NOTES ON THE DEGREE FORMULA

MARKUS ROST

The major aim of this text is to provide a proof of Remark 10.4 in [1]. I am indebted to A. Suslin for helpful and encouraging comments.

1. The invariants ρ^{i}

Let p be a prime. We work over a ground field k of characteristic different from p. Let G = μ_{p}.

Let X be a smooth variety over k, equidimensional of dimension d, together with a G-action. Let Y ⊂ X be the fix-point subscheme. We assume that Y is smooth (this is always the case, but I don’t know a reference at the moment). Let U = X \ Y.

Let \overline{X} = X/G be the quotient, let \overline{U} ⊂ \overline{X} be the image of U and let \overline{Y} = \overline{X} \ \overline{U} with the reduced subscheme structure. Certainly \overline{U} is smooth. Since char k ≠ p, the projection π: X → \overline{X} induces an isomorphism Y → \overline{Y}.

In the following we will identify \overline{Y} with Y.

The morphism π′ = π|U: U → \overline{U} is a μ_{p}-torsor. Let

α ∈ A^{0}(\overline{U}, K_{1}/p)

be the corresponding unramified p-th power class (locally π′ is presented as R = \overline{R}[t]/(t^{p} − f) for some regular function f, and α is then the p-th power class of f). Let further L be the line bundle on \overline{U} induced from π′ via the inclusion μ_{p} → G_{m} and let

β ∈ CH^{1}(\overline{U}) = A^{1}(\overline{U}, K_{1})

be the first Chern class of L.

We define a series of classes ρ^{i} ∈ CH_{d−i}(Y)/p by

ρ^{i}(X) = ρ^{i} = ∂\overline{U} ; Y(αβ^{i−1})

where

∂\overline{U} ; Y: A^{i−1}(\overline{U}, K_{i}/p) → CH_{d−i}(Y)/p

is the boundary map, cf. [2].

Proposition 1. (Degree formula) Let φ: X′ → X be a morphism of smooth Gvarieties of the same dimension with X/G irreducible. Let Y, Y′ be the G-fix-point subschemes and let ψ = φ|Y′: Y′ → Y. Then

ψ_{∗}(ρ^{i}(X′)) = (deg φ)ρ^{i}(X)

Proof. Let Z = φ^{−1}(Y). The restriction of α′ resp. β′ to (X′ \ Z)/G is equal to the pull back of α resp. β. Therefore

φ^{∗}(ρ^{i}(X)) = i_{∗}(ρ^{i}(X′))

Date: December 13, 2001.

where i: Z → X′ is the inclusion. Thus

φ_{∗}(ρ^{i}(X′)) = φ_{∗}(φ^{∗}(ρ^{i}(X))) = (deg φ)ρ^{i}(X).
□

Remark 1. It is very likely that one can extend the considerations to not necessarily smooth varieties. More generally one may consider also arbitrary branched coverings X → \overline{X} of degree p which are not necessarily cyclic. Everything should work in arbitrary characteristic (this means essentially to establish the Steenrod operations in all characteristics) . It would be interesting to understand the situation also for arbitrary finite morphisms X → \overline{X}.

2. The invariant ρ

Let us assume that Y is proper. We write I(Y) for the image of the degree map deg_{Y} : CH_{0}(Y) → Z. We have then a degree map

d̃eg_{Y} : CH_{0}(Y)/p → Z/pI(Y)

We define

ρ = d̃eg_{Y} (ρ^{d})

Note that the image of ρ^{i} in CH_{d−i}(\overline{X})/p is trivial. Therefore, if X is proper, then the image of ρ in Z/p is trivial.

3. The invariant η

In this section we assume that X is proper.

The degree map deg\overline{X} defines a degree map

deg\overline{U}: CH_{0}(\overline{U}) = CH_{0}(\overline{X})/ CH_{0}(Y) → Z/I(Y)

We define

η = deg(β^{d}) ∈ Z/I(Y).

