ON FROBENIUS, K-THEORY, AND CHARACTERISTIC NUMBERS

MARKUS ROST

                 preliminary version

Contents

1. Introduction 1 2. Preliminaries 3 3. Exploiting Riemann-Roch 10 4. Examples 10 5. Further Remarks 11 References 11

1. Introduction

This text is in a very preliminary status. The starting point was my proof of the degree formula. For the

prime 2 this used the Hilbert scheme Hilb(2, X) and its canonical line bundle \overline{L}. One has for X smooth and proper of dimension d:

(1)  deg(c_{1}(\overline{L})^{2d}) = \frac{1}{2} deg(c_{d}(−T_{X}))

This gives a simple short proof that for any X the Segre number deg(c_{d}(−T_{X})) is 2-divisible.

Let’s have a look at the situation in characteristic 2. In this case there exist a canonical smooth divisor j: \overline{P} → Hilb(2, X) which represents \overline{L}. It fits into a commutative diagram

P(Ω_{X}) ⟶^{i} Bl_{Δ}(X × X)
ρ↓  \overline{ρ}↓
\overline{P} ⟶^{j} Hilb(2, X)

Here Bl_{Δ}(X × X) is the blow up of the diagonal and i is the inclusion of the exceptional fiber which is the projective tangent bundle of X.

Date: March 16, 2006; slightly revised on Feb 21, 2008.

Moreover \overline{ρ} is the standard double cover. The morphism ρ is radicial of degree 2. The pullback of \overline{L} to Bl_{Δ}(X × X) is the canonical line bundle of the blow up, and hence the pullback of \overline{L} to P(Ω_{X}) is the canonical line bundle of the projective bundle.

One may take also the following point of view: The canonical line bundle has a connection with respect to itself with trivial 2-curvature. By general principles it descends canonically to something, and that something is \overline{P}.

Let L = j^{∗}(\overline{L}). Now, since \overline{P} represents c_{1}(\overline{L}), we get the following variant of (1):

deg(c_{1}(L)^{2d−1}) = \frac{1}{2} deg(c_{d}(−T_{X}))

Hence \frac{1}{2} of the Segre number is the degree of a zero cycle on \overline{P}. One can do somewhat better: There is a commutative diagram

X ⟵^{π} P(Ω_{X})
F ↓  ^{ρ}↓
X^{(2)} ⟵^{π^{(2)}∘η}  \overline{P}
id↓  ^{η}↓
X^{(2)} ⟵^{π^{(2)}} P(Ω_{X})^{(2)}

Here F and η ∘ ρ are the relative Frobenius morphisms. One finds

(2)  2(π^{(2)} ∘ η)_{∗}(c_{1}(L)^{2d−1}) = c_{d}(−T_{X^{(2)}}) = F_{∗}(c_{d}(−T_{X}))

Hence one gets in characteristic 2 not only the 2-divisibility of the image of the Segre class in CH_{0}(Spec k) = Z (the Segre number) but of its image in CH_{0}(X^{(2)}).

In the case of curves this is long known: The line bundle Ω_{X^{(2)}} has a canonical square root, namely F_{∗}(O_{X})/O_{X^{(2)}}. (See [2], [1], [5].)

  Note: (2) is not yet contained in the text.

There is another result in this text, Proposition 1, which together with Riemann-Roch and the Hattori-Stong theorem yields the following: Let P ∈ Z[c_{1}, …, c_{d}] be a polynomial of degree d (with deg c_{i} = i), and suppose that there is a p-power q such that for any compact almost complex manifold X of dimension 2d the number P (X)/q is integral. Then for any smooth variety X in characteristic p of dimension d, the number P (X)/q is the degree of an integral zero cycle on X^{(p^{d})}.

