The Life and Work of Alexander GrothendieckAuthor(s): Piotr Pragacz, Abe Shenitzer, John Stillwell, Roman DudaSource: The American Mathematical Monthly, Vol. 113, No. 9 (Nov., 2006), pp. 831-846Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/27642067Accessed: 14/01/2010 12:19

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=maa.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

http://www.jstor.org

THE EVOLUTION OF ... Edited by Abe Shenitzer and John Stillwell

The Life and Work of Alexander Grothendieck

Piotr Pragacz

What can be humbler than water?

But when it undermines the rocky bank

And presses violently, nothing can resist it,

No obstacle can change its course.

Lao Tzu: Tao Te Ching

1. INTRODUCTION. We tell here the story of Alexander Grothendieck, a man who

changed mathematics during some twenty years of work in functional analysis and

algebraic geometry. He turned seventy-five last year [2003]. The present article was written in April 2004 and is based on two lectures, the first

read at the Konferencja Matematyki Pogl^dowej (Conference on Conceptual Mathe

matics) in Grzegorzewice organized by M. Kordos (in August 2003), and the second

at a meeting of Impanga, ' Hommage ? Grothendieck, organized by the author of the

present article at Centrum Banacha (Banach Center) in Warsaw in January 2004. The

aim of this article is to bring closer to Polish mathematicians the fundamental ideas of

Grothendieck's work.

Alexander Grothendieck was born in Berlin in 1928. His father Alexander Shapiro

(1890-1942) was a Russian Jew from a Hasidic town located in what is now the border

area of Russia, Ukraine, and Belarus. He was a political activist (an anarchist) who

took part in all major European revolutions between 1905 and 1939. In the 1920s and

1930s he lived most of the time in Germany and was active in leftist movements that

fought against the rising tide of Nazism. He made a living as a street photographer. In

Germany he met Hanka Grothendieck, a native of Hamburg. (The name Grothendieck

is plattdeutsch (a North German dialect) for Grosser Deich (German for "big dike")). Hanka Grothendieck worked occasionally as a journalist, but her real interest was in

writing. She gave birth to her son Alexander on March 28, 1928.

Between 1928 and 1933 Alexander lived with his parents in Berlin. When Hitler

came to power, Alexander's parents emigrated to France; Alexander stayed in Ham

burg for the next five years with an adoptive family. He attended an elementary school

and then a gymnasium. In 1939 he joined his parents in France. Soon thereafter his

father was interned in a camp in Vernet. Then the Vichy authorities handed him over

to the Nazis, who murdered him in Auschwitz in 1942.

*This article was translated by Abe Shenitzer with the editorial assistance of the author and John Stillwell. It

originally appeared in the Polish journal Wiadomosci Matematyczne (2004), and we thank the editor, Professor

Roman Duda, for permission to publish this English translation.

ampanga is an abbreviation that denotes The Institute of Mathematics of the Polish Academy of Science

Devoted to Algebraic Geometry.

November 2006] the evolution of ... 831

A very young Alexander Grothendieck

Hanka and Alexander survived the occupation but not without problems. Between

1940 and 1942 they were interned?as "dangerous foreigners"?in the Rieucros camp near Mende in the south of France. Then Hanka was transferred to the Gurs camp in

the Pyrenees, while Alexander studied in a lyc?e called Coll?ge C?venol in Chambon

sur-Lignon in the Cevennes, in the southern part of the Central Massive. The lyc?e was run by local Protestants. They helped many children whose lives were threatened

during World War II to survive the occupation.

Already at that time it became clear that Alexander was exceptionally gifted. He

asked himself the question: How can one measure precisely the lengths of curves, the

areas of plane figures, and the volumes of solids? He continued his attempts to answer

such questions during his university studies in Montpellier (1945-1948) and arrived at

results equivalent to measure theory and the Lebesgue integral.2 J. Dieudonn? wrote

in [D] that the university of Montpellier, at the time when Grothendieck studied there, was not "the right place" for learning about great mathematical problems? In the fall

of 1948 Grothendieck went to Paris and for a year attended lectures as an auditor at the

famous ?cole Normale Sup?rieure (ENS), whose graduates constitute the majority of

the elite of French mathematics. In particular, Grothendieck took part in the legendary seminar of H. Cartan, which in that year dealt with problems in algebraic topology. (For more information about the period of Grothendieck's life described thus far, about

his parents, and about contemporary France, see [C2].)

2. THE FUNCTIONAL ANALYSIS PERIOD. But then Grothendieck's interests

began to focus on functional analysis. Following Cartan's advice, he went to Nancy in October 1949 and joined J. Dieudonn?, L. Schwartz, and others who worked in

functional analysis. They conducted a seminar devoted to the study of Fr?chet spaces and their direct limits and encountered a number of problems that they were unable to

solve. They asked Grothendieck to study these problems. He did. The results exceeded

21 dedicate this story to teachers who will read my article. Pay attention to students who ask themselves

important and natural mathematical questions. They will be the future "Columbuses of Mathematics."

832 ? the mathematical association of America [Monthly 113

their expectations. In less than a year Grothendieck solved the problems in question by means of ingenious constructions. When it came to his doctorate, he had at his disposal six texts, each of which could serve as an impressive doctoral dissertation. The title of

his doctoral dissertation, which he dedicated to his mother,3 was

Produits tensoriels topologiques et espaces nucl?aires

HANKA GROTHENDIECK in Verehrung und Dankbarkeit gewidmet

(dedicated to Hanka Grothendieck in reverence and gratitude). It was given final form

in 1953 and appeared in 1955 in the Memoirs of the American Mathematical Soci

ety [18].4 This dissertation is viewed as one of the most important events in postwar functional analysis.5

Grothendieck worked intensively in functional analysis between 1950 and 1955.

