Grothendieck topologies (topologies 4/6)

This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:

Corollary. Let \mathscr C be a (small) site. For a set of morphisms \mathscr U = \{U_i \to U\}_{i \in I} with the same target, write S_{\mathscr U} \subseteq h_U for the presheaf image of \coprod_{i\in I} h_{U_i} \to h_U. Then a presheaf \mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set} is a sheaf if and only if for every covering \mathscr U = \{U_i \to U\}_{i \in I} in \mathscr C, the inclusion S_{\mathscr U} \hookrightarrow h_U induces an isomorphism

    \[\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).\]

Thus, if \mathscr C is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category \mathbf{Sh}(\mathscr C) \subseteq \mathbf{PSh}(\mathscr C) of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.

Before giving the definition, note that any morphism f \colon Y \to X in \mathscr C gives a pullback \mathbf{Siv}(X) \to \mathbf{Siv}(Y) taking S \subseteq h_X to its inverse image under h_f \colon h_Y \to h_X (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to h_Y defined uniquely up to unique isomorphism). Thus, \mathbf{Siv} is itself a presheaf \mathscr C^{\operatorname{op}} \to \mathbf{Set} (it takes values in \mathbf{Set} since \mathscr C is small).

Also note the following method for producing sieves: if \mathscr F is a presheaf, \mathscr G \subseteq \mathscr F a subpresheaf, and s \in \mathscr F(X) a section over some X \in \mathscr C, we get a sieve (s \in \mathscr G) \in \mathbf{Siv}(X) by

    \[(s \in \mathscr G)(Y) = \left\{f \colon Y \to X\ \big|\ f^*(s) \in \mathscr G(Y)\right\}.\]

By the Yoneda lemma, this is just the inverse image of \mathscr G \subseteq \mathscr F along the morphism h_X \to \mathscr F classifying s. Note that (s \in \mathscr G) is the maximal sieve h_X if and only if s \in \mathscr G(X).

Definition. Let \mathscr C be a small category. Then a Grothendieck topology on \mathscr C consists of a subpresheaf J \subseteq \mathbf{Siv} such that

  1. For all X \in \mathscr C, the maximal sieve h_X \subseteq h_X is in J(X).
  2. If S \in J(X) and S' \in \mathbf{Siv}(X) with S \subseteq S', then S' \in J(X).
  3. If S \in \mathbf{Siv}(X) is a sieve such that (S \in J) \in J(X), then S \in J(X) (equivalently, then (S \in J) is the maximal sieve h_X).

The sieves S \in J(X) are called covering sieves. Since J is a presheaf, we see that for any f \colon Y \to X and any covering sieve S \subseteq h_X, the pullback f^*S \subseteq h_Y is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV_1, Exp. II, Def. 1.1:

  • If S, S' \in \mathbf{Siv}(X) with S \in J(X), such that for every morphism h_Y \to S the inverse image of S' \subseteq h_X along h_Y \to S \to h_X is in J(Y), then S' \in J(X).

Indeed, applying this criterion when S \subseteq S' immedately shows S' \in J(X) if S \in J(X), since the inverse image of S' \subseteq h_X along h_Y \to S \to h_X is the maximal sieve h_Y. Thus the local character implies criterion 2. The local character says that if (S' \in J) contains a covering sieve S, then S' is covering. Assuming criterion 2, the sieve (S' \in J) contains a covering sieve if and only if (S' \in J) is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that J(X) is closed under binary intersection, i.e. if S, T \in J(X) then (S \cap T) \in J(X). Indeed, if f \in S(Y) for some f \colon Y \to X, then

    \[f^*(S \cap T) = f^*S \cap f^*T = h_Y \cap f^*T = f^*T \in J(Y),\]

so S \subseteq ((S \cap T) \in J). Axioms 2 and 3 give (S \cap T) \in J(X).

Example. Let \mathcal Cov(\mathscr C) be a pretopology on the (small) category \mathscr C; see Tag 00VH for a list of axioms. For each X \in \mathscr C, define the subset J(X) \subseteq \mathbf{Siv}(X) as those S \subseteq h_X that contain a sieve of the form S_{\mathscr U} for some covering \mathscr U = \{U_i \to X\} in \mathcal Cov(\mathscr C). (See the corollary at the top for the definition of S_{\mathscr U}.) Concretely, this means that there exists a covering \{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C) such that f_i \in S(U_i) for all i \in I, i.e. X is covered by morphisms f_i \colon U_i \to X that are in the given sieve S.