Another example of such divisibilities has been provided by Deligne: For a smooth surface X in characteristic 2, let F: X → X^{(2)} be the relative Frobenius. Then one has in CH_{0}(X^{(2)}):

(3)  4c_{2}(F_{∗}(O_{X})) = (c_{1}^{2} + c_{2})(T_{X^{(2)}})

Taking degrees, one gets

deg(c_{2}(F_{∗}(O_{X}))) = 3 Todd(X)

  Note: (3) is not yet contained in the text.

I have learned Lemma 2 from Deligne. I don’t have a reference for it. I am also wondering about a reference for Lemma 1, Lemma 3 and, in particular, for Lemma 4.

2. Preliminaries

References: [4, Section 7].

For the Frobenius maps we use the following notations. Let X be a scheme in characteristic p.

The absolute Frobenius is denoted by

f = f_{X}: X → X

If X = Spec R is affine, then f is given by the p-th power map

φ: R → R
φ(a) = a^{p}

For a sheaf M of O_{X}-modules let

M^{[p]} = f^{∗}(M)

be the pull back of M along f. Similarly, in the affine case X = Spec R we denote for R-modules V

V^{[p]} = V ⊗_{R,φ} R

Here the tensor product is understood so that va ⊗ b = v ⊗ a^{p}b and the R-module structure on V^{[p]} is given by (v ⊗ a)b = v ⊗ ab.

The symmetric algebras are denoted by

S^{•}M = ⨁_{k≥0} S^{k}M, S^{•}V = ⨁_{k≥0} S^{k}V

There is a natural morphism

j_{M}: M^{[p]} → S^{p}M

given locally by

j_{V} : V^{[p]} → S^{p}V
j_{V} (v ⊗ a) = v^{p}a

Let

B(M) = S^{•}M/⟨j_{M}(M^{[p]})⟩

be the quotient of the symmetric algebra of M by the ideal sheaf generated by the image of j_{M}.

Similarly we understand B(V)....

The natural multiplication map

B(M) ⊗ B(M′) → B(M ⊕ M′)

is an isomorphism.

For a line bundle L one has

B(L) = S^{•}L/⟨L^{⊗p}⟩ = O_{X} ⊕ L^{⊗1} ⊕ L^{⊗} ⊕ ⋯ ⊕ L^{⊗(p−1)}

Lemma 1. Suppose that M is a vector bundle on X of rank n. Then

(1) j_{M} is a monomorphism.

(2) If M is invertible (n = 1), then j_{M} is an isomorphism, so that M^{[p]} = M^{⊗p}.

(3) B(M) is a vector bundle of rank p^{n}.

Proof. The question being local, we may assume that X = Spec R and that M is given by a free R-module V with basis e_{i}, i = 1, …, n. Then S^{•}V is the polynomial ring R[e_{1}, …, e_{n}]. Moreover V^{[p]} is free with basis e_{i} ⊗ 1, i = 1, …, n and j_{V} is given by j_{V} (e_{i} ⊗ 1) = e_{i}^{p}.

From this (1) and (2) are clear. As for (3), note that

B(V) = R[e_{1}, …, e_{n}]/⟨e_{1}^{p}, …, e_{n}^{p}⟩ = ⨂_{i=1}^{n} R[e_{i}]/⟨e_{i}^{p}⟩

  Better:

B(L_{1} ⊕ ⋯ ⊕ L_{n}) = B(L_{1}) ⊗ ⋯ ⊗ B(L_{n})
□

Remark 1. Let K^{0}(X) denote the Grothendieck group of vector bundles on X. There is the ring homomorphism f^{∗}: K^{0}(X) → K^{0}(X) induced by the absolute Frobenius. Since f^{∗} is on line bundles the p-th power map (cf. Lemma 1 (2)), it follows that f^{∗} is the p-th Adams operation in K-theory.

