A presheaf that has no associated sheaf

In his paper Basically bounded functors and flat sheaves (Pacific Math. J, vol. 57, no. 2, 1975, p. 597-610), William C. Waterhouse gives a nice example of a presheaf that has no associated sheaf. This is Theorem 5.5 (page 605).  I thank François Loeser for having indicated this paper to me, and for his suggestion of explaining it here!

Of course, such a beast is reputed not to exist, since it is well known that any presheaf has an associated sheaf, see for example Godement's book Topologie algébrique et théorie des faisceaux, pages 110-111.

That is, for any presheaf $F$ on a topological space, there is a sheaf $G$ with a morphism of presheaves $\alpha\colon F\to G$ which satisfies a universal property: any morphism from $F$ to a sheaf factors uniquely through $\alpha$.

Waterhouse's presheaf is a more sophisticated example of a presheaf, since it is a presheaf on the category of affine schemes for the flat topology. Thus, a presheaf $F$ on the category of affine schemes is the datum, 

• of a set $F(A)$ for every ring $A$, 
• and of a map $\phi_*\colon F(A)\to F(B)$ for every morphism of rings $\phi\colon A\to B$,

subject to the following conditions:

• if $\phi\colon A\to B$ and $\psi\colon B\to C$ are morphism of rings, then $(\psi\circ\phi)_*=\psi_*\circ\phi_*$;
• one has ${\rm id}_A)_*={\rm id}_{F(A)}$ for every ring $A$.

Any morphism of rings $\phi\colon A\to B$ gives rise to two morphisms $\psi_1,\psi_2\colon B\to B\otimes B$ respectively defined by $\psi_1(b)=b\otimes 1$ and $\psi_2(b)=1\otimes b$, and the two compositions $A\to B\to B\otimes_A B$ are equal. Consequently, for any presheaf $F$, the two associated maps $F(A) \to F(B) \to F(B\otimes_A B)$ are equal.

By definition, a presheaf $F$ is a sheaf for the flat topology if for any faithfully flat morphism of rings, the map ${\phi_*} \colon F(A)\to F(B)$ is injective and its image is the set of elements $g\in F(B)$ at which the two natural maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ coincide.

Here is Waterhouse's example.

For every ring $A$, let $F(A)$ be the set of all locally constant functions $f$ from $\mathop{\rm Spec}(A)$ to some von Neumann cardinal such that $f(\mathfrak p)<\mathop{\rm Card}(\kappa(\mathfrak p))$ for every $\mathfrak p\in\mathop{\rm Spec}(A)$.

This is a presheaf. Indeed, let $\phi\colon A\to B$ is a ring morphism, let $\phi^a\colon\mathop{\rm Spec}(B)\to \mathop{\rm Spec}(A)$ be the associated continuous map on spectra. For $f\in F(A)$, then $f\circ\phi^a$ is a locally constant map from ${\rm Spec}(B)$ to some von Neumann cardinal. Moreover, for every prime ideal $\mathfrak q$ in $B$, with inverse image $\mathfrak p=\phi^{-1}(\mathfrak q)=\phi^a(\mathfrak q)$, the morphism $\phi$ induces an injection from the residue field $\kappa(\mathfrak q)$ into $\kappa(\mathfrak p)$, so that $f\circ\phi^a$ satisfies the additional condition on $F$, hence $f\circ\phi^a\in F(B)$.

However, this presheaf has no associated sheaf for the flat topology. The proof is by contradiction. So assume that $G$ is a sheaf and $\alpha\colon F\to G$ satisfies the universal property.

First of all, we prove that the morphism $\alpha$ is injective: for any ring $A$, the map $\alpha_A\colon F(A)\to G(A)$ is injective. For any cardinal $c$ and any ring $A$, let $L_c(A)$ be the set of locally constant maps  from ${\rm Spec}(A)$ to $c$. Then $L_c$ is a presheaf, and in fact a sheaf. There is a natural morphism of presheaves $\beta_c\colon F\to L_c$, given by $\beta_c(f)(\mathfrak p)=f(\mathfrak p)$ if $f(\mathfrak p)\in c$, that is, $f(\mathfrak p)<c$, and $\beta_c(f)(\mathfrak p)=0$ otherwise. Consequently, there is a unique morphism of sheaves $\gamma_c\colon G\to L_c$ such that $\beta_c=\gamma_c\circ\alpha$. For any ring $A$, and any large enough cardinal $c$, the  map $\beta_c(A)\colon F(A)\to L_c(A)$ is injective. In particular, the map $\alpha(A)$ must be injective.