In his first papers (written at age twenty-two) he posed many questions on the struc

ture of topological linear locally convex spaces, in particular, spaces of this kind that were linear, metric, and complete. Some of these spaces are connected with the the

ory of linear partial differential equations and spaces of analytic functions. Schwartz's

kernel theorem suggested to Grothendieck the singling out of the class of nuclear

spaces.6 Roughly speaking, the kernel theorem states that "respectable" operators on

distributions are themselves distributions. Grothendieck stated this fact abstractly as an

isomorphism of appropriate injective and projective tensor products. The fundamen

tal difficulty that arises in connection with the introduction of nuclear spaces is the

question of identity of two interpretations of kernels: as elements of tensor products and as linear operators (in the case of finite-dimensional spaces there is a complete

correspondence between matrices and linear transformations). This question led to the

so-called approximation theorem (first stated in a certain form in S. Banach's famous

monograph [B]), whose deep study takes up a considerable part of his doctoral dis

sertation, the Red Book. Grothendieck discovered many beautiful equivalences (some

implications were known earlier to S. Banach and S. Mazur): he proved, among other

things, the equivalence of the problem of approximation and of Mazur's problem 153

in the Scottish Book [Ma], and the equivalence, for reflexive spaces, of the property of approximation and the so-called metric property of approximation. Nuclear spaces are also connected with the Dvoretzky-Rogers theorem of 1950 (which solved prob lem 122 in [Ma]): in every infinite-dimensional Banach space there is an uncondi

tionally convergent sequence that is not absolutely convergent. Grothendieck showed

that nuclear spaces are spaces in which the unconditional convergence of a sequence is equivalent to its absolute convergence (see [Ma], problem 122 and commentary).

One of the reasons for the fundamental significance of nuclear spaces is that almost

all locally convex spaces that occur naturally in analysis and are not Banach spaces are nuclear spaces. Specifically, we have in mind various spaces of smooth functions,

distributions, and holomorphic functions with their natural topologies. In many cases

their nuclearity was established by Grothendieck.

Another important result in the red booklet is the equivalence of the product defini

tion of nuclear spaces and their definition as inverse limits of Banach spaces with mor

3Grothendieck was very much attached to his mother. He spoke German with her. She wrote poems and

novels. Her best known work is her autobiographical novel Eine Frau.

4See the first item in the bibliography at the end of this article.

5It is known as Grothendieck's (Tittle) Red Book.

6A11 his life Grothendieck was a zealous pacifist. He thought that the word "nuclear" should be used only for abstract mathematical notions. During the Vietnam war, when American pilots bombed Hanoi, he lectured

on category theory in a wood surrounding that city.

November 2006] THE EVOLUTION OF ... 833

phisms that are nuclear or absolutely summing operators (which Grothendieck called

semi-integral on the left). The study of various classes of operators (Grothendieck was

the first person who defined them functorially in the spirit of category theory) led him

to deep results that initiated the modern local theory of Banach spaces. He published these results in two important articles [22], [26] in Bol. Soc. Mat. S?o Paulo during his

stay in that city (1953-1955). In these papers he proved, along with other things, that

operators from the space of measures to Hubert space are absolutely summing (an an

alytically equivalent form of this result is the Grothendieck inequality), and he formu

lated as a conjecture the central problem of the theory of convex fields, proved in 1959

by A. Dvoretzky. Many very difficult questions formulated in these papers were solved

by P. Enflo (negative resolution of the problem of approximation, in 1972), B. Maurey, G. Pisier, J. Taskinen ("probl?me des topologies," dealing with bounded sets in tensor

products), U. Haagerup (the noncommutative analogue of the Grothendieck inequality for C*-algebras), and Fields medalist J. Bourgain. The work of these people influenced

indirectly the results obtained by T. Gowers, another "Banach-type" Fields medalist.

Apparently, of all the problems in functional analysis posed by Grothendieck, only one

remains unsolved (see [PB, 8.5.19]). To sum up, Grothendieck's contribution to functional analysis includes the follow

ing: nuclear spaces, topological tensor products, the Grothendieck inequality and its

connection with absolutely summing operators, and many other scattered results.7

We note that probably the best-known paper [LP] on absolutely summing operators

by A. Pelczy?ski (with J. Lindenstrauss) is based on Grothendieck's (very difficult to

read) paper [22]. That Grothendieck's ideas entered the theory of Banach spaces is

largely due to this paper.

3. HOMOLOGICAL ALGEBRA AND ALGEBRAIC GEOMETRY. In 1955 Grothendieck's interests shifted to homological algebra. At that time, due to the pa

pers of H. Cartan and S. Eilenberg, homological algebra flourished as a powerful tool of algebraic topology. In 1955, during his stay at the University of Kansas,

Grothendieck developed the axiomatic theory of Abelian categories. His main result

states that sheaves of modules form an Abelian category with sufficiently many in

jective objects, which allows one to define cohomologies with values in such a sheaf

without restrictions as to the kind of sheaf or the kind of base space (this theory was

published in the Japanese journal T?hoku; see [28]). Grothendieck's next area of interest was algebraic geometry. Here his contacts with

C. Chevalley and J.-P. Serre were a significant source of influence. Grothendieck re

garded Chevalley as a personal friend and took part, in the years that followed, in his

famous seminar at ENS by giving a number of lectures on algebraic groups and the

theory of intersections [81]-[86]. As for Serre, who had a vast knowledge of algebraic

geometry, Grothendieck treated him as an inexhaustible source of relevant information

and plied him with questions (recently, the French Mathematical Society published a

substantial extract of the correspondence between these two mathematicians; from this

book one can learn more algebraic geometry than from many a monograph). Serre's

paper [SI], in which he built the foundations of the theory of sheaves and their coho

mologies in algebraic geometry, was of key significance for Grothendieck.

