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Surface duality for twists and a skyscraper on the projective plane

Example

Assume AC. On the smooth projective surface $X = P_{k}^{2}$ , $ω_{X} = O (- 3)$ . For $m \geq 0$ , duality pairs the $(\binom{m + 2}{2})$ monomials in $H^{0} (X, O (m))$ with $H^{2} (X, O (- m - 3))$ . It also gives ${Ext}_{X}^{2} (k_{p}, ω_{X}) = k$ and ${Ext}_{X}^{1} (k_{p}, ω_{X}) = {Hom}_{X} (k_{p}, ω_{X}) = 0$ at any $k$ -rational point $p$ .

Verification

Given: $k, m \geq 0$ , $X$ and $p \in X (k)$ , with AC.

[F1] The A theorem and its smooth specialization are Serre duality for coherent sheaves on a projective Cohen–Macaulay scheme and The smooth projective locally free theorem is the special case.

[F2] Twisting cohomology and the Laurent residue coefficient pairing are Cohomology of O(d) on projective space and Residue pairing between H^0 and top cohomology of projective space; flasque sheaves have no higher cohomology (Flasque abelian sheaves are Γ-acyclic).

1.1F1F2givenalgebra

The three affine charts are polynomial planes, so $X$ is smooth, projective and pure of dimension two. The smooth specialization in [F1] gives $ω_{X} = O (- 3)$ . A basis of $H^{0} (O (m))$ consists of $x_{0}^{a_{0}} x_{1}^{a_{1}} x_{2}^{a_{2}}$ with nonnegative exponents summing to $m$ . By [F2], the dual basis of $H^{2} (O (- m - 3))$ is $x_{0}^{- a_{0} - 1} x_{1}^{- a_{1} - 1} x_{2}^{- a_{2} - 1}$ . Multiplication followed by the coefficient of $(x_{0} x_{1} x_{2})^{- 1}$ gives the Kronecker pairing. This is the A theorem's $i = 0$ pairing for $F = O (m)$ under the locally free Ext identification.

2.1F1F2step 1.1algebra∎

The point sheaf has $H^{0} = k$ and no higher cohomology, since it is flasque and [F2] applies. Applying the A theorem [F1] with $F = k_{p}$ in degrees $i = 0, 1, 2$ gives respectively the stated Ext degree two, degree one, and degree zero groups. Thus the same surface theorem handles a coherent sheaf which is not locally free, as well as the twisting bundles. AC is inherited through [F1]–[F2].

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Sources

Stacks, Lemma 48.27.5: surface coherent Ext duality (standard reference, not scraped)
Vakil 2025, 29.2.2–3: twisting and locally free specializations (standard reference, not scraped)