The Perfectoid Concept: Test Case for an Absent Theory
MICHAEL HARRIS
Department of Mathematics
Columbia University
Perfectoid prologue
It's not often that contemporary mathematics provides such a clear-cut
example of concept formation as the one I am about to present: Peter Scholze's
introduction of the new notion of perfectoid space. The 23-year old Scholze firstunveiled the concept in the spring of 2011 in a conference talk at the Institute forAdvanced Study in Princeton. I know because I was there. This was soon followedby an extended visit to the Institut des Hautes Études Scientifiques (IHES) at Bûressur-Yvette, outside Paris — I was there too. Scholze's six-lecture series culminatedwith a spectacular application of the new method, already announced in Princeton, toan outstanding problem left over from the days when the IHES was the destination ofpilgrims come to hear Alexander Grothendieck, and later Pierre Deligne, report on thecreation of the new geometries of their day. Scholze's exceptionally clear lecturenotes were read in mathematics departments around the world within days of his
lecture — not passed hand-to-hand as in Grothendieck's day — and the videos of histalks were immediately made available on the IHES website. Meanwhile, more killer
apps followed in rapid succession in a series of papers written by Scholze, sometimes
in collaboration with other mathematicians under 30 (or just slightly older), often
alone. By the time he reached the age of 24, high-level conference invitations to talkabout the uses of perfectoid spaces (I was at a number of those too) had enshrined
Scholze as one of the youngest elder statesmen ever of arithmetic geometry, thebranch of mathematics where number theory meets algebraic geometry.) Two years
later, a week-long meeting in 2014 on Perfectoid Spaces and Their Applications at the
Mathematical Sciences Research Institute in Berkeley broke all attendance records for"Hot Topics" conferences.
Four years after its birth, perfectoid geometry, the theory of perfectoid spaces,
is a textbook example of a progressive research program in the Lakatos sense. It isseen, retrospectively, as the right theory toward which several strands of arithmetic
geometry were independently striving. It has launched a thousand graduate student
seminars (if I were a historian I would tell you exactly how many); the students'
advisors struggle to keep up. It has a characteristic terminology, notation, and style ofargument; a growing cohort of (overwhelmingly) young experts, with Scholze and his
direct collaborators at the center; a domain of applications whose scope continues toexpand to encompass new branches of mathematics; an implicit mandate to unify andsimplify the fields in its immediate vicinity. Last, but certainly not least, there is thegenerous, smiling figure of Peter Scholze himself, in the numerous online recordingsof his lectures or in person, patiently answering every question until his questioner issatisfied, still just 27 years old, an inexhaustible source of revolutionary new ideas.
For the historian of concepts (which I am not), there is only one problem withthis picture: the concept of a perfectoid space is one of the most difficult notions ever
introduced in arithmetic geometry, which has a long tradition of difficult notions. The

reader of this essay faces a second problem: not only am I not a historian of concepts,
I am by no means an expert on the kinds of mathematics that have been put together
to create the new concept of perfectoid spaces. So I will have to skim the surface, atthe risk of distortion; but even the surface will be unfamiliar for most readers. I hopethat the technical details will nevertheless provide the reader with the means ofperceiving how the somewhat extreme case of perfectoid spaces exemplifies typicalaspects of mathematical concept formation: the reference to a historical background(in this case, the long practice of studying properties of equations in number theory by
relating them to properties of geometric objects), the identification of a specific range
of open questions in this historical context and the expectation that they would be
solved by the introduction of new techniques of a specific type, and the reception andgeneral acceptance of the new concept as the right one because it met expectations inspite of its many novel features and because it lent itself immediately to solving
outstanding open problems.

