The two adjunctions of the preimage

Sometimes in mathematics, you are told about very elementary things of which you hadn't even thought.

I was well aware of some “duality” between image and preimage, but I just learned from Anatole Dedecker (who learned it from Patrick Massot) about another “duality” between preimage and some other notion. Moreover, it appears that this new notion can be used for making slightly more natural a proof in general topology!

Here, “duality” is taken in an informal meaning, the correct word is “adjunction”, in the sense of category theory, and I will try to explain that.

1. Image and preimage

So consider a map $f\colon X \to Y$ between two sets. It induces two other maps relating the sets $\mathcal P(X)$ and $\mathcal P(Y)$ of subsets of $X$ and $Y$. Note that the inclusion relation between subsets these two sets $\mathcal P(X)$ and $\mathcal P(Y)$ allows to view them as ordered sets.

First, we have the direct image operation $f_{*}$, that maps a subset $A\subseteq X$ to the subset $f_{*}(A)$ of $Y$, the set of all images $f(a)\in X$, for $a\in A$. The classical notation would be $f(A)$, but it is ambiguous in the case where a subset $A$ of $X$ is also an element of $X$, and introducing a specific notation will help to clarify some statements later on. This map $f_{*}\colon \mathcal P(X) \to \mathcal P(Y)$ is increasing: for $A$ and $A'\in\mathcal P(X)$ such that $A\subseteq A'$, one has $f_{*}(A) \subseteq f_{*}(A')$.

Then we have the preimage operation $f^{*}$, that maps a subset $B\subseteq X$ to the subset $f^{*}(B)$ of $X$ consisting of all preimages of elements of $B$, namely all $a\in A$ such that $f(a) \in B$. The classical notation is rather $f^{-1}(B)$, but it has the same ambiguity as the direct image. Bizarrely, Bourbaki found the need to invent a another notation for that one, and they put the symbol “$-1$” on top of the letter $f$. The notation $f ^{*}$ is chosen by symmetry with the direct image $f_{*}$. Again, the map $f^{*}\colon \mathcal P(Y) \to\mathcal P(X)$ is increasing: for $B$ and $B'\in\mathcal P(Y)$ such that $B\subseteq B'$, one has $f^{*}(B) \subseteq f^{*}(B')$.

Finally, there is a compatibility between these two operations $f_{*}$ and $f ^{*}$: for $A\in\mathcal P(X)$ and $B\in\mathcal P(Y)$, one has $f_{*}(A) \subseteq B$ if and only if $A \subseteq f^{*}(B)$. Indeed, both of these expressions mean that if $f(a) \in B$ for all $a\in A$. We summarize this property by  saying that the operation $f_{*}$ is left adjoint to the operation $f ^{*}$, or that the operation $f^{*}$ is right adjoint to the operation $f_{*}$.

This terminology comes from category theory, in which adjunctions of functors play an important role since the paper of Daniel Kan (1958), Adjoint functors.

In our case, the categories are just the ordered sets $\mathcal P(X)$ and $\mathcal P(Y)$, with the corresponding sets as sets of objects, and where the set of arrows $A$ to $A'\in\mathcal P(X)$ is a singleton when $A\subseteq A'$, and is empty otherwise. The book of Emily Riehl (2016), Category Theory in Context, is a nice introduction to this topic, with illuminating elementary examples. The property that the operations $f_{*}$ and $f^{*}$ are increasing means that they are *functors* between these categories, and the equivalence $f_{*}(A) \subseteq B \Leftrightarrow A \subseteq f^{*}(B)$ induces the category-theoretical adjunction.

In this case, an adjunction pair is also called a Galois connection. There, the terminology comes from  Galois theory, the two ordered sets are the set of subextensions of a Galois extension $K\to L$ and the set of subgroups of the Galois group $\operatorname{Gal}(L/K)$, the maps are decreasing and correspond to mapping a subextension $E$ of $L$ to the subgroup of $\operatorname{Gal}(L/E)$ of $\operatorname{Gal}(L/K)$, and a subgroup $H\subseteq \operatorname{Gal}(L/K)$ to the fixed-field $L^H$. In Galois theory, these two maps are even bijective.