Lemma 1. The image of η under the inclusion

Z/I(Y) ⟶^{p} Z/pI(Y)

is equal to −ρ.

Proof. Suppose first that X is a proper smooth curve. Let f ∈ k(\overline{X})^{∗} be a function representing α. Then

d(f) = pA + B

for a zero cycle A on \overline{X} and a zero cycle B on \overline{Y}. Since \sqrt[p]{f} is a section of the line bundle L, on has

β = [A]

On the other hand

∂\overline{U} ; \overline{Y}(α) = [B]

by the very definition of ∂\overline{U} ; \overline{Y}. Since

0 = deg\overline{X}(d(f)) = p deg\overline{X}(A) + deg\overline{X}(B)

one has

p deg\overline{X}(β) = p deg\overline{X}(A) = − deg\overline{X}(B) = − deg\overline{Y} (B) = − deg\overline{Y} (∂\overline{U} ; \overline{Y}(α))

This proves the claim for d = 1.

For the general case one chooses a proper smooth 1-dimensional variety \overline{C} and a morphism f: \overline{C} → \overline{X} such that β^{d−1} ∈ CH_{1}(\overline{U}) is represented by the image of the fundamental cycle of \overline{C}. This way one easily reduces the problem to the case of a curve. □

4. Relations with η_{p}

Let Y be a smooth proper equidimensional variety, let X = Y^{p} and let G = Z/p = μ_{p} act on X by cyclic permutations. We then have the invariant

η ∈ Z/I(Y)

as above. This class is denoted by η_{p} in [1].

The computation of η_{p} stated in Remark 10.4 in [1] will follow from Lemma 1 and the computations of the classes ρ^{i} below. The technical advantage of the more general setting of the classes ρ^{i} is that one does not need to assume that X is proper.

5. Reduction to the case of G-vector bundles

Lemma 2. Let V be the normal bundle of Y in X. Then

ρ^{i}(X) = ρ^{i}(V)

Proof. This follows easily by applying our construction to the deformation to the normal cone. To be a bit more specific, let M → A^{1} be the deformation to the normal cone of Y in X with fibre V over 0 ∈ A^{1} and fibre X otherwise. Consider the class

ρ^{i}(M) ∈ CH_{d+1−i}(Y × A^{1})/p

It specializes to ρ^{i}(V) at Y × {0} and to ρ^{i}(X) at Y × {1}. □

6. Preliminaries on Chern classes

Let R = Z/p[[c_{1}, c_{2}, …]] be the completed ring of mod p Chern classes.

For each r ∈ Z/p, r ≠ 0 define the series

b^{[r]} = 1 + b_{1}^{[r]} + b_{2}^{[r]} + ⋯ ∈ R

as follows. Write formally

1 + c_{1} + ⋯ + c_{n} = ∏_{i=1}^{n}(1 + x_{i})

and then define

b^{[r]} = ∏_{i=1}^{n}(1 + r^{−1}x_{i}) mod terms of degree > n

Remark 2. Note that b^{[1]} = 1 + c_{1} + c_{2} + · · ·.

Remark 3. Put

b = ∏_{r} b^{[r]}

Since

b = ∏_{i=1}^{n}(1 + x_{i}^{p−1})^{−1} mod terms of degree > n

the series b lies in the subring of R of elements of degree divisible by p − 1. Thus b is of the form

b = 1 + b_{p−1} + b_{2(p−1)} + ⋯

with b_{n} of degree n. The classes b_{k(p−1)} are closely related to the Steenrod powers P^{2k(p−1)}.