One of Grothendieck's first results in algebraic geometry was his classification

of holomorphic bundles over the Riemann sphere [25]. This result states that ev

ery such bundle is a direct sum of a certain number of tensor powers of a tauto

logical line bundle. Some time after the publication of this paper it turned out that

7The information about Grothendieck's contribution to functional analysis is derived mainly from [P].

834 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113

other "embodiments" of this result were known earlier to mathematicians such as

G. Birkhoff and D. Hilbert, as well as R. Dedekind and H. Weber (1892). This story illustrates two points: Grothendieck's intuitive grasp of what are important mathemat

ical problems and his lack of knowledge of the classical mathematical literature. In

fact, Grothendieck was anything but a bookworm?he preferred to learn mathematics

by talking to fellow mathematicians. Be that as it may, Grothendieck's paper initiated

the systematic study and classification of bundles over projective spaces and other va

rieties.

Grothendieck worked in algebraic geometry between 1956 and 1970. At the begin

ning of this period his leading objective was the transformation of "absolute" theorems

(about varieties) to relative theorems (about morphisms). Here is an example of an ab

solute theorem:8

If X is a complete (say projective) variety and T is a coherent sheaf on X (for example, a

sheaf of sections of a vector bundle), then dim Hj (X, T) < oo.

And here is the relative version of this result:

If f : X ?> Y is a proper morphism (for example, a morphism between two projective

varieties) and T is coherent on X, then 7Zj f+T is coherent on Y.

Grothendieck's main achievement during that period was connected with the rel

ative theorem of Hirzebruch-Riemann-Roch. The initial problem that motivated the

work on this issue can be formulated as follows: given a smooth connected projective

variety X and a vector bundle E over X, compute dim H?iX, E), that is, the dimen

sion of the space of global sections of E. Serre's sophisticated intuition suggested to

him the need to reformulate this problem by inclusion of higher cohomology groups.

Specifically, Serre formulated the hypothesis that the number

Y^(-iy dimH?(X,E)

must be expressible in terms of topological invariants connected with X and E. Of

course, Serre's starting point was his reformulation of the classical Riemann-Roch

theorem for the curve X: for a divisor D and the line bundle CiD) associated with it

we have:

dim H?iX, CiD)) - dim Hl (X, CiD)) = deg D + ~x(X) (1)

(a similar formula was also known for surfaces). This [Serre's] conjecture was proved (in 1953) by F. Hirzebruch, who was inspired

by earlier inventive computations of J. A. Todd. Here is the formula discovered by Hirzebruch for a variety X of dimension n :

?(-iy dimHliX, E) = deg(ch(?)td(X))2n,

where (?)2n denotes a component of degree 2n of an element in the cohomology ring

H*(X) and

ch(E) = ?>. tdiX) =

Y\r^j 8In the sequel I will use standard concepts and notations of algebraic geometry (see [H]). Unless otherwise

indicated, a variety is a complex algebraic variety. If no other sheaf of coefficients is clearly indicated, the field

of coefficients of the cohomology groups of such a variety will be the field of rational numbers.

November 2006] the evolution of ... 835

(the a? are the Chern roots9 of E, and the Xj are the Chern roots of the tangent bundle

TX). To formulate the relative version of this result we assume that we have a proper mor

phism / : X -> F between smooth varieties. We want to understand the connection between

cM-)td(X)

and

Chy(-)td(F),

"induced" by /. If / : X -> Y = point, we should obtain the Hirzebruch-Riemann

Roch theorem. The relativization of the right side of (1) is simple: there is a well defined additive mapping of the cohomology groups, /* : HiX) -> //(F), and

deg(z)2/7 corresponds to /*(z) for z in HiX). How do we relativize the left side of (1)? The W f*T are coherent modules that

vanish for j ^> 0, and these are the relative version of Hj (X, T). To relativize the al

ternating sum, Grothendieck defined the group KiY) (now known as the Grothendieck

group). This is a factor group of the "very large" free Abelian group generated by the

isomorphism classes [J7] of coherent sheaves by F, modulo the relations

tn = it'] + it"}

when there exists an exact sequence

0 -? T' -? T -? T" -> 0. (2)

The group K(Y) satisfies the following universality property: an arbitrary mapping <p from 0 Z[T] to an Abelian group that satisfies

<p([T]) = <p([T']) + q>([T"]) (3)

is factored by K(Y). In our case, we define

<pi[T]) = ]?(-l)W/*^

e K(Y).

Note that (3) follows from the long exact sequence of derived functors

-> WUT' -> WUT -? W UT" ->

1Zj+xUT' -* -

associated with the short exact sequence (2) (see [H, chap. 3]). Thus we have an addi tive mapping

fi:K(X)-+ K(Y).

Now the relative Hirzebruch-Riemann-Roch theorem, discovered by Grothendieck

[102], [BS] and bearing the mark of genius, asserts the commutativity of the diagram

K(X) ?^-? K(Y)

chx(-)td X chy(-)td Y

HiX) ?^ //(F) 9These are classes of divisors associated with line bundles that split the bundle E (see [H, p. 430]).

836 ? the mathematical association of America [Monthly 113

(Note that, owing to additivity, the Chern character ch(-) is well-defined in K- theory.) More information about various aspects of the theory of intersections, whose crowning

result is the Grothendieck-Riemann-Roch theorem just described, can be found in [H,

Supplement A]. This theorem has found many applications in concrete computations of characteristic classes.