Starting points
Perfectoid spaces stand at the crossroads where topology, Galois theory, andthe study of equations by means of congruences meet. It's a busy crossroads, alreadyoccupied by étale cohomology and crystalline cohomology, both created by
Grothendieck and his school; by the new number systems (Fontaine rings) created by
Jean-Marc Fontaine in his program to devise a mysterious functor, a canonicaloperation relating the two kinds of cohomology; and by the many technicalinnovations introduced in order to complete Fontaine's program. The most relevant ofthese for Scholze's work is the almost mathematics of Gerd Faltings, which in turnbuilds on the foundational work of John Tate, 90 years old and still active. Tate is also
the creator of rigid analytic spaces, the first of several successful attempts to create ap-adic geometry. Perfectoid geometry doesn't transcend these previous attempts but
rather extracts those of their features that are relevant to the problems Scholze set out
to solve.
I don't expect readers to possess any of the technical vocabulary introduced in
the previous paragraph. Each of the technical terms, however, can be seen as theprecise analogue of a familiar notion from higher-dimensional geometry in Euclidean
space; for the purposes of this essay, much of the meaning of Scholze's innovations
can be understood by reference to these analogies. I will start with the way
differential calculus encodes topology, and specifically with Green's theorem incalculus in two variables (as a topic in second year calculus, no other starting point isnearly as elementary). Green's theorem relates the line integral of a differential
fdx + gdy
on a simple closed curve C with the double integral of an expression involving thepartial derivatives of f and g on the region D bounded by the curve. This breaks downwhen D is no longer a region bounded by a simple closed curve, but rather a region
with holes. The breakdown is an expression of the complicated topology of D — thepresence of holes — but Green's theorem has a generalization — a version of Stokes'
theorem that accounts for the holes. The generalization of these theorems to amanifold M of arbitrary dimension relates the topology of M — and more precisely,
the cohomology of M, which is the way of keeping track of generalized holes — tothe possibility or impossibility of integrating differential forms on M. For general(compact) manifolds the relation is the statement of de Rham's theorem. When M isthe space of complex solutions (in projective space) to a system of polynomial
equations — a (smooth, projective, complex) algebraic variety — then Hodge theory
gives a more precise relation, where the distinction between holomorphic differentials(like dz) and antiholomorphic differentials (like dz ) is central. For the purposes ofthis discussion, the important fact about Hodge theory is that it uses differential
calculus to compute the topology, or more precisely the cohomology, of a complex
algebraic variety M. Another fruitful way of saying the same thing is to say that M€ the topological one (that keeps track ofhas two kinds of cohomology groups —holes), often called Betti cohomology, and the analytic one, based on the algebra ofdifferential forms, called de Rham cohomology. Then the comparison theorem ofHodge theory — at this stage it's really de Rham's theorem — says that these are twodifferent ways of computing the same thing.
Now forget the algebraic variety M but keep the polynomial equations used todefine M. What happened to the holes? In the 1960s Grothendieck and his
collaborators constructed a purely algebraic invariant1 of an algebraic variety — notnecessarily smooth, not necessarily projective, and over any coefficient field F — thatrecovered the topological (Betti) cohomology groups, in the process transforming thenotion of topological space practically beyond recognition. Something was gained
and something was lost. What was gained was that the cohomology groups acquired
an action of the Galois group of the coefficient field F — and when F is the field ofrational numbers, or a finite field, this defines an invariant of exceptional richness.
The Tate Conjecture, which has been proved in only a few situations (notably by Tateand Faltings), asserts that this Galois representation encodes a good deal of theinformation about solutions of the polynomial equations and higher-dimensional
generalizations of solutions (whatever one might mean by that). What was lost is that,
whereas Betti cohomology groups actually count holes (more precisely, they are
finitely generated abelian groups), the algebraic invariants introduced by
Grothendieck and his collaborators, the étale cohomology groups, only count holes
modulo N for all integers N — with the important proviso that N is not divisible by
the characteristic of the coefficient field F. It is more useful to package thisinformation by prime powers and to look at the l-adic (étale) cohomology, which puts
together the étale cohomology modulo ln for all n, where l runs through the set ofprime numbers that are invertible in the field F.2 The underlying assumption is thatthe number of holes in an algebraic variety, or even a 2-dimensional manifold, is afundamental geometric property — an invariant.
Let us pause to consider that set of prime numbers l invertible in a fixed fieldF, since this is the most characteristic property of the field, and because the raisond'être of Scholze's work is to answer questions about equations over p-adic fields
1

André Weil had constructed such an invariant for varieties of dimension 1, speculated that such atheory should exist in general, and conjectured some of its desirable properties. Grothendieck proved
the Weil conjectures, with the exception of the last one, which was solved in 1973 by Pierre Deligne. I
was at the IHES, not yet a graduate student, the day the 29-year-old Deligne announced his solution. It
was seen, correctly, as a very big deal.
2
For a philosophically enlightening introduction to the topological approach to number theory, see §3of C. McLarty,"What does it take to prove Fermat's Last Theorem? Grothendieck and the logic ofnumber theory." Bulletin of Symbolic Logic 16.03 (2010): 359-377.

