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Second Chern class need not be additive on the first piece

PDF p. 11: Remark 5.4: second Chern nonadditivity

Remark

Assume the Axiom of Choice and the compatible universe and site-size bounds of the supplied constructions. The second motivic Chern class need not be a group homomorphism on the first image-filtration group $F_{m o t}^{1} K_{0} (X)$ . In fact, for virtual vector-bundle classes $x, y$ the Whitney identity gives $c_{2} (x + y) = c_{2} (x) + c_{2} (y) + c_{1} (x) c_{1} (y) .$ Here $K_{0} (X) = π_{0} K (X)$ is the degree-zero group of the supplied nonconnective perfect-complex $K$ -theory spectrum, and $F_{m o t}^{1} K_{0} (X)$ is the image of the first filtration stage in it. The classes $c_{i}$ lie in integral motivic cohomology $H_{m o t}^{2 i, i} (X, Z)$ , and the product is its cup product. A virtual vector-bundle class is a difference of classes of finite locally free sheaves; its Chern series is the quotient of their total Chern series.

For a concrete example, take an algebraically closed field $k$ , $X = P_{k}^{2}$ , $L = O_{X} (1)$ and $x = [L] - [O_{X}]$ . The class $x$ belongs to $F_{m o t}^{1} K_{0} (X)$ , while $c_{2} (x) = 0$ and $c_{2} (2 x) = c_{1} (L)^{2} \neq 0$ .

Facts & Assumptions

[A1]

The stated Choice, universe and site-size assumptions are inherited from the smooth first-stage comparison in [F1], used in step 1.2; the actual and virtual bundle constructions in [F2] and perfect-K-space constructions in [F5], used in steps 1.1 and 2.1; and the projective-bundle and derived coefficient constructions in [F3]–[F4], used in step 3.1. The Axiom of Choice

Given: The Axiom of Choice, the supplied compatible universe and site-size bounds, an algebraically closed field $k$ , and $X = P_{k}^{2}$ with its quotient-tautological line bundle $L = O_{X} (1)$ .

Goal: Establish the cross-term identity and the example, using the actual first image filtration and the supplied motivic product.

[F1]

The first image filtration is the image of the first stage in perfect-complex $K_{0}$ . For a smooth finite-type scheme over a field, that image is the kernel of the locally constant rank map, via the actual first-stage augmentation. Motivic filtration and image groups, Smooth rank, first Chern, determinant and supported positive leading cycle

[F2]

Actual Chern series satisfy Whitney's identity and extend multiplicatively to the vector-bundle Grothendieck group. Their degree-two coefficient obeys the displayed cross-term formula. A line bundle has no Chern classes above degree one, and the trivial line has total series $1$ . These assertions concern the vector-bundle group; they do not require a general identification of that group with perfect $K_{0}$ . Whitney identities and virtual Chern arithmetic, Extending a Whitney class to K zero, Motivic Chern classes

[F3]

For a rank-three trivial bundle over $Spec k$ , the projective-bundle formula identifies the direct sum of the three coefficient groups with the motivic cohomology of $P_{k}^{2}$ by $(a_{0}, a_{1}, a_{2}) \mapsto π^{*} a_{0} + π^{*} a_{1} h + π^{*} a_{2} h^{2}$ , where $h = c_{1} (O (1))$ . Pullback preserves the unit and products. Motivic projective bundle formula

[F4]

For a smooth finite-type scheme, $H_{m o t}^{0, 0}$ is isomorphic as an abelian group to its codimension-zero Chow group. For $Spec k$ that group is $Z$ , generated by its unique point. Smooth motivic comparison

[F5]

The actual universal perfect-K-space construction supplies natural pointed operations $c_{i} : K_{0} (X) \to H_{m o t}^{2 i, i} (X, Z)$ for $i \geq 1$ on every qcqs field-scheme, retaining the given Chern classes on actual bundle object classes. Its full affine-to-local perfect-K-space comparison includes components and the actual bundle object map. The actual perfect Cartan lemma supplies coefficientwise multiplicativity and inversion through degree two, with $c_{0} = 1$ and $c_{i} (0) = 0$ for $i > 0$ . Thus $c_{\leq 2} (u) = 1 + c_{1} (u) t + c_{2} (u) t^{2}$ is multiplicative modulo $t^{3}$ on actual perfect $K_{0}$ , and agrees there with the virtual bundle series through this degree. Neither premise requires global bundle-difference representatives for arbitrary perfect classes. External formal universal Chern maps and supported construction, Actual perfect Cartan through degree three on all qcqs field-schemes

Proof

1.1F2F5givenA1

For virtual vector-bundle classes the multiplicative series in [F2] satisfies $c_{t} (u + v) = c_{t} (u) c_{t} (v)$ . Comparing coefficients of $t^{2}$ gives $c_{2} (u + v) = c_{2} (u) + c_{1} (u) c_{1} (v) + c_{2} (v)$ . Independently, the actual operations in [F5] satisfy the same coefficient identity for every $u, v \in K_{0} (X)$ , by their Cartan formula in degree two. These are the operations on perfect $K_{0}$ whose restriction to the first image filtration is at issue.

1.2F1givenA1

Projective space is smooth and finite type over $k$ . The vector bundles $L$ and $O_{X}$ each have rank one, so the perfect-complex class $x = [L] - [O_{X}]$ has rank zero. Applying [F1] places this actual class in $F_{m o t}^{1} K_{0} (X)$ . This uses the canonical map sending a vector bundle to its degree-zero perfect complex, not a general vector-bundle-to-perfect $K_{0}$ isomorphism.

2.1F2F5step 1.1step 1.2A1

Put $h = c_{1} (L)$ . By [F2], the actual bundle series are $c_{t} (L) = 1 + h t$ and $c_{t} (O_{X}) = 1$ . Their virtual quotient is $1 + h t$ . By [F5], the actual perfect operations have precisely these values on the bundle object classes and are multiplicative with inverses modulo $t^{3}$ . Consequently, on the perfect class $x = [L] - [O_{X}]$ of step 1.2, they give $c_{\leq 2} (x) = (1 + h t) \cdot 1^{- 1} = 1 + h t (m o d t^{3})$ . Thus the actual values are $c_{1} (x) = h$ and $c_{2} (x) = 0$ . Apply the perfect-class identity from step 1.1 with both arguments equal to $x$ to obtain $c_{2} (2 x) = h^{2}$ . This proves compatibility for the displayed virtual representatives using the supplied operation itself.

3.1F3F4step 2.1A1∎

By [F4], the underlying abelian group $H_{m o t}^{0, 0} (Spec k, Z)$ is nonzero. Its ring unit is therefore nonzero: a unital ring whose unit is zero is the zero ring. In [F3], the element with this unit in the third summand and zero in the others maps to $h^{2}$ . Since the displayed projective-bundle map is injective, $h^{2} \neq 0$ . Thus $c_{2} (2 x) \neq 2 c_{2} (x)$ , proving the example and the claimed failure of additivity.

Depends on

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 11, Remark 5.4
Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 journal version · Definition 5.1 and Section 5 Steps 1–3, Proposition 5.3 and formula (5.3.1), printed pp. 269–271
Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Definition 4.4.1 and Lemma 4.4.3 with its displayed proof, printed pp. 30–31