2. The adjoint functor theorem

While, as MacLane wrote, “adjoint functors arise everywhere”, not every functor can be part of an adjunction. Indeed, if a functor $F$ is left adjoint to a functor $G$, then $F$ preserves colimits and $G$ preserves limits.

Category theory considers limits and colimits of arbitrary diagrams, but in the restricted setting of ordered sets, where there can be at most one arrow from one object to another, diagrams boil down to subsets of objects, limits correspond to infimums (greatest lower bound) and colimits to supremums (least upper bound), which may exist, or not, in particular ordered sets.In our even more restricted case of the set $\mathcal P(X)$ of subsets of a given set $X$, infimum corresponds to intersection, supremum to union, and we have $f_{*}(\bigcup A_i) = \bigcup f_{*}(A_i)$ for every family $(A_i)$ of subsets of $X$, and $f^{*}(\bigcap B_i) = \bigcap f^{*}(B_i)$ for every family $(B_i)$ of subsets of $Y$.

There is an abstract theorem in category theory, the “general adjoint functor theorem”, that says that these property are essentially sufficient for a functor $F$ to be a left adjoint to some functor $G$, or for a functor $G$ to be a right adjoint to some functor $G$. One has to be more careful for the actual statement, but this is the idea.

For an increasing map $G\colon T \to S$ between ordered sets $S$ and $T$, the existence of a left adjoint $F$ can be understood from: for $s\in S$ and $t\in T$, one should have $F(s)\leq t$ if and only if $s\leq G(t)$: consequently, it suffices to take for $F$ the infimum, assuming it exists, of all $t$ such that $s\leq G(t)$. Dually,  the right adjoint $G$ to a functor $F$ would map $t$ to the supremum, assuming it exists,  of all $s$ such that $t\leq F(s)$.

In the case of the image $f_{*}\colon \mathcal P(Y)\to \mathcal P(X)$, this rule defines the right adjoint as mapping $B \in\mathcal P(Y)$ to the union of all subsets $A\in\mathcal P(X)$ such that $f _{*}(A) \subseteq B$. This is exactly the preimage of $B$!

Conversely, in the case of the preimage $f^{*}\colon \mathcal P(Y)\to \mathcal P(X)$, this procedure defines the left adjoint as mapping $A \in\mathcal P(X)$ to the intersection of all subsets $B$ such that $A \subseteq f^{*}(B)$. Again, this is just the image $f _{*}(A)$ of $A$, but I find it slightly more difficult to prove without using that we already know this image and the already known adjunction between $f _{*}$ and $f ^{*}$.

3. The other adjunction

We have seen that preimages respect intersections. As a matter of fact, they also respect unions: $f ^{*}(\bigcup B_i)= \bigcup f ^{*}(B_i)$. Given the adjoint functor theorem, this implies that there is an increasing map $f_! \colon \mathcal P(X) \to \mathcal P(Y)$ which is a right adjoint to $f ^{*}$. What is this operation?

The adjoint functor theorem gives a way to compute it: for $A\in\mathcal P(X)$, the set $f_!(A)\in\mathcal P(Y)$ is the union of all subsets $B\in\mathcal P(Y)$ such that $f^{*}(B) \subseteq A$. It suffices to consider such sets $B$ which are singletons $\{b\}$ and we get that a point $b\in Y$ belongs to $f_!(A)$ if and only if all preimages of $b$ belong to $A$.

Here are two more ways to get a grip on this new adjunction.

Note that a point $b\in Y$ belongs to $f_{*}(A)$ if and only if there exists $a\in A$ such that $b = f (a)$, which means that there exists $a\in A$ in the preimage $f^{*}(\{b\})$, relating $f_{*}$ with the existential quantifier. Similarly, a point $b\in Y$ belongs to $f_! (A)$ if and only if for every $a\in f^{*}(\{b\})$, one has $a\in A$, relating $f_!$ with the universal quantifier.

The other way comes by taking complements: a point $b$ does not belong to $f_!(A)$ if it has a preimage that does not belong to $a$. In other words, $f_!(A) = \complement f_{*}(\complement A)$. This leads to considering the complement map from $\mathcal P(X)$ to itself as an order-reversing involution, and similarly on $\mathcal P(Y)$, and observing that they commute with preimage, in the sense that $f^{*}(\complement B) = \complement f^{*}(B)$ for all $B\subseteq Y$. Consequently, this operation transfers the left adjoint $f _{*}$ of $f ^{*}$ to a right adjoint, and conversely, which is exactly what we had observed.