Grothendieck's ?T-group initiated the evolution of K-theory, with papers by D. Quillen and many other mathematicians. We note that ?T-theory plays a signif icant role in many areas of mathematics, beginning with the theory of differential

operators (the Atiyah-Singer index theorem) and ending with the modular theory of

representations of finite groups (the Brauer theorem).10 After this spectacular result Grothendieck was described as a superstar of alge

braic geometry and invited to the International Mathematical Congress in Edinburgh in 1958, where he sketched a program for defining a theory of homology over a field

of positive characteristic that might lead to a proof of A. Weil's conjectures (see [32]). The Weil conjectures [W] suggested the existence of deep connections between the

arithmetic of algebraic varieties over finite fields and the topology of algebraic vari

eties over the field of complex numbers. Let k = ?q be a finite field with q elements

and k its algebraic closure. Take a finite system of homogeneous polynomials in n + 1

variables with coefficients in k. Let X (respectively, X) be the set of zeros of this

system in n-dimensional projective space over k (respectively, k). Let Nr denote the

number of points of X whose coordinates are in the field ?c/r with qr elements for r ? 1, 2,_We "organize" the numbers Nr into a generating function Z called the

zeta-function of X,

(oo tr\

For a smooth manifold X the Weil conjectures speak of the properties of Z(t) and of

the connection between the classical Betti numbers and the complex variety "asso

ciated" with X. The formulation of Weil's conjectures is dealt with in 1.1-1.4 of

[H, Addendum C] and in W1-W5 of [M, chap. 6, sec. 12] (both lists begin with the

conjecture of the rationality of the function Z(t)). In these lists one also finds more

information leading into the body of problems associated with Weil's conjectures. In addition to Weil and the Grothendieck school, B. Dwork, J.-P. Serre, S. Lubkin, S. Lang, Yu. Manin, and many others, have worked on the Weil conjectures.

4. IHES. The Weil conjectures became the primary motivation for Grothendieck's

work in algebraic geometry during his stay at IHES.11 Grothendieck began work at

IHES in 1959. He brought into being the S?minaire de g?om?trie alg?brique du Bois

Marie (IHES was located in a wood by that name). During the next decade the seminar

became the world center of algebraic geometry. Grothendieck worked on mathematics

twelve hours a day. He generously shared his mathematical ideas with his collabora

tors. The atmosphere of this extraordinary seminar is well described in an interview

1()M. Atiyah (item [A] in the bibliography) stresses the important role of Grothendieck's pioneering introduction of A'-theory into mathematics. Contrary to Atiyah's suggestion in [A], Grothendieck's work

proves that there is no fundamental dichotomy between algebra and geometry (one should also mention that

Grothendieck's mathematical inspirations did not come from physics and were, for the most part, "of an alge braic nature").

1 ' IHES = Institut des Hautes ?tudes Scientifiques in Bures-sur-Yvette near Paris. A great place for the study of mathematics, partly because of its charming canteen, which never lacked for bread and wine.

November 2006] the evolution of ... 837

Music pavilion at IHES: this is where the first seminars in algebraic geometry took place.

with J. Giraud, one of Grothendieck's students. Let us give an account of some of

Grothendieck's most important ideas during that period.12 "Schemes" are objects that make possible the unification of geometry, commutative

algebra, and number theory. Let X be a set and F a field. Consider the ring

Fx = {functions / : X -* F]

with multiplication "by values." For jc in I we define ax : Fx -> F by / h? fix). The kernel of ax is a maximal ideal; this enables us to identify X with the set of

maximal ideals in Fx. Variants of this idea appeared earlier in M. Stone's papers on

Boolean lattices and in I. M. Gelfand's papers on commutative Banach algebras. In

commutative algebra ideas of this kind appeared for the first time in papers by M. Na

gata and E. Kahler. In the late 1950s many mathematicians in Paris (for example, Cartan, Chevalley, Weil, and others) tried to generalize the idea of an algebraic variety over an algebraically closed field.

Serre showed that the notion of localization of a commutative ring leads to a sheaf over the spectrum Specm of maximal ideals of an (arbitrary) commutative ring. We

note that A i-> Specm i A) is not a functor (the preimage of a maximal ideal need not

be maximal). On the other hand, the correspondence

A h^ Spec A := {prime ideals in A]

is a functor. It seems that P. Cartier was the first to suggest, in 1957, that the appropriate

generalization of the classical algebraic variety is the ringed space (X, Ox) locally

isomorphic to Spec A (this suggestion was the fruit of speculations of many algebraic

geometers). Such an object was named a scheme.

12 See also [D], which contains a more detailed account of the theory of schemes.

838 ? the mathematical association of America [Monthly 113

Grothendieck intended to write a thirteen-volume account of algebraic geometry? EGA13?based on schemes and ending with a proof of Weil's conjectures. Grothen

dieck and Dieudonn? wrote and published four volumes of EGA. A great deal of the

material that was to appear in the remaining volumes appeared in SGA14?publications of the seminar on algebraic geometry at IHES. (The textbook [H], which we refer to

often, is a didactic condensation of the most useful information on EGA bearing on

schemes and cohomology.) We will now deal with constructions in algebraic geometry that make use of repre

sentaba functors. Consider an object X in a category C. We associate with it a con

travariant functor from C to the category of sets,

hxiY)=MovciY,X).

