# 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.

# Union of hyperplanes over a finite field

The following lemma is a (presumably well-known) result that Raymond Cheng and I happened upon while writing our paper Unbounded negativity on rational surfaces in positive characteristic (arXiv, DOI). Well, Raymond probably knew what he was doing, but to me it was a pleasant surprise.

Lemma. Let be a power of a prime , and let . Then satisfy a linear relation over if and only if

Proof. If for , then for all since . As is a ring homomorphism, we find

so the determinant is zero. Conversely, the union of -rational hyperplanes is a hypersurface of degree (where denotes the dual projective space parametrising hyperplanes in ). Since the determinant above is a polynomial of the same degree that vanishes on all -rational hyperplanes, we conclude that it is the polynomial cutting out , so any for which the determinant vanishes lies on one of the hyperplanes.

Of course when the determinant is zero, one immediately gets a vector in the kernel. There may well be an immediate argument why this vector is proportional to an element of , but the above cleverly circumvents this problem.

For concreteness, we can work out what this determinant is in small cases:

• : a point only satisfies a linear relation over if it is zero.
• : the polynomial cuts out the -rational points of .
• : the polynomial

cuts out the union of -rational lines in . This is the case considered in the paper.

# Hodge diamonds that cannot be realised

In Paulsen–Schreieder [PS19] and vDdB–Paulsen [DBP20], the authors/we show that any block of numbers

satisfying , , and (characteristic only) can be realised as the modulo reduction of a Hodge diamond of a smooth projective variety.

While preparing for a talk on [DBP20], I came up with the following easy example of a Hodge diamond that cannot be realised integrally, while not obviously violating any of the conditions (symmetry, nonnegativity, hard Lefschetz, …).

Lemma. There is no smooth projective variety (in any characteristic) whose Hodge diamond is

Proof. If , we have , with equality for all if and only if the Hodge–de Rham spectral sequence degenerates and is torsion-free for all . Because contains an ample class, we must have equality on , hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that , so and the same for . Thus, is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption .

Remark. In characteristic zero, the Hodge diamonds

cannot occur for any , by essentially the same argument. Indeed, the only thing left to prove is that the image cannot be a surface. If it were, then would have a global 2-form; see e.g. [Beau96, Lemma V.18].

This argument does not work in positive characteristic due to the possibility of an inseparable Albanese map. It seems to follow from Bombieri–Mumford’s classification of surfaces in positive characteristic that the above Hodge diamond does not occur in positive characteristic either, but the analysis is a little intricate.

Remark. On the other hand, the nearly identical Hodge diamond

is realised by , where is a curve of genus . This is some evidence that the full inverse Hodge problem is very difficult, and I do not expect a full classification of which Hodge diamonds are possible (even for surfaces this might be out of reach).

References.

[Beau96] A. Beauville, Complex algebraic surfaces. London Mathematical Society Student Texts 34 (1996).

[DBP20] R. van Dobben de Bruyn and M. Paulsen, The construction problem for Hodge numbers modulo an integer in positive characteristic. Forum Math. Sigma (to appear).

[PS19] M. Paulsen and S. Schreieder, The construction problem for Hodge numbers modulo an integer. Algebra Number Theory 13.10, p. 2427–2434 (2019).

# An interesting Noether–Lefschetz phenomenon

The classical Noether–Lefschetz theorem is the following:

Theorem. Let be a very general smooth surface of degree . Then the natural map is an isomorphism.

If is a smooth proper family over some base (usually of finite type over a field), then a property holds for a very general if there exists a countable intersection of nonempty Zariski opens such that holds for for all .

In general, Hilbert scheme arguments show that the locus where the Picard rank is ‘bigger than expected’ is a countable union of closed subvarieties of (the Noether–Lefschetz loci), but it could be the case that this actually happens everywhere (i.e. ). The hard part of the Noether–Lefschetz theorem is that the jumping loci are strict subvarieties of the full space of degree hypersurfaces.

If is a family of varieties over an uncountable field , then there always exists a very general member with . But over countable fields, very general elements might not exist, because it is possible that even when .

The following interesting phenomenon was brought to my attention by Daniel Bragg (if I recall correctly):

Example. Let (the algebraic closure of the field of elements, but the bar is not so visible in MathJax), let (or some scheme covering it if that makes you happier) with universal family of elliptic curves, and let be the family of product abelian surfaces . Then the locus

is exactly the set of -points (so it misses only the generic point).

Indeed, , and every elliptic curve over has . But the generic elliptic curve only has .

We see that the Noether–Lefschetz loci might cover all -points without covering , even in very natural situations.