(where some standard methods of algebraic geometry don't apply) by "tilting" theequations so they are now over fields of characteristic p. In fact, the characteristic of
F is decreed to be either the set of primes that are not invertible in F, if there are such,or else zero, if all primes are invertible in F. A field is of characteristic zero if andonly if it contains the rational field Q; for example, the familiar fields of real orcomplex numbers are of characteristic zero. Otherwise, there is exactly one prime,
traditionally denoted p, which is not invertible in F, and F is said to be of (positive)
characteristic p. Since every non-zero element of F is invertible, this means that p = 0
in F, and F contains the prime field Fp = Z/pZ of p elements, the field whosearithmetic corresponds to the study of congruences modulo p. Scholze's initial
motivation, very roughly speaking, was to study equations over certain kinds of fields
of characteristic zero by reinterpreting them as equations over fields of prime
characteristic p, where other methods are available.
A field F of characteristic p > 0 has two closely related properties that it does
not share with fields of characteristic zero.
(i) First, polynomials of positive degree can have derivative zero: indeed, ifpp-1
P(X) = X , then P'(X) = pX = 0 because p = 0.3(ii) On the other hand, the map Frob (for Frobenius) that sends a ∈ F to its pp
th power Frob(a) = a is a homomorphism of rings:
Frob(ab) = Frob(a)Frob(b) and Frob(a+b) = Frob(a) + Frob(b)for any a, b ∈ F. The multiplicative property is obvious, the additive propertypfollows from the binomial formula for (a+b) , because all the intermediate binomial
coefficients are divisible by p. Because F is a field, only 0 has p-th power zero, andthus Frob is injective. The field F is called perfect if Frob is also surjective: in otherwords, if every element of F has pth roots (and therefore pnth roots for all n).
Scholze's term perfectoid is derived from this property of perfect fields.
Algebraic varieties over any field still have differential forms, as in our earlier
discussion and there is an algebraic version of de Rham cohomology (whose main
properties were outlined in a letter4 from Grothendieck to Atiyah). But to work
properly with differentials of degree n, one needs to be able to divide by n!, and this is
impossible if the characteristic of F is a prime dividing n; alternatively, property (i)p-1above shows that one can't integrate the differential X dX. Expanding the notion of
space a bit more, Grothendieck outlined the properties of yet another algebraic
cohomology theory (for smooth projective varieties), crystalline cohomology, whose
construction was carried out in the thesis of his student Pierre Berthelot and his
collaborators. It was around this time that Grothendieck abandoned the IHES and themathematical community, but not before he had asked for an analogue of Hodge
3

Didn't I just say that p > 0? So how can I say p = 0? The prime number p is and remains a positive
number in the field of rational numbers, but for the purposes of arithmetic in a field of characteristic p
it is treated as if it were equal to 0, and indeed this is the meaning of characteristic p. There is no
logical inconsistency.
4
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. math. de l’I.H.É.S. tome
29 (1966), p. 95-103.