Let k be a field of characteristic p and let X be a scheme over k. The structure morphism of X is denoted by π_{X}: X → Spec k. The relative Frobenius is described by the following commutative diagram

X ⟶^{F} X^{(p)} ⟶^{W} X
π_{X} ↓  ^{π_{X^{(p)}}} ↓  ^{π_{X}} ↓
Spec k ⟶^{id} Spec k ⟶^{f_{k}} Spec k

Here the right square is Cartesian and defines X^{(p)} as the fiber product of X and k over k with respect to the absolute Frobenius, with W and π_{X^{(p)}} the corresponding maps. One has f_{k} ∘π_{X} = π_{X} ∘f_{X}. The relative Frobenius F is the unique map with W ∘ F = f_{X} and π_{X^{(p)}} ∘ F = π_{X}.

The morphism W: X^{(p)} → X is flat, since it is the pull back of the flat morphism f_{k}. If X is a localization of a scheme of finite type over k, then F is finite.

Lemma 2. Suppose that X is smooth over k of dimension d. Then F is flat and finite of rank p^{d}.

For the induced maps F_{∗}: K_{0}(X) → K_{0}(X^{(p)}), F^{∗}: K_{0}(X^{(p)}) → K_{0}(X) one has

F^{∗}(F_{∗}(x)) = [B(Ω_{X/k})] · x

for x ∈ K_{0}(X).

  Better:

F^{∗} ∘ F_{∗} is multiplication by [B(Ω_{X/k})].

Proof. For the first claim see [4]....

We consider the following commutative diagram of short exact sequences:

0 ⟶ I ⟶ O_{X} ⊗_{k} O_{X}  ⟶^{μ} O_{X} ⟶ 0
h↓  ↓  ^{id}↓
0 ⟶ J ⟶ O_{X} ⊗_{O_{X^{(p)}}} O_{X} ⟶^{μ} O_{X} ⟶ 0

Here μ denotes the multiplication maps with kernels I, J and the vertical arrows are the natural maps. By definition one has I/I^{2} = Ω_{X/k} and J/J^{2} = Ω_{X/X^{(p)}}. Moreover h induces an isomorphism I/I^{2} → J/J^{2}. Let

gr_{I}(O_{X} ⊗_{k} O_{X}) = ⨁_{n≥0} I^{n}/I^{n+1}, gr_{J}(O_{X} ⊗_{O_{X^{(p)}}} O_{X}) = ⨁_{n≥0} J^{n}/J^{n+1},

be the sheaves of graded rings associated to the filtrations induced by I, J, respectively, and let

gr(h): gr_{I}(O_{X} ⊗_{k} O_{X}) → gr_{J}(O_{X} ⊗_{O_{X^{(p)}}} O_{X})

be the homomorphism induced from h. Since X is smooth, the natural ring homomorphism

α: S^{•}(I/I^{2}) → gr_{I}(O_{X} ⊗_{k} O_{X})

with α the identity on I/I^{2} is an isomorphism. For x ∈ J one has x^{p} = 0; namely, for a ∈ O_{X} one has (a⊗1−1⊗a)^{p} = a^{p} ⊗1−1⊗a^{p} = 0 since a^{p} is in O_{X^{(p)}}. Hence gr(h)∘α factors trough a ring homomorphism

β: B(I/I^{2}) → gr_{J}(O_{X} ⊗_{O_{X^{(p)}}} O_{X})

It is not difficult to see by local considerations that β is an isomorphism. Composing β with the inverse of B(h): B(I/I^{2}) → B(J/J^{2}) we obtain an isomorphism

β: B(J/J^{2}) → gr_{J}(O_{X} ⊗_{O_{X^{(p)}}} O_{X})

of O_{X}-modules.