At first sight it is difficult to see the value of such a simple association, but knowledge of this functor determines uniquely (up to isomorphism) the object X that represents it (this is the content of Yoneda's lemma). Hence it is natural to adopt the following

definition: a contravariant functor from C to the category of sets is said to be repre sentable (by X) if it is of the form hx for a certain object X in C. Grothendieck used

the properties of representable functors in a masterly way to construct various param eter spaces. We frequently encounter such spaces in algebraic geometry. A splendid

example is the Grassmanian, which parametrizes the linear subspaces of specified di

mension in a specified projective space. Here it is natural to ask if there exist more

general schemes that parametrize the subvarieties of a specified projective space with

specified numerical invariants. Let S be a scheme over a field k. Let X cP" xk S be a closed subscheme with the

natural morphism X -> S. We will call it a family of closed subschemes of the space Fn with base S. Let P be a numerical polynomial. Grothendieck considered the func

tor typ from the category of schemes to the category of sets that associates with the

scheme S the set *I>P(S) of flat closed subschemes P" with base S and Hubert poly nomial P. If / : S' -+ S is a morphism, then ^Pif) : ̂ PiS) -> typ(S') associates

with the family X ?> S the family X' = X xs S' -> S'. Grothendieck proved that the

functor typ is represented by a scheme (called the Hubert scheme) that is projective [74].15 This result is (very) noneffective?for example, there remains the open prob lem of [finding] the number of irreducible components of Hubert schemes of curves

in three-dimensional projective space with specified genus and degree. Nevertheless, in many geometric arguments it is enough to know that such an object exists, and that

is why Grothendieck's theorem has many applications. More generally, Grothendieck

constructed a so-called Quot scheme that parametrizes (flat) quotient schemes of a

specified coherent scheme with a specified Hilbert polynomial. Quot schemes have

many applications in the construction of spaces of modules of vector bundles. Another

scheme constructed by Grothendieck along these lines is the Picard scheme [75], [76]. In 1966 Grothendieck was awarded the Fields medal for his contribution to func

tional analysis, for the Grothendieck-Riemann-Roch theorem, and for his contribution to the theory of schemes (see [S2]).

5. ?TALE COHOMOLOGY. The most important topic of Grothendieck's investi

gations at IHES was the theory of ?tale cohomologies. We recall that Weil's conjee

]3EGA-?l?ments de g?om?trie alg?brique, published by IHES and Springer-Verlag [57]-[64].

14SGA-S?minaire de g?om?trie alg?brique, published in the series Springer Lecture Notes in Mathematics

and (SGA 2) by North-Holland [97]-[103]. 15In fact, Grothendieck published a far more general result for projective schemes over a Noetherian base

scheme.

November 2006] the evolution of ... 839

Alexander Grothendieck

tures required the construction of a counterpart to the cohomology theory of complex varieties for algebraic varieties over a field with positive characteristic (but with coef

ficients in a field of characteristic zero, so that one could compute the number of fixed

points of a morphism as the sum of traces on cohomology groups ? la Lefschetz). There

were unsuccessful earlier attempts to utilize for this purpose "classical" topology, the

Zariski topology used in algebraic geometry (closed subsets = algebraic subvarieties),

which turned out to be too coarse for homological purposes. Grothendieck observed

that one could construct a "good" homology theory by considering a variety with all

its unramified coverings (see [32], which contains a description of the detailed context

of this discovery). This was the beginning of the theory of ?tale topology, developed

jointly with M. Artin and J.-L. Verdier. Grothendieck's brilliant idea was his revolu

tionary generalization of the notion of topology that differed from the classical notion

of a topological space in that "open sets" are not all contained in a single set but have

basic properties that make it possible to construct a "satisfactory" theory of the coho

mology of sheaves.

A sketch of the source of these ideas is found in the following discussion by Cartier

[CI]. When we use sheaves on a variety X, or study cohomologies of X with coeffi

cients in sheaves, a key role is played by the lattice of open sets in X (the points of

X play a secondary role). Thus, in this discussion, we can "replace," without much

loss, the variety with its lattice of open sets. Grothendieck's idea was an adaptation of Riemann's idea that, strictly speaking, many-valued holomorphic functions inhabit

not open sets in the complex plane but appropriate Riemann surfaces that cover it

(Cartier uses the suggestive description "les surfaces de Riemann ?tal?es"). There are

projections between Riemann surfaces, and in this connection they form the objects of a certain category. A lattice is an example of a category in which there is at most

one morphism between any two objects. Hence Grothendieck proposed replacing the

lattice of open sets with the category of open ?tale sets. After its adaptation to alge braic geometry this idea resolved the fundamental difficulty of the lack of an implicit function theorem for algebraic functions. It also made possible a functorial way of

considering ?tale sheaves.

840 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113

Let us continue our discussion in a mathematically more formal manner. Let C be a

given category in which there exist fiber products. To define a Grothendieck topology on C is to define for each object X of C a set CovZ of families of morphisms {f :

Xi -> X}iei called coverings of X that satisfy the following conditions:

(1) {id : X -> X] belongs to CovZ;

(2) if {fi : X -> X} is in CovZ, then the family [Xt xx Y -> F}, obtained from it

by a change of basis F ?> X belongs to CovF;

(3) if {Zz- -^ Z} is a member of CovZ and if {Z/;- -> Z/} is in CovZ for each /,

then the doubly-indexed family {Zijf- -> X} belongs to CovZ.

If direct sums are defined in C?assume this to be the case?then the family {Xt -> Z} can be replaced with the single morphism

i

Specification of coverings makes it possible to speak of sheaves and their cohomolo

gies. The contravariant functor F from C to the category of sets is called a sheaf of sets

if for an arbitrary covering X' ?> X the following condition holds:

FiX) = {s' e FiX') : p*(sf)

= p*(sf)};

here px and p2 are two projections of X' xx X! onto X'. By the canonical topology in

the category C we mean the topology "richest in coverings," in which all representable functors are sheaves. If, conversely, an arbitrary sheaf in the canonical topology is a representable functor, then the category C is called a topos. More information on

Grothendieck topologies is found, among other places, in [BD]. Let us go back to geometry. It is important?in fact, very important?that the mor

phisms fi need not be embeddings! The most important example of a Grothendieck

topology is ?tale topology, in which f : X? -> X are ?tale morphisms16 that induce a surjection LLZ;