theory in the algebraic setting. The original Hodge theory was defined for varieties
with complex coefficients; Grothendieck expected an analogue for varieties over padic local fields. For our purposes a field F is p-adic local if it is a topological fieldwith an open subring O = OF such that F is obtained from O by allowing division byp, and such that O is a local ring (it has a unique maximal ideal m) and a discrete
valuation ring (every ideal is a power of m such that k = O/m is a finite field). When
F is a p-adic local field, the algebraic de Rham cohomology, crystalline cohomology,
and p-adic étale cohomology all have coefficients in the same kind of field, namely
either F or a p-adic local field closely related to F. Grothendieck asked for a"mysterious functor" that played the role of the de Rham and Hodge theorems in
relating the p-adic étale cohomology of an algebraic variety over F to its algebraic deRham cohomology, and (when this makes sense) to the crystalline cohomology of its
reduction over k. A few years earlier, Tate had introduced what are now called
Hodge-Tate structures and, together with Jean-Pierre Serre, had formulated a different
conjectural p-adic version of the Hodge theorem; Grothendieck's proposal would
necessarily imply Tate's conjecture (not to be confused with "the Tate conjecture"
mentioned earlier).
Fontaine introduced his new rings, including the ones called BdR and Bcris, as away of relating the Galois structure and the differential structure, and used them todefine a candidate for Grothendieck's mysterious functor. There ensued aninternational contest, lasting nearly two decades, to prove that Fontaine's approachprovided the mysterious functor. Along the way the field saw the growth of the newresearch program of p-adic cohomology, characterized by the proliferation of newcohomology theories with p-adic coefficients, including syntomic cohomology, deRham-Witt cohomology, and rigid cohomology, and centered around the problem ofrealizing Grothendieck's mysterious functor.5
The main goal of the program was to find better ways to understand the Galois
representations constructed by Grothendieck, because they are among the central
objects of algebraic number theory. Results of this research program were offundamental importance in Andrew Wiles's proof of Fermat's Last Theorem and in
subsequent work that derived from that of Wiles. Especially in the hands of Richard
Taylor and his students and collaborators, this line of research has developed into oneof the most active research programs in algebraic number theory, overlapping with theLanglands program on automorphic forms — and only incidentally in geometry.
This is largely responsible for the growing familiarity of number theorists with
Fontaine's p-adic constructions in the years preceding Scholze's perfectoid
announcements.
A word about Fontaine's rings is in order, because of their importance for thetheory of perfectoid spaces. These rings, which can be thought of as alternative padic analogues of the complex number field, are constructed by an elaborate series ofsteps,6 of which the first invariably involves taking pnth roots for all n in a field of
characteristic zero in order to define a new and perfect field of characteristic p. In
5

It was mysterious because Grothendieck didn't know how to define it but assumed it must exist.
Then Fontaine defined a functor and over the years he and others showed that it had the expected
properties.
6
See, for example, http://en.wikipedia.org/wiki/Ring_of_p-adic_periods.

retrospect, this way of moving from characteristic 0 to characteristic p can be seen asa prototypically perfectoid thing to do. As recently as 15 years ago, any talk at anumber theory seminar involving one of Fontaine's rings would normally begin with areminder of the ring's construction. I never found this very enlightening, because it
was never explained why this particular sequence of otherwise unmotivated steps
yielded an interesting object — in other words, the construction did not illuminate theconcept of the Fontaine ring. These days, the rings tend to be introduced without
explanation, which I take to mean that the seminar audience has become sufficiently
familiar with the rings not to need to be reminded that the speaker knows how they
are put together. In practice, this means that a typical (algebraic) number theorist-intraining is expected to know a few standard properties of Fontaine's rings; one might(or might not) want to say in this connection that the number theorist-in-training has
been initiated into one or more of the language games in which Fontaine's rings
feature prominently. More to our purpose, this familiarity means that the communityof participants in number theory seminars was prepared for Scholze's theory ofperfectoid spaces by repeated exposure to the theory and use of Fontaine's rings,
largely in connection with outgrowths of the ideas of Wiles and Taylor.
To put an end to this lengthy introduction to the background to Scholze's
work, Fontaine's hope to have realized the mysterious functor was confirmed, notonce but several times, giving rise to what is now generally known as p-adic Hodge
theory. The first complete proof was obtained by Faltings; a rather different method,
extending ideas due to Fontaine and William Messing, was developed by Japanese
specialists, culminating in a second (and initially more complete) complete proof by
T. Tsuji. Two relevant features of the Faltings approach deserve to be mentioned
here. The first feature is the introduction of what Faltings called almost
mathematics,7 which is (very roughly) a form of commutative algebra in which
certain kinds of error are systematically ignored, and in which it is shown rigorouslythat they do not matter to the final proof of Fontaine's comparison. The second
feature is that Faltings made systematic use of constructions involving the taking ofpnth roots for all n. Faltings said at the time that his constructions were inspired by
Tate's original work on p-adic Hodge theory.