Lemma. The association X \mapsto J(X) is a topology. It is the coarsest topology on \mathscr C for which each S_{\mathscr U} for \mathscr U \in \mathcal Cov(\mathscr C) is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If S \in J(X), then there exists \mathscr U = \{U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C) with S_{\mathscr U} \subseteq S. If f \colon Y \to X is any morphism in \mathscr C, then f^*\mathscr U = \{U_i \times_X Y \to Y\}_{i \in I} \in \mathcal Cov(\mathscr C) by criterion 3 of Tag 00VH. But S_{f^*\mathscr U} = f^*S_{\mathscr U}, because a morphism g \colon U \to Y factors through U_i \times_X Y if and only if fg \colon U \to X factors through U_i. Thus, S_{f^*\mathscr U} = f^*S_{\mathscr U} \subseteq f^*S, so f^*S \in J(Y), and J is a subpresheaf of \mathbf{Siv}.

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose S \in \mathbf{Siv}(X) satisfies (S \in J) \in J(X). Then there exists \mathscr U = \{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C) with S_{\mathscr U} \subseteq (S \in J). This means that f_i \in (S \in J)(U_i) for all i, i.e. f_i^*S \in J(U_i) for all i. Thus, for each i \in I there exists \mathscr V_i = \{g_{ij} \colon V_{ij} \to U_i\}_{j \in J_i} in \mathcal Cov(\mathscr C) such that S_{\mathscr V_i} \subseteq f_i^*S, i.e. f_ig_{ij} \in S(X) for all i \in I and all j \in J_i. Thus, if \mathscr V denotes \{f_ig_{ij} \colon V_{ij} \to X\}_{i \in I, j \in J_i}, then we get S_{\mathscr V} \subseteq S. But \mathscr V is a covering by criterion 2 of Tag 00VH, so S \in J(X).

If J' is any other Grothendieck topology for which each S_{\mathscr U} for \mathscr U \in \mathcal Cov(\mathscr C) is covering, then J' contains J by criterion 2. \qedsymbol

To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:

Definition. Let \mathscr C be a small category, and let J \subseteq \mathbf{Siv} be a Grothendieck topology. Then a presheaf \mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set} is a sheaf if for any X \in \mathscr C and any S \in J(X), the map S \hookrightarrow h_X induces an isomorphism

    \[\operatorname{Hom}(h_X,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S,\mathscr F).\]

Thus, a Grothendieck topology is an internal characterisation (inside \mathbf{PSh}(\mathscr C)) of which morphisms S \to h_X one needs to localise to get \mathbf{Sh}(\mathscr C,J). In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.

Covering sieves and the sheaf condition (topologies 3/6)

In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) \mathbf 1_X on a small category (resp. small site) with terminal object X correspond to left closed (resp. local) properties on \mathscr C. In this post, I will explain the main examples that intervene in setting up topoi, and show how to define the sheaf condition using sieves (instead of coverings).

For simplicity, assume \mathscr C is a small category with fibre products.

Definition. Given a set of morphisms \mathscr U = \{f_i \colon U_i \to U\}_{i \in I} with the same target U \in \mathscr C, define the sieve S_{\mathscr U} \subseteq h_U generated by \mathscr U as the sieve on U of those morphisms V \to U that factor through some f_i \colon U_i \to U.

It is in a sense the right ideal in \operatorname{Hom}(-,U) generated by the f_i. What does this look like as a subobject of h_U?

Example. If I has one element, i.e. \mathscr U = \{V \to U\}, then S_{\mathscr U} is the image of the morphism of representable presheaves h_V \to h_U. In the case where V \to U is already a monomorphism (this is always the case when \mathscr C is a poset, such as \operatorname{Open}(X) for some topological space X), then h_V \to h_U is itself injective (this is the definition of a monomorphism!), so S_{\mathscr U} is just h_V.

