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

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 q be a power of a prime p, and let x_0,\ldots,x_n \in \bar{\mathbf F}_q. Then x_0,\ldots,x_n satisfy a linear relation over \mathbf F_q if and only if

    \[\det \begin{pmatrix} x_0 & x_1 & \cdots & x_n \\ x_0^q & x_1^q & \cdots & x_n^q \\ \vdots & \vdots & \ddots & \vdots \\ x_0^{q^n} & x_1^{q^n} & \cdots & x_n^{q^n} \end{pmatrix} = 0.\]

Proof. If \sum_{i=0}^n c_ix_i = 0 for (c_0,\ldots,c_n) \in \mathbf F_q^n - \{0\}, then c_i^{q^j} = c_i for all i,j \in \{0,\ldots,n\} since c_i \in \mathbf F_q. As (-)^q \colon \bar{\mathbf F}_q \to \bar{\mathbf F}_q is a ring homomorphism, we find

    \[\begin{pmatrix} x_0 & x_1 & \cdots & x_n \\ x_0^q & x_1^q & \cdots & x_n^q \\ \vdots & \vdots & \ddots & \vdots \\ x_0^{q^n} & x_1^{q^n} & \cdots & x_n^{q^n} \end{pmatrix}\begin{pmatrix} c_0 \\ c_1 \\ \vdots \\ c_n \end{pmatrix} = 0,\]

so the determinant is zero. Conversely, the union of \mathbf F_q-rational hyperplanes H \subseteq \mathbf P^n_{\mathbf F_q} is a hypersurface Y of degree |\check{\mathbf P}^n(\mathbf F_q)| = q^n + \ldots + q + 1 (where \check{\mathbf P}^n denotes the dual projective space parametrising hyperplanes in \mathbf P^n). Since the determinant above is a polynomial of the same degree q^n + \ldots + q + 1 that vanishes on all \mathbf F_q-rational hyperplanes, we conclude that it is the polynomial cutting out Y, so any [x_0:\ldots:x_n] \in \mathbf P^n(\bar{\mathbf F_q}) for which the determinant vanishes lies on one of the hyperplanes. \qedsymbol

Of course when the determinant is zero, one immediately gets a vector (c_0,\ldots,c_n) \in \bar{\mathbf F}_q^{n+1} - \{0\} in the kernel. There may well be an immediate argument why this vector is proportional to an element of \mathbf F_q^{n+1}, but the above cleverly circumvents this problem.

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

  • n=0: a point x_0 \in \bar{\mathbf F}_q only satisfies a linear relation over \mathbf F_q if it is zero.
  • n=1: the polynomial x_0x_1^q-x_0^qx_1 cuts out the \mathbf F_q-rational points of \mathbf P^1.
  • n=2: the polynomial


    cuts out the union of \mathbf F_q-rational lines in \mathbf P^2. 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

    \[\left(\begin{array}{ccccc} & & h^{n,n} & & \\ & \iddots & & \ddots & \\ h^{n,0} & & & & h^{0,n} \\ & \ddots & & \iddots & \\ & & h^{0,0} & & \end{array}\right) \in \big(\mathbf Z/m\big)^{(n+1)^2}\]

satisfying h^{0,0} = 1, h^{p,q} = h^{n-p,n-q}, and h^{p,q} = h^{q,p} (characteristic 0 only) can be realised as the modulo m 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

    \[\begin{array}{ccccc} & & 1 & & \\ & 1 & & 1 & \\ 0 & & 1 & & 0 \\ & 1 & & 1 & \\ & & 1.\!\! & & \end{array}\]

Proof. If \operatorname{char} k > 0, we have \sum_{p+q = i}h^{p,q} \geq h_{\operatorname{dR}}^i \geq h_{\operatorname{cris}}^i, with equality for all i if and only if the Hodge–de Rham spectral sequence degenerates and H_{\operatorname{cris}}^i is torsion-free for all i. Because H^2_{\operatorname{cris}} contains an ample class, we must have equality on h^2, hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that h^1_{\operatorname{Weil}}(X) = 2, so \dim \mathbf{Pic}_X^0 = 1 and the same for \dim \mathbf{Alb}_X. Thus, X \to \mathbf{Alb}_X is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption h^{1,1}(X) = 1. \qedsymbol