Let M be a O_{X}-module. Then

F^{∗}(F_{∗}(M)) = M ⊗_{O_{X^{(p)}}} O_{X} = M ⊗_{O_{X}} O_{X} ⊗_{O_{X^{(p)}}} O_{X}

The J-filtration induces a filtration on F^{∗}(F_{∗}(M)) with associated graded module

M ⊗_{O_{X}} gr_{J}(O_{X} ⊗_{O_{X^{(p)}}} O_{X})

... □

Let E be a finitely generated field over k. We denote by E^{p} ⊂ E the image of the p-th power homomorphism E → E and by

\overline{E} = kE^{p} ⊂ E

the subfield generated by E^{p} and k. If E is generated as a field over k by x_{1}, …, x_{N}, then the same is true over \overline{E}. Since x_{i}^{p} ∈ \overline{E}, it follows that E is finite over \overline{E}.

Let

E^{(p)} = E^{p} ⊗_{k^{p}} k

We consider the maps

E^{p} ⟶^{f} E^{(p)} ⟶^{F} E

with f(a) = a ⊗ 1 and F (a ⊗ b) = ab. Thus F ∘ f is the natural inclusion. Let 𝔪 be the kernel of F.

One has 𝔪^{p} = 0. Indeed, let x ∈ E^{(p)}. Then x^{p} is in the field

E^{p} ⊗_{k^{p}} k^{p} = E^{p}

If F (x) = 0, then F (x^{p}) = 0 and therefore x^{p} = 0.

The ideal 𝔪 is the unique maximal ideal of E^{(p)}. Its residue class field is

\overline{E} = F (E^{(p)}) = kE^{p} ⊂ E

We denote by ℓ(E/k) the length of the ring E^{(p)}.

Lemma 3. The extension E/\overline{E} is finite.

The ring E^{(p)} has finite length.

Let [E : \overline{E}] = dim\overline{E} E, let ℓ be the length of E^{(p)}, and let d be the transcendence degree of E/k. Then

[E : \overline{E}] = ℓp^{d}

Proof. Let k ⊂ F ⊂ E be an intermediate field with F/k separable and E/F finite. For instance, if x_{1}, …, x_{d} is a transcendence basis of E/k, one may take F = k(x_{1}, …, x_{d}). Consider the diagram

E^{(p)} ⟶^{r_{E}} \overline{E} ⟶ E
↑  ↑  ↑
F^{(p)} ⟶^{r_{F}} \overline{F} ⟶ F

Since F/k is separable, F^{(p)} is a field and r_{F} is an isomorphism. Hence

dim_{F^{(p)}} E^{(p)} = length(E^{(p)})[\overline{E} : \overline{F}]

On the other hand

dim_{F^{(p)}} E^{(p)} = [E^{p} : F^{p}] = [E : F]

Since F/k is separable, one has

[F : \overline{F}] = p^{d}

(For instance, if F = k(x_{1}, …, x_{d}), then \overline{F} = k(x_{1}^{p}, …, x_{d}^{p}).) Finally

[E : F][F : \overline{F}] = [E : \overline{E}][\overline{E} : \overline{F}]

The claim is now immediate. □

Lemma 4. Let X/k be of finite type. Then for W^{∗}, F_{∗}: CH_{r}(X) → CH_{r}(X^{(p)}) one has

F_{∗} = p^{r}W^{∗}

on the cycle groups

C_{r}(X) = ⨁_{x∈X_{(r)}} Z

See also [9, Proposition 2] for smooth schemes of finite type over a finite field.

Proof. Let x ∈ X be a point with dim \overline{\{x\}} = r and let y = F (x). Let further O_{X,x}, 𝔪_{x}, κ_{x} = O_{X,x}/𝔪_{x} be the local ring at x, its maximal ideal, and its residue class field, respectively. Then

F_{∗}([x]) = [κ_{x} : κ_{y}][y]

and

W^{∗}([x]) = length(O_{X^{(p)},y} ⊗_{O_{X,x}} κ_{x})[y]

Since

O_{X^{(p)},y} ⊗_{O_{X,x}} κ_{x} = (O_{X,x} ⊗_{k,φ} k) ⊗_{O_{X,x}} κ_{x} = κ_{x} ⊗_{k,φ} k = κ_{x}^{(p)}

the claim follows from Lemma 3.