We have to consider also integral representatives of the b^{[r]}. For each r ∈ Z/p, r ≠ 0 choose an integer r′ with r′r = 1 mod p. Then define

b̃^{[r]} ∈ Z[[c_{1}, c_{2}, …]]

as follows. Write formally

1 + c_{1} + ⋯ + c_{n} = ∏_{i=1}^{n}(1 + x_{i})

and then define

b̃^{[r]} = ∏_{i=1}^{n}(1 + r′x_{i}) mod terms of degree > n

7. The fundamental identity for the class β

Let X = V be a G-vector bundle of rank e over a smooth variety Y. We assume that V^{G} = 0 and identify Y with the zero section. Write

V = ⨁_{r} V_{r}

for the decomposition of V into G-eigenspaces with G acting on V_{r} by (ζ, v) ↦ ζ^{r}v with r ∈ Z/p, r ≠ 0. Let q: \overline{U} = (V \ 0)/G → Y be the projection.

Note that the class

β ∈ CH^{1}(\overline{U})

is p-torsion, since the line bundle L becomes trivial after pull back to U and since U → \overline{U} is of degree p.

Put

b̃(V) = ∏_{r} b̃^{[r]}(V_{r}) ∈ CH^{∗}(Y)

and write

b̃(V) = 1 + b̃_{1}(V) + b̃_{2}(V) + ⋯

with b̃_{i}(V) ∈ CH^{i}(Y). One has b̃_{e}(V) = Nc_{e}(V) for some integer N with N mod p not depending on the choices of the r′.

Further let

b(V) = q^{∗}(b̃(V)) ∈ CH^{∗}(\overline{U})

and write

b(V) = 1 + b_{1}(V) + b_{2}(V) + ⋯

with b_{i}(V) = q^{∗}(b̃_{i}(V)) ∈ CH^{i}(\overline{U}). Note that the vector bundle V admits a nonvanishing section after pull back to U. Since U → \overline{U} is of degree p, the Euler class q^{∗}(c_{e}(V)) is p-torsion. Therefore the classes

β^{e}, b_{1}(V)β^{e−1}, …, b_{e}(V)

do not depend on the choices of the r′.

Theorem 1. One has

(1)  β^{e} + b_{1}(V)β^{e−1} + ⋯ + b_{e}(V) = 0

in CH^{e}(\overline{U}).

Proof. Consider first the case V = V_{1}. Let \overline{U} → P(V) = (V \ 0)/G_{m} be the projection. Then β ∈ CH^{1}(\overline{U}) is the pull back of the first Chern class ξ of the tautological line bundle on P(V) (up to a sign, about which I don’t care in this exposition). Equation (1) follows then from the standard identity for ξ.

Consider next the case V = V_{r}. This case follows similarly as for r = 1. Let \overline{U} → P(V) = (V \ 0)/G_{m} be the projection. Then β ∈ CH^{1}(\overline{U}) is r-times the pull back of the first Chern class ξ of the tautological line bundle on P(V). Equation (1) follows then again from the standard identity for ξ.

For the general case one uses a standard Mayer-Vietoris argument. Write

b̃_{V} (t) = t^{e} + b̃_{1}(V)t^{e−1} + ⋯ + b̃_{e}(V) ∈ CH^{∗}(Y)[t]

and put

b_{V} (t) = q^{∗}(b̃_{V})(t) ∈ CH^{∗}(\overline{U})[t]

We further write β = β_{V} to indicate the dependence on the bundle V. By induction we may suppose that there exists a decomposition

V = W_{1} ⊕ W_{2}

of G-vector bundles such that equation (1) holds for the bundles W_{1}, W_{2}. One has

b̃_{V} (t) = b̃_{W_{1}}(t)b̃_{W_{2}}(t)

Consider the exact sequence

CH^{∗}((W_{1} \ 0)/G) ⟶^{i_{∗}} CH^{∗}((V \ 0)/G) ⟶^{j_{∗}} CH^{∗}((V \ W_{1})/G) → 0

of Chow groups induced from the closed immersion i: (W_{1} \ 0)/G → (V \ 0)/G and the open immersion j: (V \ W_{1})/G → (V \ 0)/G of its complement. The projection s: (V \ W_{1})/G → (W_{2} \ 0)/G is a vector bundle. One has

j^{∗}(β_{V}) = s^{∗}(β_{W_{2}})