Scholze's perfectoid concept
Scholze's perfectoid spaces are, in the first place, spaces. The notion of spacein algebraic geometry has evolved through several stages since André Weil
introduced abstract varieties by the gluing together of affine algebraic varieties in his
Foundations of Algebraic Geometry. The current understanding is based on
Grothendieck's framework, developed systematically by Grothendieck and Dieudonnéin the Éléments de géométrie algébrique (EGA), following Serre's introduction ofsheaf-theoretic methods. A space in this setting then consists of a topological space
7

The first complete treatment is contained in O. Gabber and L. Ramero, Almost ring theory, Lecture
Notes in Mathematics 1800, Berlin: Springer-Verlag, (2003). Faltings' proof is in G. Faltings, p-adic
Hodge theory, J. Amer. Math. Soc. 1 (1): 255–299 (1988). The "certain kinds of error" mentioned in
the next sentence can be likened to measurement errors in physics. It's as if one were using two
different scales to make multiple measurements of the geometry of an algebraic variety; the scales cangive slightly different measurements in the intermediate steps but in almost mathematics they aretreated as identical. It turns out that the final result is insensitive to this kind of error.

(the "underlying topological space," generally not Hausdorff) together with the (sheafof) functions defined on its open sets. I pause to mention that Grothendieck's
reconceptualization of topology involves a generalization of the notion of "open set"that is radically at odds with the primitive intuition of continuity that general topology
was designed to formalize but that nevertheless works magnificently.
In p-adic geometry, the topology is somehow adapted to the topology of the padic numbers, and the functions take p-adic values. Perfectoid spaces are spaces inthe setting of p-adic geometry, and more precisely p-adic analytic geometry. Like thep-adic numbers themselves, p-adic geometry is a form of geometry designed to study
solutions of congruences modulo powers of p. The p-adic numbers themselves form a
topological space, but it is totally disconnected, which means that the space can be
broken continuously into arbitrarily small pieces. This is not suitable for geometry;
there are too many continuous functions. One could disallow all continuous functions
that are not polynomials, or quotients of polynomials; but that would yield algebraic
geometry, not p-adic analytic geometry. On the other hand, allowing all analytic
functions — convergent power series — would again not be geometric. Tate was thefirst to propose a workable compromise with his theory of rigid analytic spaces.
Tate's solution to the continuity problem was versatile, ingenious, andthoroughly in the spirit of Grothendieck's reformulation of topology. It also had anumber of immediate applications to traditional problems in arithmetic geometry,notably the construction of the Tate elliptic curve, which still serves as a model for
applications of Tate's rigid geometry. The field of rigid analysis that developed on
the basis of Tate's (long unpublished) founding document has its own canon andcharacteristic vocabulary; especially after it was extended by Michel Raynaud andPierre Berthelot, it has grown into a healthy subfield of arithmetic geometry, a topic
with its own practice and a resource for those in neighboring fields. However, it was
not the last word; specifically, it lacked a native cohomology theory along the lines ofétale cohomology, and the rigid analytic version of de Rham cohomology (MonskyWashnitzer cohomology, later incorporated in Berthelot's rigid cohomology) lacked
certain desirable features. This pointed to a failure of p-adic Hodge theory to be atrue analogue of complex Hodge theory. Just as complex algebraic varieties are also
complex analytic manifolds, p-adic algebraic varieties are also rigid analytic spaces.
Hodge's Hodge theory is based on complex analysis, but the proofs of p-adic Hodge
theory were purely algebraic. In particular, there was no p-adic analytic way tointerpret the algebraic invariant of étale cohomology.
Three distinct versions of étale cohomology in the setting of p-adic analysis
were proposed in the early 1990s, due respectively to Kazuhiro Fujiwara, VladimirBerkovich, and Roland Huber. Each of these approaches involved a more or lessradical expansion of the kinds of spaces considered in p-adic geometry, and each ofthem had immediate applications.8 Huber's theory of adic spaces had relatively few
adherents, however, until it was revived by Scholze as the setting in which perfectoid
spaces naturally coexisted with other kinds of p-adic algebraic and analytic structures.
8This, by the way, is where I enter, and exit, the subject. The work discussed in Chapter 9 of myMathematics without Apologies was only possible because Berkovich had recently found a way todefine l-adic cohomology for his analytic spaces. Fujiwara's theory played a crucial role in mysubsequent work with Richard Taylor.