Let $B$ be a ring and $\phi\colon A\to B$ be a faithfully flat morphism. Let $\psi_1,\psi_2\colon B\to B\otimes_A B$ be the two natural morphisms of rings defined above. Then, the equalizer $E(A,B)$ of the two maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ must inject into the equalizer of the two corresponding maps from $G(B)$ to $G(B\otimes_A B)$. Consequently, one has an injection from $E(A,B)$ to $G(A)$.

The contradiction will become apparent once one can find rings $B$ for which $E(A,B)$ has a cardinality as large as desired. If ${\rm Spec}(B)$ is a point $\mathfrak p$, then $F(B)$ is just the set of functions $f$ from the point $\mathfrak p$ to some von Neumann cardinal $c$ such that $f(\mathfrak p)<{\rm Card}(\kappa(\mathfrak p))$. That is, $F(B)$ is the cardinal ${\rm Card}(\kappa(\mathfrak p))$ itself. And since ${\rm Spec}(B)$ is a point, the coincidence condition is necessarily satisfied, so that $E(A,B)= {\rm Card}(\kappa(\mathfrak p))\leq G(A)$.

To conclude, it suffices to take a faithfully flat morphism $A\to B$  such that $B$ is field of cardinality strictly greater than $G(A)$. For example, one can take $A$ to be a field and $B$ the field of rational functions in many indeterminates (strictly more than the cardinality of $G(A)$).

What does this example show? Why isn't there a contradiction in mathematics (yet)?

Because the definition of sheaves and presheaves for the flat topology that I gave above was definitely defective: it neglects in a too dramatic way the set theoretical issues that one must tackle to define sheaves on categories. In the standard setting of set theory provided by ZFC, everything is a set. In particular, categories, presheaves, etc. are sets or maps between sets (themselves represented by sets).  But the presheaf $F$ that Waterhouse defines does not exist as a set, since there does not exist a set $\mathbf{Ring}$ of all rings, nor a set $\mathbf{card}$ of all von Neumann cardinals.

The usual way (as explained in SGA 4) to introduce sheaves for the flat topology consists in adding the axiom of universes — there exists a set $\mathscr U$ which is a model of set theory. Then, one does not consider the (inexistent) set of all rings, or cardinals, but only those belonging to the universe $\mathscr U$—one talks of $\mathscr U$-categories, $\mathscr U$-(pre)sheaves, etc.. In that framework, the $\mathscr U$-presheaf $F$ defined by Waterhouse (where one restricts oneself to algebras and von Neumann cardinals in $\mathscr U$) has an associated sheaf $G_{\mathscr U}$. But this sheaf depends on the chosen universe: if $\mathscr V$ is an universe containing $\mathscr U$, the restriction of $G_{\mathscr V}$ to algebras in $\mathscr U$ will no longer be a $\mathscr U$-presheaf.

S. Bloch on Milne's Étale cohomology

A long time ago, Spencer Bloch wrote a review of James Milne's Étale cohomology for the Bulletin of AMS which can freely be obtained from the AMS web site. This review explains why one should study étale cohomology, how it grew up, and discusses some aspects of Milne's book. It features an unusual combination of humor and seriousness. Reading highly recommended once you have 15 minutes free!

The Poisson summation formula, Minkowski's first theorem, and the density of sphere packings

I opened by accident a paper by Henry Cohn and Noam Elkies, New upper bounds on sphere packings I, Annals of Mathematics, 157 (2003), 689–714, and I had the good surprise to see a beautiful application of the Poisson summation formula to upper bounds for the density of sphere packings.