  Better:

For a morphism h: Z → X there are the Cartesian diagrams

Z^{(p)} ⟶^{h^{(p)}} X^{(p)} ⟶^{π_{X^{(p)}}} Spec k
W ↓  ^{W} ↓  ^{f}↓
Z ⟶^{h} X ⟶^{π_{X}} Spec k

If h is a closed immersion, then W^{∗} ∘ h_{∗} = h_{∗} ∘ W^{∗}. If h is a open immersion, then h^{∗} ∘ W^{∗} = W^{∗} ∘ h^{∗}, see [3], [8]. Thus we may replace X by Spec κ_{x}. This case follows easily from Lemma 3. □

  This can be generalized as follows:

Let S be a scheme over a field of characteristic p and let X be a scheme over S. The structure morphism of X is denoted by π_{X}: X → S. The relative Frobenius is described by the following commutative diagram

X ⟶^{F} (X/S)^{(p)} ⟶^{W} X
π_{X} ↓  ^{π_{X^{(p)}}} ↓  ^{π_{X}} ↓
S ⟶^{id} S  ⟶^{f_{S}} S

Lemma 5. Let S be smooth over k of dimension e and let X/S be of finite type. Then for W^{∗}, F_{∗}: CH_{r}(X) → CH_{r}((X/S)^{(p)}) one has

F_{∗} = p^{r−e}W^{∗}  if r ≥ e
p^{e−r}F_{∗} = W^{∗}  if r ≤ e

on the cycle groups

C_{r}(X) = ⨁_{x∈X_{(r)}} Z

Proof. No proof yet.

□

Let π_{X}: X → Spec k be a scheme of finite type over k. Let K_{0}(X) denote the Grothendieck group of coherent O_{X}-module sheaves on X. Let K_{0}(X)_{(d)} ⊂ K_{0}(X) be the subgroup generated by sheaves M with dim supp M ≤ d.

Proposition 1.

(π_{X})_{∗}(K_{0}(X)_{(d)}) ⊂ (π_{X^{(p^{d})}})_{∗}(CH_{0}(X^{(p^{d})})) ⊗ Z_{(p)}

Proof. For d ≥ 0 and a O_{X}-module sheaf M on X let

Θ_{d}(M) = F_{∗}(M) − p^{d}f^{∗}(M)

This is a sheaf on X^{(p)}. Note that

(π_{X^{(p)}})_{∗}(Θ_{d}(M)) = (1 − p^{d})(π_{X})_{∗}(M)

This needs proof!!!

Clearly dim supp Θ_{d}(M) ≤ dim supp M. The two diagrams

CH_{d}(X) ⟶ K′_{0}(X)_{(d)}/K′_{0}(X)_{(d−1)} ⟶ 0
F_{∗},f^{∗}↓  ^{F_{∗},f^{∗}}↓
CH_{d}(X^{(p)}) ⟶ K′_{0}(X^{(p)})_{(d)}/K′_{0}(X^{(p)})_{(d−1)} ⟶ 0

for F_{∗}, f^{∗}, respectively commute and have exact rows. By Lemma 4 one has dim supp Θ_{d}(M) ≤ d − 1 if dim supp M ≤ d. Argue directly for sheaves!!!

Let us define for d > 0

θ_{d}: K_{0}(X)_{(d)} → K_{0}(X^{(p)})_{(d−1)} ⊗ Z_{(p)}
θ_{d}([M]) = (1 − p^{d})^{−1}[Θ_{d}(M)]

Then

(π_{X})_{∗} = (π_{X^{(p)}})_{∗} ∘ θ_{d}: K_{0}(X)_{(d)} → K_{0}(k) ⊗ Z_{(p)} = Z_{(p)}

Consider the map

θ̄_{d} = θ_{1} ∘ θ_{2} ∘ ⋯ ∘ θ_{d}: K_{0}(X)_{(d)} → K_{0}(X^{(p^{d})})_{(0)} ⊗ Z_{(p)}