-> Z. When applied to this topology, the cohomological machin

ery constructed earlier leads to the construction of ?tale cohomologies //?(Z, ?). The

basic ideas are relatively simple but the verification of the many technical details of

the properties of ?tale cohomologies required years of work by Grothendieck's "coho

mological" students: P. Berthelot, P. Deligne, L. Illusie, J. P. Jouanolou, J.-L. Verdier, and others, who filled in the details of ever new results sketched by Grothendieck. The

papers of the Grothendieck school dealing with ?tale cohomologies are published in

[100].17 The proof of the Weil conjectures called for a variant of ?tale cohomologies called

l-adic cohomologies. Their basic properties, and especially a formula of the Lefschetz

type, enabled Grothendieck to prove some of Weil's conjectures but not the most diffi

cult one, an analogue of the Riemann hypothesis. In its proof the role of Grothendieck

was similar to that of the biblical Moses, who led the Israelites out of Egypt towards

the promised land: he was their guide for a large part of the road but was not to reach

the final goal. In the case of the Weil-Riemann conjecture the final goal was achieved

by Deligne, Grothendieck's most gifted student. (Grothendieck's plan of proving the

,6These are smooth morphisms of relative dimension zero. For smooth varieties, these are morphisms that

induce isomorphisms on spaces tangent at all points?of course, such morphisms need not be monomorphisms. A general discussion of ?tale morphisms is found in [M].

17 A didactic account of ?tale cohomologies is found in [M].

November 2006] the evolution of ... 841

Weil-Riemann conjecture by proving the so-called standard conjectures has not yet been realized; these conjectures are discussed in [44].)

6. RETURN TO MONTPELLIER. In 1970 Grothendieck accidentally discovered

that some of the money for IHES came from military sources. He immediately left

IHES. He was offered a position by the prestigious College de France. However, by then (he was about forty-two) there were things that interested him more than math

ematics: he felt that one should save the world that faced a variety of threats. He

formed an ecological group called Survivre et Vivre (Survive and Live). C. Cheval

ley and P. Samuel, two great mathematicians and his friends, joined his group. Be

tween 1970 and 1975 the group published ajournai under the same title. As always, Grothendieck engaged in this work body and soul. His lectures at College de France

had little to do with mathematics and a lot with how to avoid a world war and how

to live ecologically. The result was that he had to look for another position. His alma

mater, the University of Montpellier, offered him a professorship. He settled on a farm near Montpellier and taught at the university as a kind of "didactic foot soldier." But

he also wrote a few (long) sketches of new mathematical theories, hoping to get a job at CNRS18 and good students from ENS as collaborators. He did not "get" good stu

dents from ENS, but four years before retirement (at age sixty) he obtained a position at CNRS. His sketches are now being developed by a few groups of mathematicians;

what these sketches are about calls for another article. When in Montpellier, Grothendieck wrote his "mathematical" diary R?coltes et Se

mailles iHarvests and Sowings) [Gl]. It contains fantastic fragments that describe his

way of seeing mathematics, references to the "male" and "female" in mathematics,

and scores of other fascinating things. The diary contains an extensive description of

his relations with the mathematical community and a highly critical evaluation of his

former students. But let's talk about pleasant things. Without hesitation Grothendieck names E. Galois his model mathematician. It should be noted that the life story of

Galois by the physicist L. Infeld, Whom the Gods Love,]9 made a strong impression on young Grothendieck. Grothendieck speaks with great warmth of J. Leray, A. An

dreotti, and C. Chevalley. It is characteristic of Grothendieck that the human aspect of

his contacts with other mathematicians has always been extremely important for him.

At a certain point in [Gl] he writes:

Si dans R?coltes et Semailles je m'addresse ? quelqu'un d'autre encore qu'? moi m?me, ce

n'est pas ? un "public." Je m'y addresse ? toi qui me lisse comme ? une personne, et ? une

personne seule. (Roughly: If in Harvests and Sowings I address someone other than myself, then it is certainly not "the public." I address you, the reader, y o u, an individual.)

It is possible that the experience of loneliness that accompanied him all his life made

him so sensitive on this point. In 1988 Grothendieck and Deligne were jointly awarded the prestigious Crafoord

Prize of the Swedish Royal Academy of Sciences. The prize carries with it a large amount of money. Grothendieck turned it down. His letter to the Swedish Academy included the following fragment, which I think is of key importance:

Je suis persuad? que la seule ?preuve d?cisive pour la f?condit? d'id?es ou d'une vision nou

velles est celle du temps. La f?condit? se reconnait par la prog?niture, et non par les honneurs.

18CNRS-Centre National de la Recherche Scientifique employs researchers without official teaching duties. 19 Another of my appeals to teachers: this book should be recommended reading (not only) for senior "gym

nasium" students interested in mathematics.

842 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113

(To my mind, time provides the only proof of the fruitfulness of new ideas or visions. Fruit

fulness is demonstrated by the fruit and not by honors.)

Let me add that the letter includes a highly critical evaluation of the professional ethics

of the mathematical community in the 1970s and 1980s.

7. CONCLUSION. Time to sum up. Here is a list of the twelve most important topics of Grothendieck's mathematical work (translated from the French text in [Gl]; I have

included a few comments):

(1) Topological tensor products and nuclear spaces

(2) "Continuous" and "discrete" dualities (derived categories, the "six opera tions")

(3) Riemann-Roch-Grothendieck yoga (/^-theory and its relationship to intersec tion theory)

(4) Schemes

(5) Topos theory (As I have already indicated, toposes, unlike schemes, provide a "geome

try without points"?see also [Cl] and [C2]. Grothendieck bestowed greater "love" on toposes than on schemes. He valued most the topological aspects of

geometry that led to appropriate theories of cohomology.)