Remark. In characteristic zero, the Hodge diamonds

    \[\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 1 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}\]

cannot occur for any a \geq 1, by essentially the same argument. Indeed, the only thing left to prove is that the image X \to \mathbf{Alb}_X cannot be a surface. If it were, then X 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

    \[\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 2 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}\]

is realised by C \times \mathbf P^1, where C is a curve of genus a. 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).


[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 X \subseteq \mathbf P^3_{\mathbf C} be a very general smooth surface of degree d \geq 4. Then the natural map \Pic(\mathbf P^3) \to \Pic(X) is an isomorphism.

If \mathscr X \to S is a smooth proper family over some base S (usually of finite type over a field), then a property \mathcal P holds for a very general X = \mathscr X_s if there exists a countable intersection U = \bigcap_i U_i \subseteq S of nonempty Zariski opens U_i such that \mathcal P holds for X_s for all s \in U.

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

If \mathscr X \to S is a family of varieties over an uncountable field k, then there always exists a very general member \mathscr X_s with s \in S(k). But over countable fields, very general elements might not exist, because it is possible that \bigcup Z_i(k) = S(k) even when \bigcup Z_i \neq S.

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

Example. Let k = \bar{\mathbf F}_p (the algebraic closure of the field of p elements, but the bar is not so visible in MathJax), let S = \mathcal A_1 = \mathcal M_{1,1} (or some scheme covering it if that makes you happier) with universal family \mathscr E \to S of elliptic curves, and let \mathscr X = \mathscr E \times_S \mathscr E be the family of product abelian surfaces E \times E. Then the locus

    \[NL(S) = \left\{s \in S\ \big| \ \operatorname{rk} \Pic(\mathscr X_s) > 3\right\}\]

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

Indeed, \Pic(E \times E) \cong \Pic(E) \times \Pic(E) \times \End(E), and every elliptic curve E over k has \operatorname{rk} \End(E) \geq 2. But the generic elliptic curve only has \End(E) = \mathbf Z. \qedsymbol

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

Local structure of finite unramified morphisms

It is well known that a finite étale morphism f \colon X \to Y of schemes is étale locally given by a disjoint union of isomorphisms, i.e. there exists an étale cover Y' \to Y such that the pullback X' \to Y' is given by X' = \coprod_{i=1}^n Y' \to Y'. Something similar is true for finite unramified morphisms:

Lemma. Let f \colon X \to Y be a finite unramified¹ morphism of schemes. Then there exists an étale cover Y' \to Y such that the pullback X' \to Y' is given by X' = \coprod_i Z_i \to Y', where Z_i \hookrightarrow Y' are closed immersions of finite presentation.

Proof. Let y \in Y be a point, let A = \mathcal O_{Y,y}^{\operatorname{sh}} be the strict henselisation of Y at y, and let \Spec B \to \Spec A be the base change of X \to Y along \Spec A \to Y. Then A \to B is unramified, so by Tag 04GL it splits as

    \[B = A_1 \times \ldots \times A_r \times C\]

whereA \to A_i is surjective for each i and no prime of C lies above \mathfrak m_y \subseteq A. But A \to C is also finite, so by Tag 00GU the map \Spec C \to \Spec A hits the maximal ideal if \Spec C \neq \varnothing. Thus, we conclude that C = 0, hence B is a product of quotients of A.

But A is the colimit of \mathcal O_{Y',y'} for (Y',y') \to (Y,y) an étale neighbourhood inducing a separable extension \kappa(y) \to \kappa(y'). Since f is of finite presentation, each of the ideals \ker(A \to A_i) and the projections B \to A_i are defined over some étale neighbourhood (Y',y') \to (Y,y). Then the pullback X' \to Y' is given by a finite disjoint union of closed immersions in Y'.