Hence

(π_{X})_{∗}(K_{0}(X)_{(d)}) ⊂ (π_{X^{(p^{d})}})_{∗}(K_{0}(X^{(p^{d})})_{(0)}) = (π_{X^{(p^{d})}})_{∗}(CH_{0}(X^{(p^{d})}))

□

  Simplify this!  For perfect k look at
F′ = (W^{−1}) ∘ F: X → X.  Then F′ acts with eigenvalue p^{r} on

CH_{r}(X)....

Corollary 1. If k is perfect of characteristic p, then

(π_{X})_{∗}(K_{0}(X)) ⊂ (π_{X})_{∗}(CH_{0}(X)) ⊗ Z_{(p)}

In other words, the Euler characteristic of a O_{X}-module sheaf on X is the degree of a p-integral zero cycle on X.

10 MARKUS ROST

3. Exploiting Riemann-Roch

[6][7]

4. Examples

Lemma 6. Suppose dim X = 1. Then

F_{∗}(O_{X} − Ω_{X/k}) = O_{X^{(p)}} − Ω_{X^{(p)}/k}
F_{∗}(O_{X} + Ω_{X/k}) = p(O_{X^{(p)}} + Ω_{X^{(p)}/k})

Moreover, if p = 2, then

2(F_{∗}(O_{X}) − 2O_{X^{(2)}}) = Ω_{X^{(2)}/k} − O_{X^{(2)}}

and

F_{∗}(F_{∗}(O_{X}) − 2O_{X^{(2)}}) = F_{∗}(O_{X^{(2)}}) − 2O_{X^{(4)}}

and if p is odd, then

F_{∗}(O_{X}) − pO_{X^{(p)}} = \frac{p − 1}{2}(Ω_{X^{(p)}/k} − O_{X^{(p)}})

Notation:  bundles versus classes.  Better notations!

Lemma 7. Suppose dim X = 2 and p = 2. Then

F_{∗}(O_{X} − Ω_{X/k}^{2}) = 2(O_{X^{(2)}} − Ω_{X^{(2)}/k}^{2})
4F_{∗}(O_{X}) = ...

Lemma 8. Suppose dim X = 2 and p = 3. Then

....

9F_{∗}(O_{X}) = ...

Lemma 9. Suppose dim X = 2 and p > 3. Then

....

p^{2}F_{∗}(O_{X}) = ...

5. Further Remarks

[X → X^{(p)}] ∈ Ω(X^{(p)}) new natural elements in bordism (= in any oriented cohomology theory).

Compress [X]: Over Z_{(p)}, [X] is represented by sums [Y → X^{(p^{d})}].L, dim Y = 0, L= Lazard ring.

References

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[2] A. Fröhlich, J.-P. Serre, and J. Tate, A different with an odd class, J. Reine Angew. Math. 209 (1962), 6–7, MR0139601 (25 #3033).

[3] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984.

[4] N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. (1970), no. 39, 175–232, MR0291177 (45 #271).

[5] , Twisted L-functions and monodromy, Annals of Mathematics Studies, vol. 150, Princeton University Press, Princeton, NJ, 2002, MR1875130 (2003i:11087).

[6] I. Panin, Oriented cohomology theories of algebraic varieties, K-Theory 30 (2003), no. 3, 265–314, Special issue in honor of Hyman Bass on his seventieth birthday. Part III, MR2064242 (2005f:14043).

[7] , Riemann-Roch theorems for oriented cohomology, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261–333, MR2061857 (2005g:14025).

[8] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic), MR1418952 (98a:14006).

[9] C. Soulé, Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268 (1984), no. 3, 317–345, MR751733 (86k:14017).

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

E-mail address: rost@math.uni-bielefeld.de
URL: http://www.math.uni-bielefeld.de/~rost