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Prescribing two surface Chern classes

PDF p. 12: Theorem A surjectivity construction

Statement

Assume the Axiom of Choice and the site-size and universe bounds required by the cited motivic and perfect K-theory constructions. Let $k$ be an algebraically closed field and let $X$ be a normal affine finite-type $k$ -scheme of Krull dimension two. For every pair $a_{1}$ in $H_{m o t}^{2, 1} (X, Z)$ and $a_{2}$ in $H_{m o t}^{4, 2} (X, Z)$ , there is a rank-two vector bundle $W$ on $X$ with $c_{1} (W) = a_{1}$ and $c_{2} (W) = a_{2}$ .

More precisely, one may choose a line bundle $L$ with $c_{1} (L) = a_{1}$ and a rank-two bundle $V$ with trivial determinant such that $[V] - [O_{X}^{2}]$ lies in the second motivic image filtration and has second Chern class $a_{2}$ . Then $V$ direct-summed with $L$ is isomorphic to $W$ direct-summed with $O_{X}$ , and the resulting $W$ has the prescribed classes. The virtual class $[V] - [O_{X}^{2}] + [L] - [O_{X}]$ also has first and second Chern classes $a_{1}$ and $a_{2}$ .

A vector bundle is a finite locally free sheaf, its rank is the number of copies of the structure sheaf in a local trivialization, and a line bundle is a rank-one vector bundle. Its determinant is its top exterior power. Normal means that every local ring is an integrally closed domain; finite type over $k$ means a finite affine cover by spectra of finitely generated $k$ -algebras. Krull dimension is the supremum of lengths of chains of irreducible closed subsets. The groups $H_{m o t}^{2 i, i} (X, Z)$ are the indicated cohomology groups of the integral weight- $i$ motivic complexes. The group $K_{0} (X)$ is the zeroth homotopy group of nonconnective K-theory of perfect complexes; a perfect complex is locally quasi-isomorphic to a bounded complex of vector bundles. The second motivic image filtration $F_{m o t}^{2} K_{0} (X)$ is the image in $K_{0} (X)$ of the zeroth homotopy group of the second stage of the motivic K-theory tower. Brackets denote the class of a bundle in $K_{0} (X)$ , and $c_{i}$ on virtual classes denotes the actual perfect K-theory Chern operation.

Facts & Assumptions

Given: An algebraically closed field $k$ , a normal affine finite-type $k$ -scheme $X$ of dimension two, the two prescribed motivic cohomology classes, Choice, and the stated site-size and universe bounds.

Goal: Construct $L$ , $V$ and $W$ with the stated properties and calculate both actual and virtual Chern classes.

[F1]

Under these Choice and size hypotheses, first Chern class identifies line bundles with $H_{m o t}^{2, 1} (X, Z)$ . Line bundles and first motivic Chern class

[F2]

For a normal finite-type scheme of dimension two over an algebraically closed field, with the stated Choice and size hypotheses, the actual second Chern operation maps $F_{m o t}^{2} K_{0} (X)$ onto $H_{m o t}^{4, 2} (X, Z)$ . Second Chern surjectivity in dimensions two and three

[F3]

On such a normal finite-type scheme, every class in the second image filtration has pointwise rank zero and first perfect Chern class zero. If it is represented by a difference of bundles, their determinants are isomorphic. Rank and determinant on the second piece

[F4]

On a nonempty affine finite-type scheme of dimension two over a field, Choice and suitable universes allow every pointwise rank-zero perfect $K_{0}$ class to be written $[V] - [O_{X}^{2}]$ with $V$ a rank-two bundle. Critical-rank representatives of rank-zero classes

[F5]

Under Choice, a rank-three bundle on a nonempty noetherian affine scheme of dimension two splits as a rank-two bundle plus a trivial line. Serre splitting above the dimension

[F6]

Actual perfect Chern operations satisfy the total Cartan identity on noetherian finite-dimensional schemes over a field, and the universal operation restricts to the supplied actual bundle classes. Bundle Chern classes obey Whitney's formula, a line has no Chern classes above degree one, and trivial bundles have total Chern class one. The actual perfect K-space operations satisfy total Cartan External formal universal Chern maps and supported construction Whitney identities and virtual Chern arithmetic

[F7]

The Axiom of Choice is assumed for the cited constructions and affine splitting. The Axiom of Choice

Proof

technique · Choose the two classes separately, represent the second by a critical-rank bundle, and split one trivial line from the resulting rank-three bundle

1.1F1F2F7givenchoose

The dimension hypothesis makes $X$ nonempty. It is noetherian because it is finite type over a field. By [F1], choose a line bundle $L$ with $c_{1} (L) = a_{1}$ . By [F2], choose a class $a l p h a$ in $F_{m o t}^{2} K_{0} (X)$ with $c_{2} (a l p h a) = a_{2}$ .

2.1F3F4step 1.1

The class $a l p h a$ has pointwise rank zero by [F3]. Applying [F4], write $a l p h a = [V] - [O_{X}^{2}]$ for a rank-two bundle $V$ . The determinant assertion of [F3] gives an isomorphism from the determinant of $V$ to the determinant of $O_{X}^{2}$ , which is the trivial line. Also $c_{1} (a l p h a) = 0$ by [F3].

3.1F6step 1.1step 2.1

The actual perfect total Cartan identity and bundle restriction in [F6] give $c_{t} (a l p h a) c_{t} (O_{X}^{2}) = c_{t} (V)$ . Since the total class of $O_{X}^{2}$ is one, $c_{1} (V) = c_{1} (a l p h a) = 0$ and $c_{2} (V) = c_{2} (a l p h a) = a_{2}$ . Here $c_{t}$ denotes the formal total Chern series with constant term one; the equalities are its coefficients in degrees one and two.

4.1F6step 1.1step 2.1step 3.1construct

Put $b e t a = a l p h a + [L] - [O_{X}]$ . By [F6], the total class of $[L] - [O_{X}]$ is $1 + a_{1} t$ . Multiplying by the total class of $a l p h a$ , whose first coefficient vanishes, gives $c_{1} (b e t a) = a_{1}$ and $c_{2} (b e t a) = a_{2} + 0$ ; the cross term is $c_{1} (a l p h a) a_{1} = 0$ . This establishes the virtual-class assertion.

5.1F5F6step 1.1step 3.1∎

The bundle $V$ direct-summed with $L$ has constant rank three. Apply [F5] to obtain an isomorphism $V$ direct-summed with $L$ isomorphic to $W$ direct-summed with $O_{X}$ , where $W$ has rank two. Whitney's formula and the trivial-line normalization in [F6] give $c_{i} (W) = c_{i} (V \oplus L)$ for $i = 1, 2$ . The first class of the latter is $0 + a_{1}$ ; its second is $a_{2} + 0 a_{1} + 0$ , using step 3.1 and the vanishing of the second class of a line. Hence $c_{1} (W) = a_{1}$ and $c_{2} (W) = a_{2}$ , as required.

Depends on

Used by · 1

Motivic Chern classification on normal affine surfaces

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 12, final paragraph of the proof of Theorem A

Every pair of first and second motivic Chern classes on the stated surface is represented by a rank-two vector bundle.

Hypotheses: A normal affine surface of finite type over an algebraically closed field, under the article conventions.

Reading limits: The cited paragraph was read in the supplied extracted article text. The proof below uses the stated completed supplier contracts; their underlying proofs were not independently reconstructed here.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 12, final paragraph of the proof of Theorem A