Higher pushforwards along finite morphisms

This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:

Question. If f \colon X \to Y is a finite morphism of schemes, is the pushforward f_* \colon \Sh(X) \to \Sh(Y) exact?

Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.

However, we show that the answer to the question above is negative.

Example. Let Y be the spectrum of a DVR (R,\mathfrak m), let R \to S be a finite extension of domains such that S has exactly two primes \mathfrak p, \mathfrak q above \mathfrak m, and let X = \Spec S. For example, R = \Z_{(5)} and S = \Z_{(5)}[i], or R = k[x]_{(x)} and S = k[x]_{(x)}[\sqrt{x+1}] if you prefer a more geometric example.

By my previous post, the global sections functor \Gamma \colon \Sh(Y) \to \Ab is exact. If the same were true for f_* \colon \Sh(X) \to \Sh(Y), then the global sections functor on X would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection \mathscr F \to \mathscr G of sheaves on X such that the map on global sections is not surjective.

The topological space of X consists of closed points x,y and a generic point \eta. Let U = \{\eta\} and Z = U^{\operatorname{c}} = \{x,y\}; then U is open and Z is closed. Hence, for any sheaf \mathscr F on X, we have a short exact sequence (see e.g. Tag 02UT)

    \[0 \to j_! (\mathscr F|_U) \to \mathscr F \to i_* (\mathscr F|_Z) \to 0,\]

where j \colon U \to X and i \colon Z \to X are the inclusions. Let \mathscr F be the constant sheaf \Z; then the same goes for \mathscr F|_U and \mathscr F|_Z. Then the map

    \[H^0(X,\mathscr F) \to H^0(X,i_*(\mathscr F|_Z)) = H^0(Z,\mathscr F|_Z)\]

is given by the diagonal map \Z \to \Z \oplus \Z, since X is connected by Z has two connected components. This is visibly not surjective. \qedsymbol

Odd degree Betti numbers are even

In characteristic 0, it follows from the Hodge decomposition and Hodge symmetry that the Betti numbers h^i(X) = \dim H^i(X^{\operatorname{an}},\mathbb C) of a smooth proper complex variety X/\mathbb C are even when i is odd. In characteristic p however, both Hodge-de Rham degeneration and Hodge symmetry fail (and de Rham cohomology is not a Weil cohomology theory), so we cannot use this method to obtain the result.

On the other hand, in the projective case, we can use hard Lefschetz plus the explicit description of the Poincaré pairing to conclude (we get a perfect alternating pairing, so the dimension has to be even). This leaves open the proper (non-projective) case in positive characteristic. This was settled by Junecue Suh [1]. I will explain the case for finite fields; one can easily reduce the general case to this case.

Notation. Throughout, K will be a p-adic field with ring of integers W = \O_K, residue field k of size q, and (normalised) valuation v such that v(q) = 1 (this is the q-valuation on K).

Throughout, X will be a smooth proper variety over k. We will write h^i(X) for the Betti numbers of X. It can be computed either as the dimension of H^i\et(\bar X, \Q_\ell), or that of H^i_{\operatorname{crys}}(X/W)[\frac{1}{p}].

Remark. Recall that if f is the characteristic polynomial of Frobenius acting on H^i\et(\bar X, \mathbb Q_\ell) for \ell \neq p, and \alpha \in \bar{\mathbb Q} is the reciprocal of a root of f, then for every complex embedding \sigma \colon \bar \Q \to \C we have

(1)   \begin{equation*} |\sigma(\alpha)| = q^{\frac{i}{2}}. \end{equation*}

The same holds for the eigenvalues of Frobenius on crystalline cohomology (in fact, the characteristic polynomials agree). All reciprocal roots are algebraic integers, and f \in \mathbb Z[t].

Defintion. An algebraic integer \alpha \in \bar \Q is a q^i-Weil integer if it satisfies (1) (for every embedding \sigma \colon \bar \Q \to \C).

Lemma. Let f \in \mathbb Q[t] be a polynomial, and let S be the multiset of reciprocal roots of f. Assume all \alpha \in S are q^i-Weil integers. Then v(S) = i - v(S) (counted with multiplicity).

Proof. If \alpha \in S, then \frac{q^i}{\alpha} is the complex conjugate with respect to every embedding \sigma \colon \bar \Q \to \C. Thus, it is conjugate to \alpha, hence a root of f as well (with the same multiplicity). Taking valuations gives the result. \qedsymbol

Theorem. Let X be smooth proper over k, and let i be odd. Then h^i(X) is even.

Proof. The Frobenius-eigenvalues whose valuation is not \frac{i}{2} come naturally in pairs (\alpha, \frac{q^i}{\alpha}). Now consider valuation \frac{i}{2}. Note that the p-valuation of the semilinear Frobenius F equals the q-valuation of the K-linear Frobenius F^r (which is the one used in computing the characteristic polynomial f). The sum of the p-valuations of the roots should be an integer, because f has rational coefficients. Thus, there needs to be an even number of valuation \frac{i}{2} eigenvalues, for otherwise their product would not be a rational number. \qedsymbol

References.

[1] Suh, Junecue, Symmetry and parity in Frobenius action on cohomology. Compos. Math. 148 (2012), no. 1, 295–303. MR2881317.