Before I introduce Scholze's perfectoid spaces, I need to devote a few words tothe curious fact that algebraic geometry is so thoroughly algebraic that it cantheoretically be developed without any reference to geometry whatsoever. The basic
spaces in any version of algebraic geometry are the affine spaces. For Grothendieck,any commutative ring R defines an affine space — an affine scheme — denoted
Spec(R), with the tautological property that R is the ring of functions on Spec(R).
The correspondence between R and Spec(R) is an equivalence of categories, whichmeans that everything you need to know about Spec(R) can be deduced from what
you know about R, and vice versa.9 Because Spec(R) is a kind of space, it has points,
which are very convenient when you want to compare Spec(R) to Spec(R') for two
rings R and R'. The general space in Grothendieck's version10 of algebraic geometry
is a scheme which is a topological space X that is covered by open sets that are affine
schemes; in other words, each of whose points belongs to some Spec(R), or rather to a(generally infinite) collection of Spec(R'), compared in an appropriate way. One
usually says that X has been constructed by gluing the affine schemes it contains.
The geometric properties of schemes that can be reinterpreted as properties of theiraffine coverings — in other words, as properties of rings — are called local.
The prototype for constructing complicated spaces from simpler spaces is thetheory of (topological or differentiable) manifolds in topology or differential
geometry; the simpler spaces are just miniature versions of Euclidean space ofdimension n, and a manifold is defined as a topological space that "looks locally" like
Euclidean space. Differential geometry is to differential calculus as algebraic
geometry is to ring theory.
In the various versions of p-adic geometry derived from Tate's rigid analytic
geometry, spaces are constructed out of rings in the same way, except that the word
affine is replaced by affinoid. Here is Scholze's definition of a perfectoid algebra(ring), copied from his article in the Proceedings of the 2014 International Congressof Mathematicians (ICM) at Seoul:

Everything in the definition is important, but the most important part is the last line:
it says that one can take pnth roots for all n in this ring R°/p. To begin with, thisallows Scholze to introduce the tilt of R, denoted R♭; the process of tilting turns aperfectoid algebra of characteristic zero into a perfectoid algebra of characteristic p insuch a way that all the algebraic structures relevant to étale cohomology (among
others) are preserved under the operation. The geometric objects Scholze attaches tothe rings R and R♭ are adic spaces in Huber's sense; these are the affinoid perfectoid
spaces that can then be glued together to form general perfectoid spaces.

9

However, this equivalence is contravariant; a comparison map (ring homomorphism) R → R' isequivalent to a comparison map of schemes Spec(R') → Spec(R).
10
There are still more general versions but this will do for now.

The tilting operation originates in the early stages of p-adic Hodge theory, andspecifically in a theorem of Fontaine and Jean-Pierre Wintenberger that identifies the
Galois theory of certain infinite extensions of p-adic fields with the Galois theory ofwhat are now called their tilts, which are fields of characteristic p. This operation isapplied as the first step in constructing the most familiar of the Fontaine rings.
Having learned from Grothendieck that any operation on a class of rings should beviewed as a local version of an operation on the corresponding class of spaces,arithmetic geometers naturally want to define a class of spaces to which the FontaineWintenberger tilt extends that will for this reason be recognized as the right class ofspaces. Several attempts were made, more or less rooted in the methods Faltings
introduced in his work on p-adic Hodge theory; it is now acknowledged that Scholze's
perfectoid concept is the right one for rings, and the right one for gluing the local
pieces together into global geometric objects. The proofs made extensive use of theFaltings "almost ring theory."11"Category" is the formalized mathematical concept that currently best captures
what is understood by the word "concept." Scholze defined perfectoid spaces as acategory of geometric spaces with all the expected trappings, and thus there's noreason to deny it the status of "concept." I will fight the temptation12 to explain in any
more detail just why Scholze's perfectoid concept was seen to be the right one assoon as he explained the proofs in the (symbolically charged) suburban setting of the
IHES. But I do want to disabuse the reader of any hope that the revelation was asstraightforward as a collective process of feeling the scales fall from our eyes.
Scholze's lectures and expository writing are of a rare clarity, but they can't conceal
the fact that his proofs are extremely subtle and difficult. Perfectoid rings lackfamiliar finiteness properties — the term of art is that they are not noetherian13. Thismeans that the unwary will be systematically led astray by the familiar intuitions ofalgebra. The most virtuosic pages in Scholze's papers generally involve finding ways
to reduce constructions that appear to be hopelessly infinite to comprehensible (finitetype) ring theory. This is my contribution to speculation about why Scholze
succeeded so brilliantly where so many outstanding mathematicians failed.