In fact, their argument is very close to a proof of the first theorem of Minkowski about lattice points in convex bodies which I had discovered in 2009. However, a final remark in their paper, appendix C, shows that this proof is not really new. Anyway, the whole story is nice enough to prompt me to discuss it in this blog.

1. The Poisson formula

I first recall the Poisson formula: let $f\colon\mathbf R^n\to \mathbf R$ be a continuous function whose Fourier transform $\hat f$ is integrable, let $L$ be a lattice in $\mathbf R^n$, and let $L'$ be the dual lattice of $L$. Then, 

\[ \sum_{x\in L} f(x) = \frac1{\mu(\mathbf R^n/L)}\sum_{y\in L'} \hat f(y). \]

In fact, one needs a bit more about $f$ that the above hypotheses (but we'll ignore this in the sequel). For example, it is sufficient that $f(x)$ and $\hat f(x)$ be bounded from above by a multiple of $1/(1+\| x\|)^{n+\epsilon}$, for some strictly positive real number $\epsilon$.

2. Minkowski's first theorem

Let $B$ a closed symmetric convex neighborhood of $0$. Minkowski's first theorem bounds from below the cardinality of $B\cap L$. A natural idea would be to apply the Poisson summation formula to the indicator function $f_B$ of $B$. However, $f_B$ is not continuous, so we need to replace $f_B$ by some function $f$ which satisfies the following properties:

• $f\leq f_B$, so that $\#(B\cap L)= \sum_{x\in L} f_B(x)\geq \sum_{x\in L} f(x) $;
• $\hat f\geq 0$, so that $\sum_{y\in L'} \hat f(y) \geq \hat f(0)$.

The second condition suggest to take for $f$ a function of the form $g*\check g$, so that $\hat f=|\hat g| ^2$. Then, the first condition will hold outside of $B$ if $g$ is supported in $\frac12 B$. Let's try $g=cf_{B/2}$ for some positive real number $c$. For all $x\in\mathbf R^n$,  

\[ f(x) \leq f(0) =c^2 \int_{\mathbf R^n} f_{B/2}(x) f_{B/2}(-x) = c^2 \mu(B/2) = c^2 \mu(B)/2^n. \]

It suffices to choose $c=(2^n/\mu(B))^{1/2}$.

On the other hand, 

\[ \hat g(0) = \int_{\mathbf R^n} f_{B/2}(x) = \mu(B/2)=c\mu(B)/2^n=1/c. \]

Finally, the Poisson formula implies

$\displaystyle \#(B\cap L) \geq \sum_{x\in L} f(x)  = \frac1{\mu(\mathbf R^n/L)} \sum_{y\in L'}\hat f(y)  \geq \frac1{\mu(\mathbf R^n/L)} \hat f(0) $

$\displaystyle = \frac1{\mu(\mathbf R^n/L)} | \hat g(0)|^2  = \frac{2^n}{\mu(B) \mu(\mathbf R^n/L)}.$

This is exactly Minkowski's first theorem!

Of course, one may consider apply this argument to other functions $f$. In the case where $B$ is an Euclidean ball, a natural choice consists in taking a Gaussian $f(x)=a\exp(-b\|x\|^2)$, since then $\hat f$ has the same form. This is what has essentially been done by Schoof and van der Geer in their paper Effectivity of Arakelov divisors and the analogue of the theta divisor of a number field, Selecta Math. New Ser. 6 (2000), 377–398, and Damian Rössler independently. Indeed, up to normalization factors, the left hand side of the Poisson summation formula is then interpreted as the exponential of the $h^0$ of a line bundle over an arithmetic curve, and the Poisson summation formula itself is the analogue of Serre's duality theorem in Arakelov geometry. As Jean-Benoît Bost explained to me, Gaussian functions provide an inequality for $\#(B\cap L)$ which is sharper than Minkowski's first theorem. The details of the computation can be found in my notes about Arakelov geometry (beware, these are mostly a work in slow progress). I have not tried to look for optimal functions beyond that case.

