerodivisor, and induction on $n$ down to degree $0$ yields $F \in (x v - y u)$ , while conversely $x v - y u \mapsto x \cdot y t - y \cdot x t = 0$ ; hence $R (I) ≅ A [u, v] / (x v - y u)$ as graded $A$ -algebras.

1.2F1F2

The element $f : = x v - y u \in A [u, v]$ is a nonzerodivisor: as a polynomial in $v$ over $A [u]$ its leading coefficient is the nonzerodivisor $x$ , so $f g = 0$ forces the top $v$ -coefficient of $g$ to vanish and descending induction gives $g = 0$ ; the dehomogenisation $f / u = x (v / u) - y$ is a nonzerodivisor of $A [v / u]$ for the same reason, and $y U - x$ is a nonzerodivisor of $A [U]$ , since comparing coefficients of $U^{i}$ in $(y U - x) g = 0$ gives $- x g_{0} = 0$ and $y g_{i - 1} = x g_{i}$ for $i \geq 1$ , so $g_{0} = 0$ and induction gives all $g_{i} = 0$ ; hence by [F2] the blowup is identified with the closed subscheme ${Bl}_{I} Spec A = Proj (A [u, v] / (x v - y u)) = V_{+} (x v - y u) \subseteq P_{A}^{1}$ , whose standard charts $D_{+} (u) \cap V_{+} (f) = Spec (A [v / u] / (x (v / u) - y)) = X_{0}$ and $D_{+} (v) \cap V_{+} (f) = Spec (A [u / v] / (y (u / v) - x)) = X_{1}$ are affine and cover it, matching the standard blowup charts of [F1] under the identification of $u, v$ with $x t, y t$ .

1.3F4

On $P_{A}^{1}$ the groups $H^{0} (O) = A$ , $H^{1} (O) = 0$ , $H^{q} (O) = 0$ for $q \geq 2$ and $H^{0} (O (- 1)) = H^{1} (O (- 1)) = 0$ hold by [F4] with $n = 1$ .

2.1F3step 1.2

Applying [F3] to the homogeneous degree-one element $f = x v - y u$ , whose chart dehomogenisations are nonzerodivisors by step 1.2, gives a short exact sequence of $O_{P_{A}^{1}}$ -modules $0 \to O_{P_{A}^{1}} (- 1) \overset{\cdot f}{\to} O_{P_{A}^{1}} \to i_{*} O_{{Bl}_{I} Spec A} \to 0$ , where $i$ is the closed immersion of step 1.2.

3.1F4step 2.1step 1.3

The long exact cohomology sequence of step 2.1, with $H^{q} (P_{A}^{1}, i_{*} O_{Bl}) ≅ H^{q} ({Bl}_{I} Spec A, O)$ for the closed immersion $i$ and the groups of step 1.3, gives $H^{0} ({Bl}_{I} Spec A, O) ≅ coker (H^{0} (O (- 1)) \to H^{0} (O)) = A$ , then $H^{1} ({Bl}_{I} Spec A, O) = 0$ because it is squeezed between $H^{1} (O) = 0$ and $H^{2} (O (- 1)) = 0$ , and for $q \geq 2$ the group $H^{q} ({Bl}_{I} Spec A, O)$ is squeezed between $H^{q} (O) = 0$ and $H^{q + 1} (O (- 1)) = 0$ .

4.1F5F6F8step 1.2step 3.1

By [F8] the blowup ${Bl}_{I} Spec A$ is a closed subscheme of the quasi-compact separated $P_{A}^{1}$ , hence quasi-compact and separated, so the two-element affine cover of step 1.2 has Čech cohomology computing the sheaf cohomology of $O$ by [F5] and, by [F6], a Čech complex concentrated in degrees $0$ and $1$ with ${\overset{ˇ}{H}}^{0} = \ker δ^{0}$ and ${\overset{ˇ}{H}}^{1} = coker δ^{0}$ , so its Čech groups are $A$ , $0$ and $0$ in degrees $0$ , $1$ and $\geq 2$ , consistently with step 3.1.

5.1F7F8step 3.1∎

The structural morphism $π$ is quasi-compact and separated, being the composite of the closed immersion of step 1.2 with the quasi-compact separated structure morphism of $P_{A}^{1}$ from [F8], and $O_{Bl}$ is quasi-coherent, so [F7] with $V = Spec A$ identifies $R^{q} π_{*} O$ with the sheaf associated to the $A$ -module $H^{q} ({Bl}_{I} Spec A, O)$ , which is $A$ in degree $0$ and $0$ for $q > 0$ by step 3.1; hence $π_{*} O = \tilde{A} = O_{Spec A}$ and $R^{q} π_{*} O = 0$ for every $q > 0$ .

Remarks

Regularity of the sequence $(x, y)$ enters twice: to identify the Rees algebra with the incidence algebra $A [u, v] / (x v - y u)$ (step 1.1) and to make the dehomogenisations of $x v - y u$ nonzerodivisors (step 1.2). For a general pair of generators the map $A [u, v] \to R (I)$ has a nontrivial kernel in general, and the blowup need not be a hypersurface in $P_{A}^{1}$ .
The theorem is stated for an affine base precisely so that the direct image computation reduces to the two cohomology modules $A$ and $0$ ; over a nonaffine base the same proof applies over each affine open, and the sheaf statement is the affine-local one of [F7].
The two-chart computation never inverts $x$ or $y$ in $A$ : the overlap identification inverts the ratio $v / u$ only, consistent with [F1].

Depends on

Used by

Dependency tree · two levels

93 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

The Stacks Project, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)
The Stacks Project, Commutative Algebra, Section 10.70 (Blow up algebras) (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, June 27, 2011 draft (author-hosted 'Early (out-of-date) version of The Rising Sea') (standard reference, not scraped)
Roman Bezrukavnikov et al., MIT 18.725 Algebraic Geometry (Fall 2015) consolidated lecture notes (standard reference, not scraped)