Number of points modulo q is a stable birational invariant

This post is about a (very weak) shadow in characteristic p of the Larsen–Lunts theorem. See my previous post for the statement and sketch of the proof of Larsen–Lunts.

Remark. In characteristic p, we do not even know the weakest form of resolution of singularities (e.g. find a smooth proper model for any function field). Thus, we certainly do not know the Larsen–Lunts theorem. However, we can still try to prove corollaries (and if they fail, we know that resolution must fail).

Today, I want to talk about the following statement:

Theorem. (Ekedahl) Let k = \mathbb F_q. Let X and Y be smooth proper varieties, and assume X and Y are stably birational. Then |X(k)| \equiv |Y(k)| \pmod{q}.

Remark. This would follow immediately from Larsen–Lunts if we knew a sufficiently strong form of resolution of singularities. Indeed, the map

    \[K_0(\operatorname{Var}_k) \to \Z/q\Z\]

given by counting \F_q-points modulo q factors through K_0(\operatorname{Var}_k)/(\mathbb L) since |\mathbb A^1(\F_q)| = q. Hence, by Larsen–Lunts, it factors through \mathbb Z[\operatorname{SB}].

It turns out that the theorem is true without assuming resolution of singularities, and the proof is due to Ekedahl (although in his paper he never explicitly states it in this form). The reader should definitely check out Ekedahl’s article (see references below), because his proof is more beautiful than the one I present here, and actually proves a bit more.

We will need one fairly deep theorem:

Theorem. Let X be a variety of dimension n over k = \F_q. Let \alpha be an eigenvalue of Frobenius on H^i_c(\bar X\et, \Q_\ell). Then \alpha and q^n\alpha^{-1} are both algebraic integers.

The first part (integrality of \alpha) is fairly well-known. For the second part (integrality of q^n\alpha^{-1}), see SGA 7_{\text{II}}, Exp. XXI, Corollary 5.5.3(iii).

The statement that appears in Ekedahl’s article is the following:

Theorem. (Ekedahl’s version) Let k = \mathbb F_q. Let X and Y be smooth connected varieties (not necessarily proper!), and assume X and Y are birational. If \alpha is an eigenvalue of Frobenius on H^i(\bar X, \Q_\ell) which is not an eigenvalue on H^i(\bar Y, \Q_\ell), then \alpha is divisible by q.

This statement should be taken to include multiplicities; e.g. a double eigenvalue for X which is a simple eigenvalue for Y is also divisible by q. By symmetry, we also get the opposite statement (with X and Y swapped). Thus, the eigenvalues (with multiplicities) that are not divisible by q are the same for X and Y.

Proof. We immediately reduce to the case where X \sbq Y is an open immersion, with complement Z. We have a long exact sequence for étale cohomology with compact support:

    \[\cdots \to H^{i-1}_c(Z) \to H^i_c(X) \to H^i_c(Y) \to H^i_c(Z) \to H^{i+1}(X) \to \cdots.\]

If \alpha is an eigenvalue on some H^i_c(Z), then q^{n-1}\alpha^{-1} is an algebraic integer (see above). Hence, for any valuation v on \bar \Q with v(q) = 1, we have v(\alpha) \leq n-1. We conclude that the eigenvalues for which some valuation is > n-1 on H^i_c(X) and H^i_c(Y) agree. Hence, by Poincaré duality, the eigenvalues of H^{2n-i}(X) and H^{2n-i}(Y) for which some valuation is < 1 agree. These are exactly the ones that are not divisible by q. \qedsymbol

The theorem I stated above immediately follows from this one:

Proof. Since |X\times\P^n(\F_q)| = (q^n + \ldots + 1) |X(\F_q)|, we may replace X by X \times \P^n. Thus, we can assume X and Y are birational; both of dimension n.

By the Weil conjectures, we know that

    \[|X(\F_q)| = \sum_{i=0}^{2n} \sum_\alpha \alpha^i,\]

where the inner sum runs over all eigenvalues of Frobenius. If we reduce mod q, then we only need to consider eigenvalues that are not divisible by q. By Ekedahl’s version of the theorem, the set (with multiplicities) of such \alpha are the same for X and Y. \qedsymbol

Historical remark. Although the theorem above was essentially proven in 1983 (but not explicitly stated), a separate proof for threefolds appeared in a paper by Gilles Lachaud and Marc Perret in 2000. It uses Abhyankar’s results on resolution of singularities, and is much closer to the proof of Larsen–Lunts than Ekedahl’s proof was. In 2002, Bruno Kahn provided a different proof for the general case using some (fairly advanced?) motive machinery (‘almost without cohomology’).

References.

Torsten Ekedahl, Sur le groupe fondamental d’une variété unirationelle. Comptes rendus de l’académie des sciences de Paris, Serie I: mathématiques, 297(12), p. 627-629 (1983).

Bruno Kahn, Number of points of function fields over finite fields. arXiv:math/0210202

Gilles Lachaud and Marc Perret, Un invariant birationnel des variétés de dimension 3 sur un corps fini. Journal of Algebraic Geometry 9 (2000), p. 451-458.