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Hilbert polynomial of projective space

Statement

Assume the Axiom of Choice as inherited from the cohomology and counting suppliers (The Axiom of Choice).

Let $k$ be a field (Field), let $n \geq 0$ , and let $X = P_{k}^{n}$ carry its standard embedding $i = id : X ↪ P_{k}^{n}$ in the convention of Hilbert function and Euler characteristic on a projective scheme, with twisting sheaves $O_{X} (d)$ (Relative projective space from standard charts, Twisting sheaf on Proj) and twists $G (m) = G \otimes_{O_{X}} O_{X} (1)^{\otimes m}$ (Invertible sheaves). Define the binomial polynomial $(\binom{t + n}{n}) : = \frac{(t + n) (t + n - 1) \dots (t + 1)}{n!} \in Q [t],$ the empty product being $1$ when $n = 0$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ). Then for the structure sheaf $O_{X}$ , with $h_{O_{X}} (m) = \dim_{k} H^{0} (X, O_{X} (m))$ and $P_{O_{X}} (m) = χ (X, O_{X} (m))$ (Sheaf cohomology as right derived global sections, Euler characteristic of a coherent sheaf):

$P_{O_{X}} (m) = χ (X, O_{X} (m)) = (\binom{m + n}{n})$ for every $m \in Z$ , where the right-hand side is the value of the displayed polynomial; that is, $(\binom{t + n}{n})$ is the Hilbert polynomial of the structure sheaf;
$h_{O_{X}} (m) = (\binom{m + n}{n})$ for every $m \geq 0$ , and when $n \geq 1$ , $h_{O_{X}} (m) = 0$ for $- n \leq m \leq - 1$ ; when $n \geq 1$ and $m \leq - n - 1$ one has $h_{O_{X}} (m) = 0$ while $P_{O_{X}} (m) = (- 1)^{n} (\binom{- m - 1}{n})$ .

The field $k$ is arbitrary (including $F_{2}$ ); $n = 0$ gives $P_{k}^{0} = Spec k$ with $P_{O_{X}} \equiv 1$ ; the values $m = 0$ and $m = - n$ are included; the natural number $(\binom{m + n}{n})$ of The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ agrees with the polynomial value at every $m \geq 0$ by the closed formula $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ .

Facts & Assumptions

Given: The Axiom of Choice as inherited, a field $k$ , an integer $n \geq 0$ , the projective space $X = P_{k}^{n}$ with its standard embedding and twisting sheaves $O_{X} (d)$ .

[F1]

Conventions: with the standard embedding of the statement the twisting sheaf $O_{X} (1)$ is invertible, every twist $G (m) = G \otimes O_{X} (1)^{\otimes m}$ of a coherent $G$ is coherent, and $h_{G} (m) = \dim_{k} H^{0} (X, G (m))$ and $P_{G} (m) = χ (X, G (m))$ are defined for every $m \in Z$ . (Hilbert function and Euler characteristic on a projective scheme, Relative projective space from standard charts, Twisting sheaf on Proj, Invertible sheaves, Coherent module sheaves, Sheaf cohomology as right derived global sections, Euler characteristic of a coherent sheaf)

[F2]

Cohomology of the twists: for every commutative ring with $1$ in place of $k$ and all $n \geq 0$ , $d \in Z$ , $H^{q} (X, O_{X} (d)) = 0$ unless $q = 0$ or $q = n$ ; if $n > 0$ then $H^{0} (X, O_{X} (d)) ≅ k [x_{0}, \dots, x_{n}]_{d}$ for $d \geq 0$ and $H^{0} (X, O_{X} (d)) = 0$ for $d < 0$ , where $k [x_{0}, \dots, x_{n}]_{d}$ is the degree- $d$ graded piece (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); and $H^{n} (X, O_{X} (d))$ is the free $k$ -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 $n > 0$ and a nonzero coefficient ring, it is nonzero precisely when $d \leq - n - 1$ . For $n = 0$ one has $X = Spec k$ and $H^{0} (X, O_{X} (d)) ≅ k$ for every $d \in Z$ , with all higher groups zero. (Cohomology of O(d) on projective space)