The concept's reception
Scholze's first lectures on perfectoid spaces already included a stunningapplication, namely the proof of the weight-monodromy conjecture, in the special
case of complete intersections in projective space (extended in the published version
to complete intersections in toric varieties) over p-adic fields. As was mentioned in
the Prologue, this question was left unsolved by Grothendieck and Deligne. The latter
had indeed proved the complete conjecture over the characteristic p analogues of p11

As formalized and systematized by O.Gabber and L. Ramero in Almost ring theory, volume 1800 of
Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. Gabber, who is based at the IHES,
attended Scholze's lectures, and I can't resist mentioning how much arithmetic geometry is indebted toGabber's constantly insightful questioning, which has kept the field honest for over 30 years. At
several points during his lectures Scholze thanked Gabber for forcing him to clarify his ideas.
12
I don't have to fight very hard, because I only barely understand the proofs myself. The interested
reader can find the details in Scholze, P., Perfectoid spaces. Publ. Math. Inst. Hautes É́tudes Sci. 116(2012), 245–313.
13
After Emmy Noether, the founder of modern algebra.

adic fields — I can assure the reader that this was a very hard thing to do — andScholze's methods allowed him to "tilt up" Deligne's proof for the cases he was able
to settle.
Six months before his IAS lectures, Scholze was seen as an exceptional talent
by many colleagues, but not by everyone; a prestigious journal foolishly rejected oneof his pre-perfectoid papers (which was later published in an even more prestigious
journal). His proof of the weight-monodromy conjecture was enough to guaranteethat this would not happen again. My understanding was that Scholze invented
perfectoid spaces in order to "tilt up" Deligne's proof of the weight-monodromy
conjecture, but it was clear from the outset that the concept would have many more
applications.14 A few months after his IHES lectures, a French graduate student asked
whether I would be willing to be his thesis advisor; things started conventionallyenough, but very soon the student in question was bitten by the perfectoid bug andproduced a Mémoire M2 — a mainly expository paper equivalent to a minor thesis —that was much too complicated for his helpless advisor.15 By then Scholze had found
two new spectacular applications that the precocious student managed to cram into his
Mémoire M2, making it by far the longest Mémoire it has even been my pleasure todirect.
Although there have been more applications in the meantime — and I will
mention at least one of them — I want to devote a moment to the two that werediscovered in the year immediately following the IHES lectures, because they served
to cement the idea that perfectoid spaces provide the right framework to adapt theconstructions of complex Hodge theory in the p-adic setting. The first application
was an (unexpectedly general) extension of the main theorems of p-adic Hodge theoryto (proper) rigid-analytic varieties. This includes a new proof of the theorem of
Faltings and Tsuji mentioned above; of the four existing proofs (the fourth, slightly
later, is due to W. Niziol), Scholze's, in its large structure and in its local arguments, isthe one that most closely resembles the proofs of the main theorems of complex
Hodge theory. The second application, joint with Jared Weinstein, was aclassification of p-divisible groups over a complete algebraically closed p-adic field.
All I want to say about p-divisible groups, is that they are objects of fundamental
importance in arithmetic geometry and number theory, that they are distantly related
to the Abel-Jacobi theory of period integrals on algebraic curves, and that there has
been a dizzying variety of classification schemes for p-divisible groups, each one
useful for one purpose or another. Again, of all the classifications, the one of Scholze
and Weinstein, apart from being perhaps the easiest to remember (though certainly
not the easiest to prove), is the one that most closely resembles its complex
counterpart. In both cases, the familiarity of the complex analogue, which is anadvanced but necessary part of the training of practically any mathematician,reinforces the impression that Scholze has found the right way to think about p-adic