4. An application in general topology

As an application, this adjunction can be used in topology to characterize open or closed maps. By definition, a map $f \colon X\to Y$ between topological spaces is open if it maps an open subset to an open subset, and it is closed if it maps a closed subset to a closed subset.

The definition of $f_!$ using complement, and the fact that a set is closed if and only if its complement is open implies the following lemma:

Lemma. — A map $f\colon X \to Y$ is closed (resp. open) if and only if for every open (resp. closed) subset $U\subseteq X$, the set $f_! (U)$ is closed (resp. open).

It also allows to give a natural proof of the classical characterization of closed maps:

Proposition. — Let  $f\colon X \to Y$ be a map between topological spaces. The following properties are equivalent:

1. The map $f$ is closed;
2. For any subset $B$ of $Y$, the filter of neighborhoods of $f^{*}(B)$ is coarser than the preimage of the filter of neighborhoods of $B$;
3. For any subset $B$ of $Y$ and any neighborhood $U$ of $f^{*}(B)$, there exists a neighborhood $V$ of $B$ such that $f^{*}(V)\subseteq U$;
4. For any point $b\in Y$, the filter of neighborhoods of $f^{*}(\{b\})$ is coarser than the preimage of the filter of neighborhoods of $b$;
5. For any point $b\in Y$ and any neighborhood $U$ of $f^{*}(\{b\})$, there exists a neighborhood $V$ of $b$ such that $f^{*}(V) \subseteq U$.

Given the definitions of the preimage of a filter and the comparison relation on filters,
the assertions (2) and (3) are equivalent, as well as the assertions (4) and (5).

Obviously, (3) implies (5).

Let us assume (1), that $f$ is closed, and let us prove (3). Let $B$ be a subset of $Y$ and let $U$ be a neighborhood of $f^{*}B$ in $X$. By definition, there exists an open subset $U'$ of $X$ such that $f^{*}B \subseteq U' \subseteq U$. Taking adjunction, we get $B\subseteq f_! U' \subseteq f_! U$. Since $f$ is closed, the set $f_! U'$ is open, so that $f_! U$ is a neighborhood of $B$. It remains to prove that $f^{*}f_! U\subseteq U$.  To prove this inclusion, we apply the adjunction $(f^{*}, f_!)$ once more, and see that it is equivalent to the obvious inclusion  $f_! U \subseteq f_! U$.

Finally, let us assume (5) and let us prove that $f$ is closed. Let $U$ be an open subset of $X$ and let us prove that $f_! U$ is open in $Y$. It suffices to prove that for every $b\in f_! U$, the set $f_! U$ is a neighborhood of $b$. By the construction of $f_!$, the set $f^{*}(\{b\}) $ is contained in $U$ so that $U$ is a neighborhood of $f^{*}(\{b\})$. Applying (5), we get a neighborhood $V$ of $b$ in $Y$ such that $f^{*}V \subseteq U$. Applying the adjunction $(f^{*}, f_!)$, we get the inclusion $V \subseteq f_! U$. In particular, $f_! U$ is a neighborhood of $b$, as was to be shown.
 

Contemporary homological algebra — Ignoramus et ignorabimus (?)

The title of this post is a quotation of Emil Dubois-Reymond (1818-1896), a 19th century German physiologist, and the elder brother of the mathematician Paul Dubois-Reymond. Meaning we are ignorant, and we will remain ignorant, it adopts a pessimistic point of view on science, which would have intrinsic limitations. As such, this slogan has been quite opposed by David Hilbert who declared, in 1900, at the International congress of mathematicians, that there is no ignorabimus in mathematics. (In fact, there is some ignorabimus, because of Gödel's incompleteness theorem, but that is not the subject of this post.)

I would like to discuss here, in a particularly informal way, some frustration of myself relative to homological algebra, in particular to its most recent developments. I am certainly ill-informed on those matters, and one of my goals is to clarify my own ideas, my expectations, my hopes,...

This mere existence of this post is due to the kind invitation of a colleague of the computer science department working in (higher) category theory, namely François Metayer, who was interested to understand my motivation for willing to understand this topic.