# Local structure of finite unramified morphisms

It is well known that a finite étale morphism of schemes is étale locally given by a disjoint union of isomorphisms, i.e. there exists an étale cover such that the pullback is given by . Something similar is true for finite unramified morphisms:

Lemma. Let be a finite unramified¹ morphism of schemes. Then there exists an étale cover such that the pullback is given by , where are closed immersions of finite presentation.

Proof. Let be a point, let be the strict henselisation of at , and let be the base change of along . Then is unramified, so by Tag 04GL it splits as

where is surjective for each and no prime of lies above . But is also finite, so by Tag 00GU the map hits the maximal ideal if . Thus, we conclude that , hence is a product of quotients of .

But is the colimit of for an étale neighbourhood inducing a separable extension . Since is of finite presentation, each of the ideals and the projections are defined over some étale neighbourhood . Then the pullback is given by a finite disjoint union of closed immersions in .

Then might not be a covering, but since was arbitrary we can do this for each point separately and take a disjoint union.

Remark. The number of needed is locally bounded, but if is not quasi-compact it might be infinite. For example, we can take an infinite disjoint union of points, and such that the fibre over for has points.

Remark. In the étale case, we may actually take finite étale, by taking to be the Galois closure of , which exists in reasonable cases². For example, if is normal, we may take to be the integral closure of in the field extension corresponding to the Galois closure of . In general, if is connected it follows from Tag 0BN2 that a suitable component of the -fold fibre product of over is a Galois closure of . If the connected components of are open, apply this construction to each component.

In the unramified case, this is too much to hope for. For example, if , then we may take to be a nontrivial finite étale cover of an elliptic curve . This is finite and unramified, but does not split over any finite étale cover of since there aren’t any. In fact, it cannot split over any connected étale cover whose image contains , since that implies the image only misses finitely many points (as is ample), which is again impossible since .

¹For the purposes of this post, unramified means in the sense of Grothendieck, i.e. including the finite presentation hypothesis. In Raynaud’s work on henselisations, this was weakened to finite type. See Tag 00US for definitions.

²I’m not sure what happens in general.

# Rings that are localisations of each other

This is a post about an answer I gave on MathOverflow in 2016. Most people who have ever clicked on my profile will probably have seen it.

Question. If and are rings that are localisations of each other, are they necessarily isomorphic?

In other words, does the category of rings whose morphisms are localisations form a partial order?

In my previous post, I explained why and are not isomorphic, even as rings. With this example in mind, it’s tempting to try the following:

Example. Let be a field, and let . Let

be an infinite-dimensional polynomial ring over , and let

Then is a localisation of , and we can localise further to obtain the ring

isomorphic to by shifting all the indices by 1. To see that and are not isomorphic as rings, note that is closed under addition, and the same is not true in .

Is there a moral to this story? Not sure. Maybe the lesson is to do mathematics your own stupid way, because the weird arguments you come up with yourself may help you solve other problems in the future. The process is more important than the outcome.

# Is the affine line isomorphic to the punctured affine line?

This is the story of Johan Commelin and myself working through the first sections of Hartshorne almost 10 years ago (nothing creates a bond like reading Hartshorne together…). This post is about problem I.1.1(b), which is essentially the following:

Exercise. Let be a field. Show that and are not isomorphic.

In my next post, I will explain why I’m coming back to exactly this problem. There are many ways to solve it, for example:

Solution 1. The -algebra represents the forgetful functor , whereas represents the unit group functor . These functors are not isomorphic, for example because the inclusion induces an isomorphism on unit groups, but not on additive groups.

A less fancy way to say the same thing is that all -algebra maps factor through , while the same evidently does not hold for -algebra maps .

However, we didn’t like this because it only shows that and are not isomorphic as -algebras (rather than as rings). Literal as we were (because we’re undergraduates? Lenstra’s influence?), we thought that this does not answer the question. After finishing all unstarred problems from section I.1 and a few days of being unhappy about this particular problem, we finally came up with:

Solution 2. The set is closed under addition, whereas is not.

This shows more generally that and are never isomorphic as rings for any fields and .

# P¹ is simply connected

This is a cute proof that I ran into of the simple connectedness of . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let be an algebraically closed field. Then is simply connected.

Proof. Let be a finite étale Galois cover with Galois group . We have to show that is an isomorphism. The diagonal is ample, so the same goes for the pullback to [Hart, Exc. III.5.7(d)]. In particular, is connected [Hart, Cor. III.7.9].

But is isomorphic to copies of because the action

is an isomorphism. If is connected, this forces , so is an isomorphism.

The proof actually shows that if is a smooth projective variety such that is a set-theoretic complete intersection of ample divisors, then is simply connected.

Example. For a smooth projective curve of genus , the diagonal cannot be ample, as . We already knew this by computing the self-intersection , but the argument above is more elementary.

References.

[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.