# 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 be a (small) site. For a set of morphisms with the same target, write for the presheaf image of . Then a presheaf is a sheaf if and only if for every covering in , the inclusion induces an isomorphism

Thus, if is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category 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 in gives a pullback taking to its inverse image under (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to defined uniquely up to unique isomorphism). Thus, is itself a presheaf (it takes values in since is small).

Also note the following method for producing sieves: if is a presheaf, a subpresheaf, and a section over some , we get a sieve by

By the Yoneda lemma, this is just the inverse image of along the morphism classifying . Note that is the maximal sieve if and only if .

Definition. Let be a small category. Then a Grothendieck topology on consists of a subpresheaf such that

1. For all , the maximal sieve is in .
2. If and with , then .
3. If is a sieve such that , then (equivalently, then is the maximal sieve ).

The sieves are called covering sieves. Since is a presheaf, we see that for any and any covering sieve , the pullback 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, Exp. II, Def. 1.1:

• If with , such that for every morphism the inverse image of along is in , then .

Indeed, applying this criterion when immedately shows if , since the inverse image of along is the maximal sieve . Thus the local character implies criterion 2. The local character says that if contains a covering sieve , then is covering. Assuming criterion 2, the sieve contains a covering sieve if and only if is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that is closed under binary intersection, i.e. if then . Indeed, if for some , then

so . Axioms 2 and 3 give .

Example. Let be a pretopology on the (small) category ; see Tag 00VH for a list of axioms. For each , define the subset as those that contain a sieve of the form for some covering in . (See the corollary at the top for the definition of .) Concretely, this means that there exists a covering such that for all , i.e. is covered by morphisms that are in the given sieve .

Lemma. The association is a topology. It is the coarsest topology on for which each for is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If , then there exists with . If is any morphism in , then by criterion 3 of Tag 00VH. But , because a morphism factors through if and only if factors through . Thus, , so , and is a subpresheaf of .

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose satisfies . Then there exists with . This means that for all , i.e. for all . Thus, for each there exists in such that , i.e. for all and all . Thus, if denotes , then we get . But is a covering by criterion 2 of Tag 00VH, so .

If is any other Grothendieck topology for which each for is covering, then contains by criterion 2.

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 be a small category, and let be a Grothendieck topology. Then a presheaf is a sheaf if for any and any , the map induces an isomorphism

Thus, a Grothendieck topology is an internal characterisation (inside ) of which morphisms one needs to localise to get . 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) on a small category (resp. small site) with terminal object correspond to left closed (resp. local) properties on . 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 is a small category with fibre products.

Definition. Given a set of morphisms with the same target , define the sieve generated by as the sieve on of those morphisms that factor through some .

It is in a sense the right ideal in generated by the . What does this look like as a subobject of ?

Example. If has one element, i.e. , then is the image of the morphism of representable presheaves . In the case where is already a monomorphism (this is always the case when is a poset, such as for some topological space ), then is itself injective (this is the definition of a monomorphism!), so is just .

In general, is the image of the map

induced by the maps . Indeed, an element of is a morphism , and it comes from some if and only if factors through .

This shows that, in fact, every sieve is of this form for some set : take as index set (the objects of) the slice category , which as in the previous post gives a surjection . This corresponds to generating an ideal by all its elements.

But we can also characterise without using the word ‘image’ (which somehow computes its first syzygy):

Lemma. Let be a set of morphisms with common target, and the sieve generated by . Then is the coequaliser of the diagram

where the maps are induced by the two projections .

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

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

since is a monomorphism. But is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in , epimorphisms are regular).

Proof 2. By the previous post, the presheaf is the colimit over of (see post for precise statement). Let be the diagram of the two projections, and let be the category of elements of , as in this post. There is a natural functor taking to and to , taking the morphisms in to the projections . We claim that is cofinal, hence the colimit can be computed over instead (see Tag 04E7).

To verify this, we use the criteria of Tag 04E6. If , then by definition the composition is given by a morphism that is contained in . Since is generated by the , this factors through some over , giving a map .

If and are two such maps, they factor uniquely through . The general result for and for (either of the form or of the form ) follows since elements of the form always map to the elements and , showing that the category is weakly connected.

Corollary. Let as above, and let be a presheaf on . Then

Proof. By the lemma above, we compute

so the result follows from the Yoneda lemma.

Corollary. Let be a (small) site. Then a presheaf is a sheaf if and only if for every object and every covering in the site, the inclusion induces an isomorphism

Proof. Immediate from the previous corollary.

Thus, the category of sheaves on can be recovered from if we know at which subobjects 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 be a small category, and write for the functor category .

Definition. A sieve on an object is a subpresheaf of the representable presheaf .

Concretely, this means that each is a set of morphisms with the property that if is any morphism, then is in . Thus, this is like a “right ideal in “. Since is small, we see that sieves on form a set, which we will denote .

Lemma. Let and be small categories, an object, and a functor. Then there is a pullback map

defined by

If and is the forgetful functor, then gives a bijection

Proof. If is a sieve, then so is since and implies , so . For the second statement, given a sieve on , define the sieve on by

Then is a sieve on , and is the unique sieve on such that .