Let me begin with a brief historical summary of the development of homological algebra, partly borrowed from Charles Weibel's History of homological algebra.

• B. Riemann (1857), E. Betti (1871), H. Poincaré (1895) define homology numbers. 
• E. Noether (1925) introduces abelian groups, whose elementary divisors, recover the previously defined homology numbers.
• J. Leray (1946) introduces sheaves, their cohomology, the spectral sequence... 
• During the years 1940–1955, under the hands of Cartan, Serre, Borel, etc., the theory develops itself in various directions (cohomology of groups, new spectral sequences, etc.).
• In their foundational book, Homological algebra, H. Cartan and S. Eilenberg (1956) introduce derived functors, projective/injective resolutions,...
• Around 1950, A. Dold, D. Kan, J. Moore, D. Puppe introduce simplicial methods. D. Kan introduces adjoint functors.
• A. Grothendieck, in Sur quelques points d'algèbre homologique (1957), introduces general abelian categories, as well as convenient axioms that guarantee the existence of enough injective objects, thus giving birth to a generalized homological algebra.
• P. Gabriel and M. Zisman (1967) developed the abstract calculus of fractions in categories, and proved that the homotopy category of topological spaces coincides with that of simplicial sets.
• J.-L. Verdier (1963) defines derived categories. This acknowledges that objects give rise to, say, injective resolutions which are canonical up to homotopy, and that the corresponding complex is an object in its own right, that has to be seen as equivalent to the initial object.  The framework is that of triangulated categories. Progressively, derived categories came to play an important rôle in algebraic geometry (Grothendieck duality, Verdier duality, deformation theory, intersection cohomology and perverse sheaves, the Riemann–Hilbert correspondence, mirror symmetry,...) and representation theory.
• D. Quillen (1967) introduces model categories, who allow a parallel treatment of homological algebra in linear contexts (modules, sheaves of modules...) and non-linear ones (algebraic topology)... This is completed by A. Grothendieck's (1991) notion of derivators.
• At some point, the theory of dg-categories appears, but I can't locate it precisely, nor do I understand precisely its relation with other approaches.
• A. Joyal (2002) begins the study of quasi-categories (which were previously defined by J. M. Boardman and R. M. Vogt, 1973). Under the name of $(\infty,1)$-categories or $\infty$-categories, these quasi-categories are used extensively in Lurie's work (his books Higher topos theory, 2006; Higher algebra, 2017; the 10+ papers on derived algebraic geometry,...).

My main object of interest (up to now) is “classical” algebraic geometry, with homological algebra as an important tool via the cohomology of sheaves, and while I have barely used anything more abstract that cohomology sheaves (almost never complexes), I do agree that there are three main options to homological algebra: derived categories, model categories, and $\infty$-categories.

While I am not absolutely ignorant of the first one (I even lectured on them), the two other approaches still look esoteric to me and I can't say I master them (yet?). Moreover, their learning curve seem to be quite steep (Lurie's books totalize more than 2000 pages, plus the innumerable papers on derived algebraic geometry, etc.) and I do not really see how an average geometer should/could embark in this journey.

However, I believe that this is now a necessary journey, and I would like to mention some recent theorems that support this idea.

First of all, and despite its usefulness, the theory of triangulated/derived categories has many defects. Here are some of them:

• There is no (and there cannot be any) functorial construction of a cone; 
• When a triangulated category is endowed with a truncation structure, there is no natural functor from the derived category of its heart to the initial triangulated category; 
• Derived categories are not well suited for non-abelian categories (filtered derived categories seem to require additional, non-trivial, work, for example);
• Unbounded derived functors are often hard to define: we now dispose of homotopically injective resolutions (Spaltenstein, Serpé, Alonso-Tarrió et al.), but unbounded Verdier duality still requires some unnatural hypotheses on the morphism, for example.

Three results, now.

The first theorem I want to mention is due to M. Greenberg (1966). Given a scheme $X$ of finite type over a  complete discrete valuation ring $R$ with uniformizer $\pi$, there exists an integer $a\geq 1$, such that for any integer $n\geq1$, a point $x\in X(R/\pi^n)$ lifts to $X(R)$ if and only if it lifts to $X(R/\pi^{an})$.