In general, S_{\mathscr U} is the image of the map

    \[\coprod_{i \in I} h_{U_i} \to h_U\]

induced by the maps U_i \to U. Indeed, an element of h_U(V) is a morphism f \colon V \to U, and it comes from some h_{U_i}(V) if and only if f factors through f_i \colon U_i \to U.

This shows that, in fact, every sieve S \subseteq h_X is of this form for some set \{U_i \to U\}_{i \in I}: take as index set (the objects of) the slice category (h \downarrow S), which as in the previous post gives a surjection \coprod_{(V,\alpha)} h_V \to S. This corresponds to generating an ideal by all its elements.

But we can also characterise S_{\mathscr U} without using the word ‘image’ (which somehow computes its first syzygy):

Lemma. Let \mathscr U = \{U_i \to U\} be a set of morphisms with common target, and S_{\mathscr U} the sieve generated by \mathscr U. Then S_{\mathscr U} is the coequaliser of the diagram

    \[\coprod_{i,j \in I} h_{U_i \underset U\times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i},\]

where the maps are induced by the two projections I^2 \to I.

We will give two proofs, one using the description of coequalisers of sets, and the other using that presheaves are colimits of representable presheaves, as discussed in the previous post.

Proof 1. The diagram

    \[\begin{array}{ccc}\displaystyle\coprod_{i,j \in I} h_{U_i \underset U\times U_j} & \to & \displaystyle\coprod_{i \in I} h_{U_i} \\ \downarrow & & \downarrow \\ \displaystyle\coprod_{j \in I} h_{U_j} & \to & h_U \end{array}\]

is a pullback, by the universal property of fibre products U_i \times_U U_j and since fibre products with a fixed set/presheaf of sets commute with coproducts. Then the same goes for the square

    \[\begin{array}{ccc}\displaystyle\coprod_{i,j \in I} h_{U_i \underset U\times U_j} & \to & \displaystyle\coprod_{i \in I} h_{U_i} \\ \downarrow & & \downarrow \\ \displaystyle\coprod_{j \in I} h_{U_j} & \to & S_{\mathscr U} \end{array}\]

since S_{\mathscr U} \to h_U is a monomorphism. But \coprod_{i \in I} h_{U_i} \to S_{\mathscr U} is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in \mathbf{Set}, epimorphisms are regular). \qedsymbol

Proof 2. By the previous post, the presheaf S_{\mathscr U} is the colimit over (V,\alpha) \in (h \downarrow S_{\mathscr U}) of h_V (see post for precise statement). Let D \colon (\bullet \rightrightarrows \bullet) \to \mathbf{Set} be the diagram I^2 \rightrightarrows I of the two projections, and let \mathcal I = \bigcup D = (h \downarrow D)^{\operatorname{op}} be the category of elements of D, as in this post. There is a natural functor F \colon \mathcal I \to (h \downarrow S_{\mathscr U}) taking (i,j) \in I^2 to (U_i \times_U U_j,h_{U_i \times_U U_j} \to S_{\mathscr U}) and i \in I to (U_i,h_{U_i} \to S_{\mathscr U}), taking the morphisms i \leftarrow (i,j) \to j in \mathcal I to the projections U_i \leftarrow U_i \times_U U_j \to U_j. We claim that F is cofinal, hence the colimit can be computed over \mathcal I instead (see Tag 04E7).

To verify this, we use the criteria of Tag 04E6. If (V,\alpha) \in (h \downarrow S_{\mathscr U}), then by definition the composition h_V \stackrel\alpha\to S_{\mathscr U} \hookrightarrow h_U is given by a morphism f \colon V \to U that is contained in S_{\mathscr U}(V). Since S_{\mathscr U} is generated by the U_i, this factors through some V \to U_i over S_{\mathscr U}, giving a map (V,\alpha) \to F(i).