3. The theorem of Cohn-Elkies

We consider a sphere packing, that is a set of points in $\mathbf R^n$ with mutual distances at least 1,

and we want to bound from above its density, that is the ratio of volume occupied by balls of radius $1/2$ centered at these spheres. One may think of a lattice packing, the particular case where the centers of these spheres are exactly the points of a lattice $L$ or, more generally, of periodic packing when the centers of the spheres are finitely many translates $v_1+L,\dots,v_N+L$ of  a lattice $L$. In fact, Cohn and Elkies argue that it suffices to study such periodic packings (repeating periodically an arbitrarily large part of the sphere packing), so we shall do like them and assume that our sphere packing is periodic. 

Now, let $f$ be a real valued function on $\mathbf R^n$ satisfying the following properties:

• $f$ is continuous, integrable on $\mathbf R^n$ as well as its Fourier transform.
• $f(0)>0$ and $f(x)\leq 0$ for $\| x\|\geq 1$;
• $\hat f(x)\geq 0$ for all $x$.

Then, the density $\Delta$ of the sphere packing satisfies 

\[ \Delta \leq 2^{-n} \mu(B) \frac{f(0)}{\hat f(0)}, \]

where $\mu(B)$ is the volume of the unit ball.

We assume that no difference $v_i-v_j$ is a point of the lattice $L$ (otherwise, we can exclude one translate from the list). With the above notation, the fundamental parallelepiped of the lattice contains exactly $N$ balls of radius $1/2$, hence 

\[ \Delta= 2^{-n}\mu(B) \frac N{\mu(\mathbf R^n/L)}. \]

For any $v\in\mathbf R^n$, the Poisson summation formula for $x\mapsto f(x+v)$ writes

\[ \sum_{x\in L} f(x+v) = \frac1{\mu(L)} \sum_{y\in L'} e^{2\pi i \langle v,y\rangle} \hat f(y), \]

hence

$\displaystyle \sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k)  = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y)  \sum_{j,k=1}^N e^{2\pi i \langle v_j-v_k,y\rangle}$ 

$\displaystyle = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y) \left| \sum_{j=1}^N e^{2\pi i \langle v_j,y\rangle} \right|^2.$

All terms of the right hand side are positive or null, so that we can bound it from below by

the term for $y=0$, hence

\[ \sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k) \geq N^2 \frac{\hat f(0)}{\mu(\mathbf R^n/L)}. \]

Now, there is a sphere of our packing centered at $x+v_j$, and another at $v_k$,

so that $\| x+v_j-v_k\|\geq 1$ unless $x+v_j=v_k$, that is $v_k-v_j=x$ hence $x=0$ and $v_j=v_k$. In the first case, the value of $f$ at $x+v_j-v_k$ is negative or null; in the latter, it equal $f(0)$. Consequently, the left hand side of the previous inequality is at most $Nf(0)$.  Finally, $\displaystyle N f(0) \geq N^2 \hat f(0)/\mu(\mathbf R^n/L)$, hence the desired inequality.

The Poisson summation formula, arithmetic and geometry

François Loeser and I just uploaded a paper on arXiv about Motivic height zeta functions. That such a thing could be possible is quite funny, so I'll take this opportunity to break a long silence on this blog.

In Diophantine geometry, an established and important game consists in saying as much as possible of the solutions of diophantine equations. In algebraic terms, this means proving qualitative or quantitative properties of the set of integer solutions of polynomial equations with integral coefficients. In fact, one can only understand something by making the geometry more apparent; then, one is interested in integral points of schemes $X$ of finite type over the ring $\mathbf Z$ of integers. There are in fact two sub-games: one in which one tries to prove that such solutions are scarce, for example when $X$ is smooth and of general type (conjecture of Mordell=Faltings's theorem, conjecture of Lang) ; the other in which one tries to prove that there are many solutions —then, one can even try to count how many solutions there are of given height, a measure of their size. There is a conjecture of Manin predicting what would happen, and our work belongs to this field of thought.