14

During the breaks in Scholze's lectures there was active speculation about the applicability of his
concept to one or another favorite problem.
15
The student in question — who has taken on by a second, more competent advisor — has not yet
finished his thesis but I would already count him as a member of the second perfectoid circle revolving
around Scholze. The first circle, as I see it, includes Scholze's immediate collaborators and a few
others; the second circle is already much broader, and there is a third circle consisting of everyone
hoping to apply the concept to one thing or another.

geometry. And Scholze has himself contributed to this impression in his expository
writing about perfectoid spaces.
Most of the blog posts, expository articles, and letters of reference I've seenagree that the outstanding applications of perfectoid geometry are contained in
Scholze's paper entitled "On torsion in the cohomology of locally symmetric spaces."
A great deal can be said about this paper, which solved several outstanding
conjectures in the course of vastly improving and generalizing an earlier paper16 on arelated topic. For the purposes of this essay, the importance of Scholze's paper is thatit confirmed that perfectoid geometry provides the right framework for thinking about
a number of central questions in algebraic number theory and, secondarily, that thecohomology of perfectoid spaces has the right (p-adic integrality) properties forapplications to such questions.17 And it confirmed the impression that number
theorists had made the right decision to devote time to learning Scholze's newframework even before it had been shown to have important applications to theirfield.

Discussion
I began writing this essay three years after Scholze's IHES lectures and one monthafter his ICM lecture in Seoul. One year earlier, I could safely assert that no one had(correctly) made use of the perfectoid concept except in close collaboration with
Scholze. The Seoul lecture made it clear that this was already no longer the case.
Now, after nearly a year has passed, the perfectoid concept has been assimilated by
the international community of arithmetic geometers and a growing group of number
theorists, in applications to questions that its creator had never considered. It is an
unqualified success.
How can this be explained? Mathematicians in fields different from mine are no
better prepared than philosophers or historians to evaluate our standards ofsignificance. One does occasionally hear dark warnings about disciplines dominatedby cliques that expand their influence by favorably reviewing one another's papers,
but by and large, when a field as established and prestigious as arithmetic geometry
asserts unanimously that a young specialist is the best one to come along in decades,
our colleagues in other fields defer to our judgment.
Doubts may linger nonetheless. I don't think that even a professional historian would
see the point in questioning whether Scholze is exceedingly bright, but is his work
really that important? How much of the fanfare around Scholze is objectively
legitimate, how much an effect of Scholze's obvious brilliance and unusuallyappealing personality, and how much just an expression of the wish to havesomething to celebrate, the "next big thing"? Is a professional historian even allowed
to believe that (some) value judgments are objective, that the notion of the rightconcept is in any way coherent? How can we make sweeping claims on behalf of16

a joint paper by Michael Harris, Kai-Wen Lan, Richard Taylor, and Jack Thorne entitled "On therigid cohomology of certain Shimura varieties."
17
The authors of the paper cited in the previous footnote — no doubt like many others working on
similar questions — were actively seeking a p-adic cohomology theory with just these properties.

perfectoid geometry when historical methodology compels us to admit that evencomplex numbers may someday be seen as a dead end? "Too soon to tell," as Zhou
En-Lai supposedly said when asked his opinion of the French revolution.
It's possible to talk sensibly about convergence without succumbing to the illusion of
inevitability. In addition to the historical background sketched above, and the active
search for the right frameworks that many feel Scholze has provided, perfectoid
geometry develops themes that were already in the air when Scholze began his
career.18 With respect to the active research programs that provide a field with its
contours, it's understandable that practitioners can come to the conclusion that a newframework provides the clearest and most comprehensive unifying perspective
available. When the value judgment is effectively unanimous, as it is in the case of
perfectoid geometry, it deserves to be considered as objective as the existence of the
field itself.
A value judgment applied to a new concept thus becomes an inflection or modality ofa judgment of existence. I don't have much time for debates over the independent
existence of the objects of mathematical practice, but I consider the existence ofmathematical practice to be as objective as any other factual judgment. In that sense,
I don't need to worry that the judgments by which my field's practice defines itself
might be seen as tautological; what else could they be?

18

These include the Fargues-Fontaine curve, a unifying construction due to Fontaine and Laurent
Fargues, announced just over a year before Scholze's IAS lectures and generalized by Scholze, andwork of K. Kedlaya and R. Liu on relative p-adic Hodge theory. Scholze described both of these indetail in his 2014 ICM talk.