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Formal Chern arithmetic through degree three

PDF p. 11: Inverse total Chern series; PDF p. 17: Exterior-power Chern computation

Statement

Let $R$ be a commutative ring with unit. A formal series $C (t) = 1 + \sum_{i \geq 1} a_{i} t^{i}$ has a multiplicative inverse, and, modulo terms of degree at least four, $C (t)^{- 1} = 1 - a_{1} t + (a_{1}^{2} - a_{2}) t^{2} + (- a_{1}^{3} + 2 a_{1} a_{2} - a_{3}) t^{3} .$ For $D (t) = 1 + \sum_{i \geq 1} b_{i} t^{i}$ , the coefficients in degrees one, two and three of $C (t) D (t)$ are respectively $a_{1} + b_{1}$ , $a_{2} + a_{1} b_{1} + b_{2}$ , and $a_{3} + a_{2} b_{1} + a_{1} b_{2} + b_{3}$ .

More generally, let $G$ be an abelian group and let $c_{t} : G \to 1 + t R [⁣ [t] ⁣]$ be a group homomorphism, where the target uses series multiplication. Write $c_{t} (x) = 1 + \sum_{i \geq 1} c_{i} (x) t^{i}$ . If $r \geq 1$ and $H$ is a subgroup of $G$ on which $c_{i}$ vanishes for every $1 \leq i < r$ , then $c_{r} : H \to (R, +)$ is a group homomorphism. In particular, under the hypotheses of the virtual Chern extension, these formulas apply to its total Chern series, with $R = ⨁_{i \geq 0} H^{2 i, i} (X)$ . If $c_{1} (x) = c_{1} (y)$ , then $c_{2} (x - y) = c_{2} (x) - c_{2} (y)$ .

Here $R [⁣ [t] ⁣]$ denotes the ring of formal sequences of coefficients, with multiplication defined by the finite convolution formula $[t^{n}] (A B) = \sum_{i = 0}^{n} [t^{i}] A [t^{n - i}] B$ . The notation $1 + t R [⁣ [t] ⁣]$ means the series with constant coefficient one, and $[t^{n}] A$ denotes the coefficient of $t^{n}$ . Equality modulo terms of degree at least four means equality of coefficients in degrees zero through three. A virtual Chern class is an element of the bundle Grothendieck group equipped with the extended series described by the cited interface.

Statement references

Extending a Whitney class to K zero

Facts & Assumptions

Given: A commutative unital ring $R$ , the series $C$ and $D$ , and, for the additivity assertion, the group homomorphism $c_{t}$ , integer $r$ and subgroup $H$ specified in the statement.

Goal: Establish the inverse and product formulas, the subgroup additivity assertion and the stated conditional virtual Chern consequences.

[F1]

Under its stated hypotheses, including the Axiom of Choice and the exact-sequence Whitney identity, the virtual Chern extension supplies a homomorphism from the bundle Grothendieck group to the multiplicative group of total series. Its coefficients lie in $H^{2 i, i} (X)$ , and $⨁_{i \geq 0} H^{2 i, i} (X)$ is commutative because the first grading of each summand is even. Extending a Whitney class to K zero

Proof

technique · Compare coefficients of formal series

1.1givenconstructalgebra

Set $q_{0} = 1$ and recursively $q_{n} = - \sum_{i = 1}^{n} a_{i} q_{n - i}$ for $n \geq 1$ . Every sum is finite, so $Q (t) = \sum_{n \geq 0} q_{n} t^{n}$ is a formal series. The coefficient of $t^{n}$ in $C Q$ is zero for $n > 0$ and one for $n = 0$ ; commutativity gives $Q C = 1$ as well. The recursion yields $q_{1} = - a_{1}$ , $q_{2} = a_{1}^{2} - a_{2}$ , and $q_{3} = - a_{1}^{3} + 2 a_{1} a_{2} - a_{3}$ , proving the inverse formula.

1.2givenalgebra

The convolution formula gives $[t^{n}] (C D) = a_{n} + b_{n} + \sum_{i = 1}^{n - 1} a_{i} b_{n - i}$ for $n \geq 1$ , taking the constant coefficients to be one. Substituting $n = 1, 2, 3$ gives the three displayed product coefficients.

2.1givenstep 1.2algebra

For $x, y \in H$ , the homomorphism property gives $c_{t} (x + y) = c_{t} (x) c_{t} (y)$ . Comparing degree $r$ as in step 1.2 gives $c_{r} (x + y) = c_{r} (x) + c_{r} (y) + \sum_{i = 1}^{r - 1} c_{i} (x) c_{r - i} (y)$ . Each summand vanishes by the hypothesis on $H$ , including the empty-sum case $r = 1$ . The identity element maps to the series $1$ , so $c_{r} (0) = 0$ ; the addition formula also gives $c_{r} (- x) = - c_{r} (x)$ . Thus $c_{r}$ is a group homomorphism.

3.1F1step 1.1step 1.2step 2.1algebra∎

Under the hypotheses in [F1], the total series satisfy the hypotheses just used, so all preceding formulas apply to virtual classes. For $x, y$ with common first coefficient $u$ , steps 1.1 and 1.2 give $c_{2} (x - y) = c_{2} (x) + (u^{2} - c_{2} (y)) - u^{2} = c_{2} (x) - c_{2} (y)$ . This proves the stated virtual consequences and completes the proof.

Depends on

Extending a Whitney class to K zero

Used by · 6

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