If (V,\alpha) \to F(i) and (V,\alpha) \to F(j) are two such maps, they factor uniquely through (V,\alpha) \to F(i,j). The general result for (V,\alpha) \to F(x) and (V,\alpha) \to F(y) for x,y \in \mathcal I (either of the form i or of the form (i,j)) follows since elements of the form (i,j) always map to the elements i and j, showing that the category ((V,\alpha) \downarrow F) is weakly connected. \qedsymbol

Corollary. Let S_{\mathscr U} as above, and let \mathscr F be a presheaf on \mathscr C. Then

    \[\operatorname{Hom}(S_{\mathscr U},\mathscr F) \stackrel\sim\to \operatorname{Eq}\left( \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\Big(U_i \underset U\times U_j\Big) \right).\]

Proof. By the lemma above, we compute

    \begin{align*}\operatorname{Hom}(S_{\mathscr U},\mathscr F) &\cong \operatorname{Hom}\left(\operatorname{Coeq}\left(\coprod_{i \in I} h_{U_i \underset U \times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i}\right), \mathscr F\right) \\&\cong \operatorname{Eq}\left(\prod_{i \in I} \operatorname{Hom}(h_{U_i},\mathscr F) \rightrightarrows \operatorname{Hom}\Big(h_{U_i \underset U\times U_j},\mathscr F\Big)\right),\end{align*}

so the result follows from the Yoneda lemma. \qedsymbol

Corollary. Let \mathscr C be a (small) site. Then a presheaf \mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set} is a sheaf if and only if for every object U \in \mathscr C and every covering \{U_i \to U\}_{i \in I} in the site, the inclusion S_{\mathscr U} \to h_U induces an isomorphism

    \[\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).\]

Proof. Immediate from the previous corollary. \qedsymbol

Thus, the category of sheaves on \mathscr C can be recovered from [\mathscr C^{\operatorname{op}},\mathbf{Set}] if we know at which subobjects S \subseteq h_U we should localise (make the inclusion invertible). Next week, we will use this to give a definition of a Grothendieck topology, abstracting and generalising the notion of a site (i.e. Grothendieck pretopology).

Subterminal presheaves and sheaves (topologies 1/6)

Grothendieck pretopologies feature prominently in algebraic geometry, but the more beautiful concept of Grothendieck topologies is rarely touched upon. In a series of short posts, I aim to introduce some of these ideas, show how key concepts like the sheaf condition get very nice categorical descriptions in this language, and give examples of why topoi have much better formal properties than sites.

Let \mathscr C be a small category, and write \mathbf{PSh}(\mathscr C) for the functor category [\mathscr C^{\operatorname{op}},\mathbf{Set}].

Definition. A sieve on an object X \in \mathscr C is a subpresheaf S \subseteq h_X of the representable presheaf h_X = \operatorname{Hom}(-,X).

Concretely, this means that each S(U) is a set of morphisms f \colon U \to X with the property that if g \colon V \to U is any morphism, then fg \colon V \to X is in S(V). Thus, this is like a “right ideal in \operatorname{Hom}(-,X)“. Since \mathscr C is small, we see that sieves on X form a set, which we will denote \mathbf{Siv}(X).

Lemma. Let \mathscr D and \mathscr C be small categories, X \in \mathscr D an object, and F \colon \mathscr D \to \mathscr C a functor. Then there is a pullback map

    \[F^* \colon \mathbf{Siv}(F(X)) \to \mathbf{Siv}(X)\]

defined by

    \[F^*S(U) = \big\{f \colon U \to X\ \big|\ F(f) \in S(F(U))\big\}.\]

If \mathscr D = \mathscr C/X and F is the forgetful functor, then F^* gives a bijection

    \[\mathbf{Siv}(X) \stackrel\sim\to \mathbf{Siv}(X \stackrel{\operatorname{id}}\to X).\]

Proof. If S is a sieve, then so is F^*S since f \in F^*S(U) and g \in \operatorname{Hom}(V,U) implies F(fg) = F(f)F(g) \in S(F(V)), so fg \in F^*S(V). For the second statement, given a sieve T on X \to X, define the sieve S on X by

    \[S(U) = \left\{ f \colon U \to X\ \ \left|\ \ \left(\begin{array}{ccccc}\!\!U\!\!\!\!\! & & \!\!\!\!\!\stackrel f\longrightarrow\!\!\!\!\! & & \!\!\!\!\!X\!\! \\ & \!\!\!{\underset{f}{}}\!\!\searrow\!\!\!\! & & \!\!\!\!\swarrow\!\!{\underset{\operatorname{id}}{}}\!\!\!\! & \\[-.3em] & & X.\! & & \end{array}\right) \in T\left(U \stackrel f\to X\right)\right\}\right..\]

Then S is a sieve on X, and is the unique sieve on X such that F^*S = T. \qedsymbol

Beware that the notation F^*S could also mean the presheaf pullback S \circ F, but we won’t use it as such.