Hence

j^{∗}(q^{∗}(b̃_{W_{2}})(β_{V})) = s^{∗}(b_{W_{2}}(β_{W_{2}})) = 0

Therefore there exist T ∈ CH^{0}((W_{1} \ 0)/G) with

i_{∗}(T) = q^{∗}(b̃_{W_{2}})(β_{V})

Thus

b_{V} (β_{V}) = q^{∗}(b̃_{W_{1}})(β_{V})q^{∗}(b̃_{W_{2}})(β_{V}) = q^{∗}(b̃_{W_{1}})(β_{V})i_{∗}(T) = i_{∗}(i^{∗}(q^{∗}(b̃_{W_{1}})(β_{V}))T )

But i^{∗}(β_{V}) = β_{W_{1}} and therefore i^{∗}(q^{∗}(b̃_{W_{1}})(β_{V})) = b_{W_{1}}(β_{W_{1}}) = 0.

□

8. Computation of ρ^{i}

We are now ready to proceed similary as in [1, Section 5]. Let f = dim Y, so that d = dim X = f + e.

Lemma 3. One has

∂(αβ^{h}) = \begin{cases}0 & for h < e − 1 \\ u[Y] & for h = e − 1 \end{cases}

for some u ∈ Z/p, u ≠ 0.

Remark 4. The number u can be easily determined from the ranks of the bundles V_{r}. We omit this.

Proof. One has ∂(αβ^{h}) ∈ CH_{f+e−(h+1)}(Y). Therefore ∂(αβ^{h}) = 0 for h < e − 1 by dimension reasons. For h = e − 1 we have ∂(αβ^{h}) ∈ CH^{0}(Y). This case follows from the case when dim Y = 0. We may therefore assume that Y = Spec k.

There exists a decomposition

V = W_{1} ⊕ W_{2}

of G-vector modules over k such that rank W_{2} = 1. Let r be the weight of W_{2} as a G-module and choose r′ with rr′ = 1 mod p. The line bundle L over \overline{U} = (V \0)/G is given by

((W_{1} × W_{2}) \ 0) ×_{G} A^{1} → ((W_{1} × W_{2}) \ 0)/G

It has the section

[w_{1}, w_{2}] ↦ [w_{1}, w_{2}, s(w_{2})^{r′}]

where s is a trivialization of the 1-dimensional vector space W_{2}. Hence

β = c_{1}(L) = r′i_{∗}([(W_{1} \ 0)/G])

By induction we are reduced to the case rank V = 1. In this case V = A^{1} and \overline{U} = (V \0)/G ≃ A^{1} \0 with α represented by t^{r} where t is the standard coordinate of A^{1} and where r is an integer representing the weight of V. Obviously

∂\overline{U} ; 0(α) = r mod p

and the claim follows. □

By equation (1) we have

β^{e−1+i} = ∑_{h=0}^{e−1} β^{e−1−h}q^{∗}(g_{i,h})

with g_{i,h} ∈ CH^{i+h}(Y). The g_{i,h} are universal polynomials in the Chern classes of the bundles V_{r}. One finds in particular

g_{0,0} + g_{1,0} + g_{2,0} + ⋯ = b̃(V)^{−1}

One has by Lemma 3

ρ^{e+i} = ∂(αβ^{e+i−1}) = ∂(αβ^{e−1})g_{i,0} = ug_{i,0} mod p

Thus we have computed

ρ = u deg(g_{dim Y,0}) ∈ Z/pI(Y)

in terms of characteristic numbers of the bundles V_{r}. It is easy to check that this yields the description of η_{p} in Remark 10.4 in [1], perhaps up to a sign.

References

[1] A. S. Merkurjev, Degree formula, notes, May 2000, http://www.math.ohio-state.edu/˜ rost/chain-lemma.html.

[2] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~rost