(6) Etale and /-adic cohomology

(7) Motives and motivic Galois groups (^-Grothendieck categories)

(8) Crystals and crystalline cohomology, yoga of the de Rham coefficients, Hodge coefficients

(9) "Topological algebra": oo-stacks, derivations, cohomological formalism of

toposes, inspiring a new conception of homotopy

(10) Mediated topology

(11) The yoga of an Abelian algebraic geometry; Galois-Teichmiiller theory (Grothendieck regarded this item as the most difficult and "deepest" one.

The most recent results connected with this topic have been obtained by F.

Pop.)

(12) Schematic or arithmetic views of regular polyhedra and, in general, of all reg ular configurations

(Grothendieck worked on such issues after his move from Paris to Mont

pellier in moments free from work on the family wine farm.)

Many mathematicians have continued to work on topics 1-12, and their work con

stitutes a substantial chunk of mathematics of the end of the twentieth century. Many of

Grothendieck's ideas are being actively developed and are bound to have an important influence on mathematics in the twenty-first century.

Let me mention the names of the most important mathematicians (some of whom are Fields medalists) who have continued the work of Grothendieck:

(1) P. Deligne, who gave a complete proof of Weil's conjectures in 1973 (using to

a considerable extent the techniques of SGA);

(2) G. Faltings, who proved the Mordell conjecture in 1983;

(3) A. Wiles, who proved Fermat's Last Theorem in 1994;

(It is hard to imagine that the last two results could have been accomplished without EGA.)

November 2006] THE EVOLUTION OF ... 843

(4) V. Drinfeld and L. Lafforgue, who established the Langlands correspondences for the general linear group over function fields;

(5) V. Voevodsky, who is responsible for the motivic theory and the proof of the

Milnor conjecture.20

The item in 5 is connected with Grothendieck's "dream" notion that there should be an

"Abelianization" of the category of algebraic varieties?a category of motives with mo

tivic cohomologies from which one could read off the Picard variety, the Chow group, and so on. A. Suslin and V. Voevodsky constructed motivic homologies satisfying the

Grothendieck postulates. In August 1991 Grothendieck suddenly left his home without informing anyone and

went to some place in the Pyrenees. He engaged in philosophical meditations (free

choice, determinism, the existence of the devil; before that he wrote an interesting item, titled La clef des songes, on how he arrived at the existence of God on the basis

of an analysis of his dreams). He also wrote about physics. He did not want to have

contacts with the outside world.

We end with a few reflections.

We quote Grothendieck's statement (in [Gl]) about what has most fascinated him

in mathematics:

C'est dire s'il a une chose math?matique qui (depuis tojours sans doute) me fascine plus que

toute autre, ce n'est ni "le nombre," ni "la grandeur," me toujours la f o r m e . Et parmi les mille-et-un visages que choisit la forme pour se r?v?ler ? nous, celui qui m'a fascin?

plus que tout autre et continue ? me fasciner, c'est la structure cach?e dans la choses

math?matiques. (If there is anything in mathematics which fascinates me more than other

things (and has always fascinated me) it is neither "number" nor "magnitude" but "form."

And of the thousand and one visages chosen by form to appear before us, the one that has

always fascinated me most, and continues to fascinate me, is the structure of mathematical

objects.)

It is amazing that the fruit of Grothendieck's reflections on form and structure are

theories that yield tools (of precision never encountered before) for the computation of

concrete numerical magnitudes and for finding explicit algebraic relations. An example of such a tool in algebraic geometry is the Grothendieck-Riemann-Roch theorem. The

following is a less well-known example: the language of Grothendieck X-rings [102] makes it possible to treat symmetric functions as operators on polynomials. In turn, this makes possible a uniform approach to a number of classical polynomials (such as symmetric and orthogonal polynomials) and formulas (for example, interpolation formulas, or formulas obtained from the representation theory of the general linear

group and the symmetric group). These polynomials and formulas are often connected

with the names of mathematicians such as E. B?zout, A. Cauchy, A. Cayley, P. Cheby shev, L. Euler, C. F. Gauss, C. G. Jacobi, J. Lagrange, E. Laguerre, A-M. Legendre,

I. Newton, I. Schur, T. J. Stieltjes, J. Stirling, J. J. Sylvester, J. M. Hoene-Wro?ski, and

others. Moreover, the language of ?-rings makes it possible to find useful algebraic combinatorial generalizations of the results of these classical mathematicians (see

[La]). Grothendieck's work shows that there is no fundamental dichotomy between

the quantitative and qualitative aspects of mathematics. There is no doubt that this viewpoint has helped Grothendieck to do a prodigious

job of unification of important topics in geometry, topology, arithmetic, and complex

analysis. This viewpoint is also connected with his fondness for studying mathematical

problems in maximal generality. 20 A detailed discussion of items 4 and 5 is found in the articles [L] and [CW] in Wiadomosci Matematyczne.

844 ? the mathematical association of America [Monthly 113

Grothendieck told a tale (see [Gl]) that reflects his style of work. Suppose we want

to prove a theorem that is a conjecture. There are two radically different ways of trying to do this. One is by brute force, the kind of thing we do when we use a nutcracker to

split a nutshell to get to the nut inside. But there is another way. We put the nut in a

glass of softening fluid and wait patiently for some time. Then slight finger pressure suffices for the nut to open by itself. Readers of Grothendieck's papers do not doubt

that the second method was his method of doing mathematics. Cartier [Cl] gives an

even more suggestive characterization of this method: it is Joshua's method of destroy

ing the walls of Jericho. If we go around the walls of Jericho sufficiently many times

and weaken their construction (by resonance), then in the end it will be enough to blow

the trumpets, give a loud shout, and the walls of Jericho will fall down.