Then Y' \to Y might not be a covering, but since y \in Y was arbitrary we can do this for each point separately and take a disjoint union. \qedsymbol

Remark. The number of Z_i needed is locally bounded, but if Y is not quasi-compact it might be infinite. For example, we can take X \cong Y = \coprod_{i \in \N} \Spec k an infinite disjoint union of points, and f \colon X \to Y such that the fibre over y_i \in Y for i \in \N has i points.

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

In the unramified case, this is too much to hope for. For example, if Y = \mathbf P^2_{\mathbf C}, then we may take X to be a nontrivial finite étale cover of an elliptic curve E \subseteq Y. This is finite and unramified, but does not split over any finite étale cover of \mathbf P^2 since there aren’t any. In fact, it cannot split over any connected étale cover Y' \to \mathbf P^2 whose image contains E, since that implies the image only misses finitely many points (as E is ample), which is again impossible since \pi_1(\mathbf P^2 \setminus \{p_1,\ldots,p_r\}) = 0.

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 A and B 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 k[x] and k[x,x^{-1}] are not isomorphic, even as rings. With this example in mind, it’s tempting to try the following:

Example. Let k be a field, and let K = k(x_1, x_2, \ldots). Let

    \[A = K[x_0,x_{-1},\ldots]\]

be an infinite-dimensional polynomial ring over K, and let

    \[B = A\left[\frac{1}{x_0}\right].\]

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


isomorphic to A by shifting all the indices by 1. To see that A and B are not isomorphic as rings, note that A^\times \cup \{0\} is closed under addition, and the same is not true in B. \qed

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 k be a field. Show that k[x] and k[x,x^{-1}] 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 k-algebra k[x] represents the forgetful functor \mathbf{Alg}_k \to \mathbf{Set}, whereas k[x,x^{-1}] represents the unit group functor R \mapsto R^\times. These functors are not isomorphic, for example because the inclusion k \to k[x] induces an isomorphism on unit groups, but not on additive groups. \qed

A less fancy way to say the same thing is that all k-algebra maps k[x,x^{-1}] \to k[x] factor through k, while the same evidently does not hold for k-algebra maps k[x] \to k[x].

However, we didn’t like this because it only shows that k[x] and k[x,x^{-1}] are not isomorphic as k-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 k[x]^\times \cup \{0\} is closed under addition, whereas k[x,x^{-1}]^\times \cup \{0\} is not. \qed

This shows more generally that k[x] and \ell[x,x^{-1}] are never isomorphic as rings for any fields k and \ell.

P¹ is simply connected

This is a cute proof that I ran into of the simple connectedness of \mathbb P^1. It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let k be an algebraically closed field. Then \mathbb P^1_k is simply connected.

Proof. Let f \colon C \to \mathbb P^1 be a finite étale Galois cover with Galois group G. We have to show that f is an isomorphism. The diagonal \Delta_{\mathbb P^1} \subseteq \mathbb P^1 \times \mathbb P^1 is ample, so the same goes for the pullback D = (f \times f)^* \Delta_{\mathbb P^1} to C \times C [Hart, Exc. III.5.7(d)]. In particular, D is connected [Hart, Cor. III.7.9].

But D \cong C \times_{\mathbb P^1} C is isomorphic to |G| copies of C because the action

    \begin{align*} G \times C &\to C \times_{\mathbb P^1} C\\ (g,c) &\mapsto (gc,c) \end{align*}

is an isomorphism. If D is connected, this forces |G| = 1, so f is an isomorphism. \qed

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

Example. For a smooth projective curve C of genus g \geq 1, the diagonal cannot be ample, as \pi_1(C) \neq 0. We already knew this by computing the self-intersection \Delta_C^2 = 2-2g \leq 0, but the argument above is more elementary.


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