[F3]

Counting multi-indices: the monomial $k$ -basis of the degree- $m$ piece $k [x_{0}, \dots, x_{n}]_{m}$ is indexed by the multi-indices $(e_{0}, \dots, e_{n}) \in N^{n + 1}$ with $\sum_{i} e_{i} = m$ , the monomials $x_{0}^{e_{0}} \dots x_{n}^{e_{n}}$ , by the uniqueness of the expansion of a polynomial (Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ , The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials). The number of $(n + 1)$ -tuples of nonnegative integers with sum $N \geq 0$ equals the number of compositions of $N + n + 1$ into exactly $n + 1$ positive parts, via $(f_{i}) \mapsto (f_{i} + 1)$ , and by Compositions of $n$ into $k$ positive parts are counted by $(\binom{n - 1}{k - 1})$ this number is $(\binom{N + n}{n})$ (with $(\binom{a}{b})$ the count of The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , and the value $0$ when $n + 1$ parts exceed $N + n + 1$ ). In particular the degree- $m$ piece of $k [x_{0}, \dots, x_{n}]$ has dimension $(\binom{m + n}{n})$ for every $m \geq 0$ , and the set ${e \in Z^{n + 1} : e_{i} < 0, \sum_{i} e_{i} = d}$ has $(\binom{- d - 1}{n})$ elements for every $d \leq - n - 1$ . (cor-compositions-with-k-parts-are-counted-by-binomial-coefficients)

[F4]

The product formula: for integers $0 \leq b \leq a$ the identity $(\binom{a}{b}) \cdot b! \cdot (a - b)! = a!$ holds in $N$ , so $(\binom{a}{b}) = \frac{a (a - 1) \dots (a - b + 1)}{b!}$ ; hence for $m \geq 0$ $(\binom{m + n}{n}) = \frac{(m + n) (m + n - 1) \dots (m + 1)}{n!},$ which is the value at $t = m$ of the polynomial $(\binom{t + n}{n})$ of the statement, and for $m \leq - n - 1$ $(- 1)^{n} (\binom{- m - 1}{n}) = (- 1)^{n} \frac{(- m - 1) (- m - 2) \dots (- m - n)}{n!} = \frac{(m + n) (m + n - 1) \dots (m + 1)}{n!},$ the last equality because each factor $m + j$ for $1 \leq j \leq n$ is the negative of $- m - j$ ; if $- n \leq m \leq - 1$ then one of the factors $m + n, \dots, m + 1$ is $0$ , so the polynomial value vanishes. ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ )

[F5]

The Axiom of Choice is the choice principle named in the statement, inherited from the cohomology computation and the counting corollary cited above. (The Axiom of Choice)

Proof

technique · direct: compute $\chi(X,\mathcal O_X(m))$ from the complete projective-space cohomology computation, count degree-$m$ monomials and strictly negative Laurent exponent vectors by the composition formula, and compare the resulting values with the polynomial $\binom{t+n}{n}$ in the three ranges of $m$

1.1F4

Values of the polynomial. By [F4] the polynomial $(\binom{t + n}{n})$ of the statement takes at an integer $m$ the value $(\binom{m + n}{n})$ when $m \geq 0$ , the value $0$ when $- n \leq m \leq - 1$ , and the value $(- 1)^{n} (\binom{- m - 1}{n})$ when $m \leq - n - 1$ ; for $n = 0$ the middle range is empty and $(\binom{t}{0}) = 1$ for all $t$ .

