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All twists on the projective line

Example

Assume the Axiom of Choice, inherited from the cohomology and finiteness suppliers cited below (The Axiom of Choice). Let $k$ be a field (Field) and let $d \in Z$ . Consider the projective line $X = P_{k}^{1}$ (Relative projective space from standard charts) with its twisting sheaves $O_{X} (d)$ (Twisting sheaf on Proj), and write $h^{q} = \dim_{k} H^{q} (P_{k}^{1}, O_{X} (d))$ for the dimensions of its sheaf cohomology (Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis). Then $h^{0} = \max (d + 1, 0), h^{1} = \max (- d - 1, 0), χ (P_{k}^{1}, O_{X} (d)) = d + 1,$ where $χ$ is the Euler characteristic (Euler characteristic of a coherent sheaf); all higher cohomology groups vanish. The field $k$ is arbitrary, the twist $d = 0$ is included with $O_{X} (0) = O_{X}$ and $χ = 1$ , and the boundary value $d = - 1$ is included with $h^{0} = h^{1} = 0$ and $χ = 0$ .

Facts & Assumptions

Given: A field $k$ , an integer $d$ , the projective line $X = P_{k}^{1}$ with its twisting sheaves $O_{X} (d)$ ; the Axiom of Choice is inherited from the cited suppliers.

[F1]

Cohomology of the twists of projective space: for a commutative ring $A$ with $1$ , an integer $n \geq 0$ , the scheme $P_{A}^{n} ≅ Proj A [x_{0}, \dots, x_{n}]$ and every $d \in Z$ , one has $H^{q} (P_{A}^{n}, O (d)) = 0$ unless $q = 0$ or $q = n$ ; for $n > 0$ , $H^{0} (P_{A}^{n}, O (d)) ≅ A [x_{0}, \dots, x_{n}]_{d}$ when $d \geq 0$ and $H^{0} = 0$ when $d < 0$ , while $H^{n} (P_{A}^{n}, O (d))$ is the free $A$ -module on the Laurent monomials $x_{0}^{e_{0}} \dots x_{n}^{e_{n}}$ with $e_{i} < 0$ for all $i$ and $\sum_{i} e_{i} = d$ , so, for $n > 0$ , it is nonzero precisely when $A \neq 0$ and $d \leq - n - 1$ . (Cohomology of O(d) on projective space, The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials, Nonnegatively graded rings and modules, homogeneous elements, and twists, Relative projective space from standard charts, Twisting sheaf on Proj)

[F2]

Finiteness and Euler characteristic: for a field $k$ , a scheme proper over $k$ and a coherent $O_{X}$ -module $F$ , each $H^{q} (X, F)$ is a finite-dimensional $k$ -vector space, only finitely many are nonzero, and $χ (X, F) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (X, F)$ is a well-defined integer. (Euler characteristic of a coherent sheaf, Finite-dimensional coherent cohomology over a field, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis)

[F3]

Properness: projective space $P_{A}^{n}$ is proper over $Spec A$ for every commutative ring $A$ and every $n \geq 0$ . (Finite-dimensional projective space is proper over every base, Proper morphisms)

[F4]

Local Noetherianity and coherence: a field is a Noetherian ring, the polynomial ring $k [t]$ is Noetherian, the standard charts of $P_{k}^{1}$ are spectra of polynomial rings in one variable, so $P_{k}^{1}$ is a locally Noetherian scheme; on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type. (A field has only the zero ideal and itself, hence is Noetherian, If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ , Relative projective space from standard charts, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves, Finite type and finitely presented module sheaves)

[F5]

The twisting sheaves of $P_{k}^{1}$ are invertible. Indeed, put $S = k [x_{0}, x_{1}]$ with its total-degree grading. On each of the two standard charts $D_{+} (x_{i})$ , multiplication by $x_{i}^{d}$ identifies $S_{(x_{i})}$ with $S (d)_{(x_{i})}$ : in $S [x_{i}^{- 1}]$ the element $x_{i}^{d}$ is a unit for every integer $d$ , and its inverse sends each degree- $d$ element to degree zero. These maps commute with localisation, so the chart description of the twisting sheaf gives $O_{X} (d) ∣_{D_{+} (x_{i})} ≅ O_{D_{+} (x_{i})}$ . The two charts cover $X$ , proving local freeness of rank one, hence invertibility, quasi-coherence and finite type. Thus each $O_{X} (d)$ is coherent by [F4]. (Invertible sheaves, Locally free sheaves of finite rank, Twisting sheaf on Proj, Finite type and finitely presented module sheaves, Coherent module sheaves)