It may be worth stating it in more concrete terms. Two particular cases of such a ring $R$ are the ring $R[[t]]$ of power series over some field $k$, then $\pi=t$, and the ring $\mathbf Z_p$ of $p$-adic numbers (for some fixed prime number $p$), in which case one has $\pi=p$. It is then important to consider the case of affine scheme. Then $X=V(f_1,\dots,f_m)$ is defined by the vanishing of a finite family $f_1,\dots,f_m$ of polynomials in $R[T_1,\dots,T_n]$ in $n$ variables, so that, for any ring $A$, $X(A)$ is the set of solutions in $A^n$ of the system $f(T_1,\dots,T_n)=\dots=f_m(T_1,\dots,T_n)=0$.  By reduction modulo $\pi^r$, a solution in $R^r$ gives rise to a solution in $R/\pi^r$, and Greenberg's result is about the converse: given a solution $x$ in $R/\pi^r$, how do decide whether it is a reduction of a solution in $R$. A necessary condition is that $x$ lifts to a solution in $R/\pi^s$, for every $s\geq r$. Greenberg's theorem asserts that it is sufficient that $x$ lift to a solution in $R/\pi^{ar}$, for some integer $a\geq 1$ which does not depend on $X$.

The proof of this theorem is non-trivial, but relatively elementary. After some preparation, it boils down to Hensel's lemma or, equivalently, Newton's method for solving equations.

However, it seems to me that there should be an extremely conceptual way to prove this theorem, based on general deformation theory such as the one developed by Illusie (1971). Namely, obstructions to lifting $x$ are encoded by various cohomology classes, and knowing that it lifts enough should be enough to see — on the nose — that these obstructions vanish.

The second one is about cohomology of Artin stacks. Y. Laszlo and M. Olsson (2006) established the 6-operations package for $\ell$-adic sheaves on Artin stacks, but their statements have some hypotheses which look a bit unnatural. For example, the base scheme $S$ needs to be such that all schemes of finite type have finite $\ell$-cohomological dimension — this forbids $S=\operatorname{Spec}(\mathbf R)$. More recently, Y. Liu and W. Zheng developed a more general theory, apparently devoid of restrictive hypotheses, and their work builds on $\infty$-categories, more precisely, a stable $\infty$-category enhancing the unbounded derived category. On page 7 of their paper, they carefully explain why derived categories are unsufficient to take care of the necessary descent datas, but I can't say I understand their explanation yet...

The last one is about the general formalism of 6-operations. While it is clear what these 6 operations should reflect (direct and inverse images; proper direct images and extraordinary inverse images; tensor product, internal hom), the list of the properties they should satisfy is not clear at all (to me). In the case of coherent sheaves, there is such a formulaire, written by A. Grothendieck itself on the occasion of a talk in 1983, but it is quite informal, and not at all a general formalism. Recently, F. Hörmann proposed such a formalism  (2015–2017), based on Grothendieck's theory of derivators.

Now, how should the average mathematician embark in learning these theories?

Who will write the analogue of Godement's book for the homological algebra of the 21st century? Can we hope that it be shorter than 3000 pages?

I hope to find, some day, some answer to these questions, and that they will allow to hear with satisfaction the words of Hilbert: Wir müssen wissen, wir werden wissen.

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.

The category of sets and its opposite

In the book Categories and sheaves by Kashiwara and Shapira, I found a nice argument for the fact that the category of sets is not equivalent to its opposite: they write « every morphism to the initial object is an isomorphism ». Of course!

In the category Sets, the initial object is the empty set, which means that for every set $A$, there is a unique map from $\emptyset$ to $A$. Now, if we reverse the process, namely, if we consider a set $A$ and a map $f\colon A\to \emptyset$, we see that $A$ must be empty and $f$ is a bijection, hence an isomorphism in the category of sets.

In the opposite category, all arrows are reversed, the initial object becomes the terminal object, etc. Isomorphisms are (reversed) maps which have an inverse, so isomorphisms are still given by bijections. A terminal object of Sets is one-element set, $\{x\}$ (you could take the set $1=\{\emptyset\}$ if, like von Neumann, you believe that numbers are sets). Indeed, there is a unique map from any set to a one-element set. Reverse the process again and consider a set $A$ and a map $f\colon \{x\} \to A$. This amounts to choosing an element of $A$, but such maps are not bijections in general, unless $A$ has itself only one element.