Remark. In particular, it suffices to study the case where \mathscr C has a terminal object, which we will denote by X (in analogy with the small Zariski and étale sites of a scheme X, which have X as a terminal object). We are thus interested in studying the subobjects of the terminal presheaf \mathbf{1}_X. We will do so both in the case of presheaves and in the case of sheaves. Note that \mathbf 1_X is a sheaf: for any set I (empty or not), the product \prod_{i \in I} \{*\} is a singleton, so the diagrams

    \[\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\left(U_i \underset U\times U_j\right)\]

are vacuously equalisers whenever \{U_i \to U\}_{i \in I} is a covering (or any collection of morphisms).

Definition. A property \mathcal P on a set A is a function \mathcal P \colon A \to P(\{*\}) to the power set of a point \{*\}. The property \mathcal P holds for a \in A if \mathcal P(a) = \{*\}, and fails if \mathcal P(a) = \varnothing.

Given a property \mathcal P on the objects of a small category \mathscr C, we say that \mathcal P is left closed if for any morphism f \colon U \to V, the implication \mathcal P(V) \Rightarrow \mathcal P(U) holds. (This terminology is my own. Below, we confusingly prove that these are equivalent to what we described earlier as “right ideals”. This change of orientation arises from the fact that diagrams are drawn in the opposite direction compared to composition of morphisms.)

If \mathscr C is a site (a small category together with a Grothendieck pretopology), we say that \mathcal P is local if it is left closed, and for any covering \{U_i \to U\}_{i \in I} in \mathscr C, if \mathcal P(U_i) holds for all i \in I, then \mathcal P(U) holds.

Lemma. Let \mathscr C be a small category with a terminal object X.

  1. Giving a subpresheaf of \mathbf 1_X is equivalent to giving a left closed property \mathcal P on the objects of \mathscr C.
  2. If \mathscr C is a site, then giving a subsheaf of the presheaf \mathbf 1_X is equivalent to a giving a local property \mathcal P.

A homotopy theorist might say that a local property is a (-1)-truncated sheaf [of spaces] on \mathscr C.

Proof. 1. The terminal presheaf \mathbf 1_X takes on values \{*\} at every U \in \mathscr C, thus any subpresheaf \mathscr F takes on the values \varnothing and \{*\}, hence is a property \mathcal P on the objects of \mathscr C. The presheaf condition means that for every morphism f \colon U \to V, there is a map f^* \colon \mathscr F(V) \to \mathscr F(U), which is exactly the implication \mathcal P(V) \Rightarrow \mathcal P(U) since there are no maps \{*\} \to \varnothing.

Alternatively, one notes immediately from the definition that a sieve on an object X \in \mathscr C is the same thing as a subcategory of \mathscr C/X which is left closed.

2. Being a subpresheaf translates to a left closed property \mathcal P by 1. Then \mathscr F is a sheaf if and only if, for every covering \{U_i \to U\}_{i \in I} in \mathscr C, the diagram

    \[\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i, j \in I} \mathscr F\Big(U_i \underset U\times U_j\Big)\]

is an equaliser. If one \mathscr F(U_i) is empty, then so is \mathscr F(U) since \mathcal P is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume \mathscr F(U_i) = \{*\} for all i \in I, i.e. \mathcal P(U_i) holds for all i \in I. Since \mathcal P is left closed, this implies that \mathscr F(U_i \times_U U_j) = \{*\} for all i, j \in I, so the two arrows agree on \prod_i \mathscr F(U_i), and the diagram is an equaliser if and only if \mathscr F(U) = \{*\}. Running over all coverings \{U_i \to U\} in \mathscr C, this is exactly the condition that \mathcal P is local. \qedsymbol

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.