1.2F1F2F31.1

Zero-dimensional projective space and nonnegative twists. If $n = 0$ , then for every integer $m$ the twist $O_{X} (m)$ is trivial on $X = P_{k}^{0} = Spec k$ by [F2], so $h_{O_{X}} (m) = χ (X, O_{X} (m)) = 1 = (\binom{m}{0})$ ; this handles all negative twists when $n = 0$ . Now assume $n \geq 1$ and let $m \geq 0$ . By [F3] the degree- $m$ piece of $k [x_{0}, \dots, x_{n}]$ has a basis indexed by the multi-indices with sum $m$ , hence has dimension $(\binom{m + n}{n})$ ; [F2] gives $H^{0} (X, O_{X} (m)) ≅ k [x_{0}, \dots, x_{n}]_{m}$ and the vanishing of all higher cohomology of $O_{X} (m)$ . Therefore $h_{O_{X}} (m) = (\binom{m + n}{n})$ and the Euler characteristic, an alternating sum with a single nonzero term, equals the same number; by 1.1 this is the value of $(\binom{t + n}{n})$ at $t = m$ ; the definitions of $h_{O_{X}}$ and of the Euler characteristic, and the coherence of the twists, are those of [F1].

1.3F21.1

Negative twists. Let $m < 0$ and first suppose $- n \leq m \leq - 1$ , which forces $n \geq 1$ . By [F2] one has $H^{0} (X, O_{X} (m)) = 0$ because $m < 0$ , and $H^{n} (X, O_{X} (m)) = 0$ because $m > - n - 1$ ; all other groups vanish, so $χ (X, O_{X} (m)) = 0$ , the value of the polynomial at $t = m$ by 1.1, and also $h_{O_{X}} (m) = 0$ .

2.1F2F31.1

Deeply negative twists. Assume $n \geq 1$ and let $m \leq - n - 1$ . Again $H^{0} (X, O_{X} (m)) = 0$ because $m < 0$ ; the only other possibly nonzero group is $H^{n} (X, O_{X} (m))$ , which by [F2] is free on the vectors $e \in Z^{n + 1}$ with $e_{i} < 0$ and $\sum_{i} e_{i} = m$ , a set whose cardinality is $(\binom{- m - 1}{n})$ by [F3]. Hence $h_{O_{X}} (m) = 0$ and $χ (X, O_{X} (m)) = (- 1)^{n} (\binom{- m - 1}{n})$ , which by 1.1 is again the value of the polynomial; the case $n = 0$ for every integer $m$ was handled in step 1.2.

2.21.21.32.1

Conclusion. For $n = 0$ , step 1.2 covers every integer $m$ . For $n \geq 1$ , combining 1.2, 1.3 and 2.1, every integer $m$ falls into exactly one of the ranges $m \geq 0$ , $- n \leq m \leq - 1$ and $m \leq - n - 1$ , and in each case $χ (X, O_{X} (m)) = (\binom{m + n}{n})$ , the value of the polynomial $(\binom{t + n}{n})$ . This proves statement 1. For $n \geq 1$ , the values of $h_{O_{X}}$ asserted in statement 2 are exactly those computed in 1.2, 1.3 and 2.1: $(\binom{m + n}{n})$ for $m \geq 0$ , and $0$ for negative $m$ ; for $n = 0$ , step 1.2 gives $h = 1$ for all integers $m$ .

3.1F2F3F51.11.21.32.2∎

Boundaries and choice. The field $k$ is arbitrary, including $k = F_{2}$ where binomial coefficients are still natural-number counts; the case $n = 0$ is $P_{k}^{0} = Spec k$ with $O_{X} (d) ≅ O_{X}$ for every $d$ , one cohomology group in degree $0$ , of dimension $1$ and $(\binom{t + 0}{0}) = 1$ , in agreement with 1.2; the value $m = 0$ lies in the range of 1.2 and gives $(\binom{n}{n}) = 1$ , and $m = - n$ lies at the endpoint of the middle range of 1.3 and gives $0$ for $n \geq 1$ . The polynomial has rational coefficients by construction and is not claimed to be integral-valued outside the ranges computed. The Axiom of Choice is inherited through [F5] and the suppliers of [F2] 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)