Verification

technique · direct: specialise the projective-space cohomology theorem to $n=1$ and the field $k$, decide the two surviving degrees by explicit monomial conditions in the three ranges $d\ge0$, $d=-1$ and $d\le-2$, count the monomial basis of the top-degree group, and read off the alternating sum

1.1F1F2F3F4F5

Setup and well-definedness of $χ$ . By [F4] the projective line $X = P_{k}^{1}$ is a locally Noetherian scheme, and by [F5] the twist $O_{X} (d)$ is a coherent $O_{X}$ -module on it; by [F3] the scheme $X$ is proper over $k$ ; hence [F2] applies, so each $H^{q} (X, O_{X} (d))$ is a finite-dimensional $k$ -vector space, only finitely many are nonzero, and $χ (X, O_{X} (d)) = h^{0} - h^{1} + \sum_{q \geq 2} (- 1)^{q} h^{q}$ is a well-defined integer. By [F1] with $n = 1$ one has $H^{q} (X, O_{X} (d)) = 0$ for every $q \notin {0, 1}$ , so $χ = h^{0} - h^{1}$ .

1.2F1algebra

The case $d \geq 0$ . By [F1] with $n = 1$ the group $H^{0}$ is $k [x_{0}, x_{1}]_{d}$ , for which the monomials $x_{0}^{d - j} x_{1}^{j}$ with $0 \leq j \leq d$ form a $k$ -basis, so $h^{0} = d + 1 = \max (d + 1, 0)$ . The degree-one group is free on the Laurent monomials with both exponents negative and sum $d$ , a set that is empty because $d \geq 0$ , so $h^{1} = 0 = \max (- d - 1, 0)$ since $- d - 1 \leq - 1$ . Hence $χ = d + 1$ .

1.3F1algebra

The case $d = - 1$ . By [F1] the group $H^{0}$ vanishes because $d < 0$ ; the group $H^{1}$ is free on the Laurent monomials with $e_{0}, e_{1} < 0$ and $e_{0} + e_{1} = - 1$ , which is impossible for integers, so $h^{1} = 0$ . Thus $h^{0} = h^{1} = 0$ , and $\max (d + 1, 0) = \max (0, 0) = 0$ , $\max (- d - 1, 0) = \max (0, 0) = 0$ , and $χ = 0 - 0 = 0 = d + 1$ .

1.4F1algebra

The case $d \leq - 2$ . By [F1] the group $H^{0}$ vanishes because $d < 0$ , so $h^{0} = 0 = \max (d + 1, 0)$ since $d + 1 \leq - 1$ . For the top-degree group, write $e_{0} = - a$ and $e_{1} = - b$ with integers $a, b \geq 1$ ; the condition $e_{0} + e_{1} = d$ becomes $a + b = - d$ , whose solutions are $a = 1, \dots, - d - 1$ with $b = - d - a$ . These are exactly $- d - 1 = \max (- d - 1, 0)$ monomials, and they form a $k$ -basis by [F1], so $h^{1} = - d - 1$ . Hence $χ = h^{0} - h^{1} = 0 - (- d - 1) = d + 1$ .

2.1F1F21.21.31.4∎

Conclusion, boundaries and choice. The three cases $d \geq 0$ , $d = - 1$ , $d \leq - 2$ exhaust $Z$ and give $h^{0} = \max (d + 1, 0)$ , $h^{1} = \max (- d - 1, 0)$ and $χ = d + 1$ in every case, with all higher groups zero by 1.1. The field $k$ is arbitrary, of any characteristic and in particular $k = F_{2}$ ; the twist $d = 0$ gives the structure sheaf with $h^{0} = 1$ , $h^{1} = 0$ , $χ = 1$ , and $d = - 1$ is the endpoint where both groups vanish, handled separately in 1.3; the case $d = 1$ gives the line bundle whose sections are the linear forms, $h^{0} = 2$ . The projective line over a field is nonempty, so no empty scheme occurs, and the alternating sums are finite because $H^{q} = 0$ for $q \geq 2$ ; the empty-sum convention is not needed. The Axiom of Choice [F1, F2] is inherited through the projective-space cohomology theorem and the finiteness corollary, and the only basis used is the explicit monomial basis of $k [x_{0}, x_{1}]_{d}$ together with the explicit monomial enumeration of 1.4, determined by $d$ with no further selection.

Depends on

Used by

h0 differs from the Euler characteristic before vanishing Counterexample
Compensating h0 and h1 jumps with constant Euler characteristic Example

Dependency tree · two levels

114 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, 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)