Many methods exist to understand rational points or integral points of varieties. When the scheme carries an action of an algebraic group, it is tempting to try to use harmonic analysis. In fact, this has been done since the beginnings of Manin's conjecture when Franke, Manin, Tschinkel showed that when the variety is a generalized flag variety ($G/P$, where $G$ is semi-simple, $P$ a parabolic subgroup, for example projective spaces, grassmannians, quadrics,...), the solution of Manin's question was already given by Langlands's theory of Eisenstein series. Later, Batyrev and Tschinkel proved the case of toric varieties, and again with Tschinkel, I studied the analogue of toric varieties when the group is not a torus but a vector space. In these two cases, the main idea consists in introducing a generating series of our counting problem, the height zeta function, and establishing its analytic properties. In fact, this zeta function is a sum over rational points of a height function defined on the adelic space of the group, and the Poisson summation formula rewrites this sum as the integral of the Fourier transform of the height function over the group of topological characters. What makes the analysis possible is the fact that, essentially, the trivial character carries all the relevant information; it is nevertheless quite technical to establish what happens for other characters, and then to check that the behavior of the whole integral is indeed governed by the trivial character.

In mathematics, analogy often leads to interesting results. The analogy between number fields and function fields suggests that diophantine equations over the integers have a geometric analogue, which consists in studying morphisms from a curve to a given variety. If the ground field of the function field is finite, the dictionary goes quite far; for example, Manin's question has been studied a lot by Bourqui who established the case of toric varieties. But when the ground field is infinite, it is no more possible to count solutions of given height since they will generally be infinite.

However, as remarked by Peyre around 2000, all these solutions, which are morphisms from a curve to a scheme, form themselves a scheme of finite type. So the question is to understand the behavior of these schemes, when the height parameter grows to infinity. In fact, in an influential but unpublished paper, Kapranov had already established the case of flag varieties (without noticing)! The height zeta function is now a formal power series whose coefficients are algebraic varieties; one viewes them as elements of the Grothendieck ring of varieties, the universal ring generated by varieties with addition given by cutting-and-pasting, and multiplication given by the product of varieties. This ring is a standard tool of motivic integration (as invented by Kontsevich and developed by Denef and Loeser, and many people since). That's why this height zeta function is called motivic.

What we proved with François is a rationality theorem for such a motivic height function, when the variety is an equivariant compactification of a vector group. This means that all this spaces of morphisms, indexed by some integer, satisfy a linear dependence relation in the Grothendieck ring of varieties! To prove this result, we rely crucially on an analogue of the Poisson summation formula in motivic integration, due to Hrushovski and Kazhdan, which allows us to perform a similar analysis to the one I had done with Tschinkel in a paper that appeared last year in Duke Math. J.

Many things remain puzzling. The most disturbing is the following. If you read Tate's thesis, or Weil's Basic Number Theory, you'll see that the Riemann Roch formula and Serre's duality theorem for curves over finite fields are consequences of the Poisson summation formula in harmonic analysis. In motivic integration, things go the other way round: if one unwinds all the definitions (we explain this in our paper with François), the motivic Poisson summation formula boils down to the Riemann-Roch and Serre theorems. So, in principle, our proof could be understood just from these two theorems. But this is not clear at all how to do this directly: passing through the looking-glass to go computing our height zeta function in the Fourier world appears to be non-trivial and efficient...

Product of two quotient spaces - an frequent and (in)famous mistake

The first edition of Bourbaki's General Topology (chapter I, §9, p. 56) contains the following theorem.

Proposition 3. Soient $E$, $F$ deux espaces topologiques, $R$ une relation d'équivalence dans $E$, $S$ une relation d'équivalence dans $F$. L'application canonique de l'espace produit $(E/R) \times (F/S)$ sur l'espace quotient $(E\times F)/(R\times S)$ est un homéomorphisme.

It is followed by a very convincing proof. However, the theorem is wrong. The subsequent editions give an example where the spaces are not homeomorphic, even when one of the equivalence relation is equality.

I finally understood where the mistake is. It is in the very statement! Indeed, there is a canonical map, say $h$, between those two spaces, but it goes the other way round, namely from $(E\times F)/(R\times S)$ to $(E/R)\times (F/S)$. This map is continuous, as it should be. But Bourbaki, assuming that the natural canonical map goes the other way round, pretended that $h^{-1}$ is continuous, and embarked in proving that its reciprocal bijection, $h$, is also continuous, what it is...