Beware that the notation could also mean the presheaf pullback , but we won’t use it as such.

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

are vacuously equalisers whenever is a covering (or any collection of morphisms).

Definition. A property on a set is a function to the power set of a point . The property holds for if , and fails if .

Given a property on the objects of a small category , we say that is left closed if for any morphism , the implication 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 is a site (a small category together with a Grothendieck pretopology), we say that is local if it is left closed, and for any covering in , if holds for all , then holds.

Lemma. Let be a small category with a terminal object .

1. Giving a subpresheaf of is equivalent to giving a left closed property on the objects of .
2. If is a site, then giving a subsheaf of the presheaf is equivalent to a giving a local property .

A homotopy theorist might say that a local property is a -truncated sheaf [of spaces] on .

Proof. 1. The terminal presheaf takes on values at every , thus any subpresheaf takes on the values and , hence is a property on the objects of . The presheaf condition means that for every morphism , there is a map , which is exactly the implication since there are no maps .

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

2. Being a subpresheaf translates to a left closed property by 1. Then is a sheaf if and only if, for every covering in , the diagram

is an equaliser. If one is empty, then so is since is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume for all , i.e. holds for all . Since is left closed, this implies that for all , so the two arrows agree on , and the diagram is an equaliser if and only if . Running over all coverings in , this is exactly the condition that is local.

# 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 is a finite morphism of schemes, is the pushforward 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 be the spectrum of a DVR , let be a finite extension of domains such that has exactly two primes above , and let . For example, and , or and if you prefer a more geometric example.

By my previous post, the global sections functor is exact. If the same were true for , then the global sections functor on would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection of sheaves on such that the map on global sections is not surjective.

The topological space of consists of closed points and a generic point . Let and ; then is open and is closed. Hence, for any sheaf on , we have a short exact sequence (see e.g. Tag 02UT)

where and are the inclusions. Let be the constant sheaf ; then the same goes for and . Then the map

is given by the diagonal map , since is connected by has two connected components. This is visibly not surjective.

# Odd degree Betti numbers are even

In characteristic 0, it follows from the Hodge decomposition and Hodge symmetry that the Betti numbers of a smooth proper complex variety are even when is odd. In characteristic 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, will be a -adic field with ring of integers , residue field of size , and (normalised) valuation such that (this is the -valuation on ).

Throughout, will be a smooth proper variety over . We will write for the Betti numbers of . It can be computed either as the dimension of , or that of .

Remark. Recall that if is the characteristic polynomial of Frobenius acting on for , and is the reciprocal of a root of , then for every complex embedding we have

(1)

The same holds for the eigenvalues of Frobenius on crystalline cohomology (in fact, the characteristic polynomials agree). All reciprocal roots are algebraic integers, and .

Defintion. An algebraic integer is a -Weil integer if it satisfies (1) (for every embedding ).

Lemma. Let be a polynomial, and let be the multiset of reciprocal roots of . Assume all are -Weil integers. Then (counted with multiplicity).

Proof. If , then is the complex conjugate with respect to every embedding . Thus, it is conjugate to , hence a root of as well (with the same multiplicity). Taking valuations gives the result.

Theorem. Let be smooth proper over , and let be odd. Then is even.

Proof. The Frobenius-eigenvalues whose valuation is not come naturally in pairs . Now consider valuation . Note that the -valuation of the semilinear Frobenius equals the -valuation of the -linear Frobenius (which is the one used in computing the characteristic polynomial ). The sum of the -valuations of the roots should be an integer, because has rational coefficients. Thus, there needs to be an even number of valuation eigenvalues, for otherwise their product would not be a rational number.

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 of the Larsen–Lunts theorem. See my previous post for the statement and sketch of the proof of Larsen–Lunts.

Remark. In characteristic , 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 . Let and be smooth proper varieties, and assume and are stably birational. Then .

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

given by counting -points modulo factors through since . Hence, by Larsen–Lunts, it factors through .

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 be a variety of dimension over . Let be an eigenvalue of Frobenius on . Then and are both algebraic integers.

The first part (integrality of ) is fairly well-known. For the second part (integrality of ), see SGA 7, Exp. XXI, Corollary 5.5.3(iii).

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

Theorem. (Ekedahl’s version) Let . Let and be smooth connected varieties (not necessarily proper!), and assume and are birational. If is an eigenvalue of Frobenius on which is not an eigenvalue on , then is divisible by .

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

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

If is an eigenvalue on some , then is an algebraic integer (see above). Hence, for any valuation on with , we have . We conclude that the eigenvalues for which some valuation is on and agree. Hence, by Poincaré duality, the eigenvalues of and for which some valuation is agree. These are exactly the ones that are not divisible by .

The theorem I stated above immediately follows from this one:

Proof. Since , we may replace by . Thus, we can assume and are birational; both of dimension .

By the Weil conjectures, we know that

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

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.