There are cases where one would like this theorem to holds, for example when one discusses topologies on the fundamental group. Indeed, the fundamental group of a pointed space $(X,x)$ is a quotient of the space of loops based at $x$ on $X$ for the pointed-homotopy relation, hence can be endowed with the quotient of the topology of compact convergence (roughly, uniform convergence on compact sets). Multiplication of loops is continuous. However, the resulting group law on $\pi_1(X,x)$ need not be.

The mistake appears in the recent litterature, see for example this paper, or that one (which has been even featured as «best AMM paper of the year» in 2000...). MathScinet is not aware of the flaws in those papers... Fortunately, MathOverflow is!

Finite choices

The axiom of choice says that an arbitrary product $\prod_{i\in I} A_i$ of non-empty sets $A_i$ indexed by a set $I$ is non-empty. It is well known that this axiom does not follow from the other axioms of Zermelo-Fraenkel theory. Even finite choices, that is, this statement restricted to the case where all sets are finite, is not a consequence. Even 2-choices, when one assumes that $A_i$ has two elements!

For each integer $n$, call  ${\rm AC}(n)$ the axiom of choice restricted to families $(A_i)$ where $A_i$ has $n$ elements. 

Tarski proved the funny following fact: ${\rm AC}(2) \Rightarrow {\rm AC}(4)$—if you know how to choose between 2 elements, you can choose between 4.

The proof is in fact quite easy. Consider a family $(A_i)$ of sets with 4 elements. I will use choice functions furnished by ${\rm AC}(2)$ to pick-up one preferred element from $A_i$. For simplicity, label the elements of $A_i$ as $\{a,b,c,d\}$ and remove the index $i$. Then, consider the set  $\{\{a,b\},\{a,c\},\{a,d\},\{b,c\},\{b,d\},\{c,d\}\}$ of all pairs of elements of $A_i$. The hypothesis ${\rm AC}(2)$ allows to choose, for each of those pairs, one preferred element. Call $n_a,n_b,n_c,n_d$ the number of times $a,b,c,d$ has been chosen; one thus has $n_a+n_b+n_c+n_d=6$ and consider those elements which have been chosen the most often, those for which $n_?$ is maximal.

• If there is only one, let's choose it. (This happens in repartitions like $(3,1,1,1)$, etc.)
• If there are three such elements (the repartition must be $(2,2,2,0)$), let's choose the unique one which has never been chosen.
• There can't be four such elements because 4 does not divides 6.
• If there are two (repartition $(2,2,1,1)$), then use your 2-choice function on this pair!

The other funny, but more difficult, thing, is that ${\rm AC}(2)$ does not imply ${\rm AC}(3)$! Why? because the group $\{\pm1\}$ can act without fixed points on a 2-elements set but cannot on a 3-elements set.  I hope to be able to say more on this another day.

Misconceptions about $K_X$

This is the title of a very short paper by Steven Kleiman published in L'enseignement mathématique, and which should be studied by every young student in scheme theory.

Here, $X$ is a scheme and $K_X$ is the sheaf of rational functions on $X$.

The misconceptions are the following, where we write $\mathop{Frac}(A)$ for the total ring of fractions of a ring $A$, namely the localized ring with respect to all element which are not zero divisors.

1. $K_X$ is not the sheaf associated to the presheaf $U\mapsto \mathop{Frac}(\Gamma(U,O_X))$; indeed, that map may not be a presheaf.
2. The germ $K_{X,x}$ of $K_X$ at a point $x$ may not be the total ring of fractions of the local ring $O_{X,x}$, it may be smaller.
3. If $U=\mathop{Spec}(A)$ is an affine open subset of $X$, then $\Gamma(U,K_X)$ is not necessarily equal to $\mathop{Frac}(A)$.

These mistakes can be found in the writings of very good authors, even Grothendieck's EGA IV... 

By chance, the first one is corrected in a straightforward way, and the other two work when the scheme $X$ is locally noetherian.

Thanks to Antoine D. for indicating to me this mistake, and to Google for leading me to Kleiman's paper.