Here is a remark I want to share with young mathematicians. Grothendieck attached

great importance to writing down his mathematical thoughts. He viewed the process of writing down and editing his mathematical texts as an integral part of his creative

work (see [He]). What follows is a comment by Dieudonn?, the faithful witness of Grothendieck's

work and a mathematician with an encyclopedic knowledge of mathematics. On the

occasion of Grothendieck's sixtieth birthday (that is, some fifteen years ago) he wrote

(see [D]):

There are few examples in mathematics of a theory so monumental and fruitful, accomplished

by one person in so short a time.

The editors of the Grothendieck Festschrift [C-R] (which includes the article [D]) echo Dieudonn?'s sentiment. They wrote:

It is difficult to grasp fully the scope of Grothendieck's contribution to, and influence on,

20th-century mathematics. He changed our way of thinking about many areas of mathematics.

Many of his ideas, revolutionary at the moment of their creation, seem now so natural as if

they had always been present in mathematics. In fact, there is a whole new generation of

mathematicians for whom Grothendieck's ideas are part of the mathematical landscape, a

generation which cannot imagine mathematics without Grothendieck's contribution.

While preparing this article, I asked a few French mathematicians with whom I

am acquainted whether Grothendieck was still alive. A brief version of their answers

is: "Unfortunately, the only news we will have about Grothendieck is the news of

his death. Since we have not received it, he must be alive." On March 28, 2004, Grothendieck turned 76.

The bibliography connected with Grothendieck's work is huge and cannot be given in my modest article. I quote only bibliographical items mentioned in the text. In these

items one can find more detailed references to the Grothendieck bibliography and to

the papers by other authors who wrote about him and his work. I warmly recommend

consulting the website

http : //www. grothendieck-circle. org/

On that page one can find many interesting items, more specifically, mathematical and

biographical data referring to Grothendieck and his parents.

ACKNOWLEDGMENTS. My warm thanks to Marcin Champnik, Pawel Doma?ski, and Adrian Langer for

their critical reading of an earlier version of this article and for their valuable remarks which have enriched the

present version; to Bronislaw Jakubczyk, Wojciech Pieczy?ski, and Jerzy Trzeciak for their help in translating

fragments from [Cl], [Gl], and [G2]; to Tadeusz Nadzieja for asking me to publish this article in Wiadomosci

November 2006] THE EVOLUTION OF ... 845

Matematyczne; and to Jan Krzysztof Kowalski for time-consuming editorial work connected with the publica tion of my article. I also wish to thank Marie-Claude Verne for allowing me to publish the photo of IHES, and

to "The Grothendieck Circle" for their permission to publish the photo of Alexander Grothendieck.

References

A numerical reference indicates a Grothendieck publication with this number in The Grothendieck Festschrift, vol. 1, P. Cartier et al., eds., Progress in Mathematics, no. 86, Birkh?user, Boston, 1990, pp. xiii-xx; see also

http://www.math.columbia.edu/lipyan/Groth.Biblio.pdf

A. M. Atiyah, Mathematics in the 20th Century, in Mathematical Evolutions, A. Shenitzer and J. Stillwell,

eds., Mathematical Association of America, Providence, 2002, pp. 1-15. [The Atiyah article was based

on a transcript of a recording of the author's Fields Lecture at The World Mathematical Year 2000

Symposium, Toronto, June 7-9, 2000.] B. S. Banach, Th?orie des op?rations lineares, Monografje Matematyczne, vol. 1, Warszawa, 1932.

BS. A. Borel and J.-P. Serre, Le th?or?me de Riemann-Roch (d'apr?s Grothendieck), Bull. Soc. Math.

France 86 (1958) 97-136.

BD. I. Bucur and I. Deleanu, Introduction to the Theory of Categories and Functors, Wiley and Sons,

London, 1968.

C-R. P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, eds., The Grothendieck

Festschrift, Progress in Mathematics, no. 86, Birkh?user, Boston, 1990.

C1. P. Cartier, Grothendieck et les motifs, in preprint, IHES/M//00/75.

C2. P. Cartier, A mad day's work: From Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001) 389-408.

CW. M. Chalupnik and A. Weber, Motywy Vladimira Voevodskiego, Wiadom. Mat. 39 (2003) 27-38.

CS. P. Colmes and J.-P. Serre, eds., Correspondance Grothendieck-Serre, Documentes Math?matiques 2, Soc. Math, de France, Paris, 2001.

D. J. Dieudonn?, De l'analyse fonctionnelle aux fondements de la g?om?trie alg?brique, in [C-R], pp. 1-14.

Du. E. Dumas, Une entrevue avec Jean Giraud, ? propos d'Alexandre Grothendieck, J. Maths. 1 (1994) 63-65.

Gl. A. Grothendieck, R?coltes et Semailles; R?flexions et t?moignages sur un pass? de math?maticien,

preprint, Universit? des Sciences et Techniques du Languedoc (Montpellier) et CNRS, 1985.

G2. -, Les d?rives de la "science officielle," Ee Monde (March 5, 1988); see also Math. Intelligencer

11(1989)34-35. H. R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., no. 52, Springer-Verlag, New York,

1977.

He. A. Herreman, D?couvrir et transmettre, in preprint IHES/M//00/75.

L. A. Langer, Program Langlandsa wedlug Lafforgue'a, Wiadom. Mat. 39 (2003) 39-46.

La. A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, American Mathe

matical Society, Providence, 2003.

LP J. Lindenstrauss and A. Pelczy?ski, Absolutely summing operators in Ep spaces and their applications, Studia Math. 29 (1988) 275-326.

Ma. R. D. Mauldin, ed., The Scottish Book. Mathematics from the Scottish Caf?, Birkh?user, Boston, 1981.

M. J. S. Milne, Etale Cohomology, Princeton University Press, Princeton, 1981.

P. A. Pelczy?ski, Eist do autora, 20.03.2004.

PB. P. P?rez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland, Amsterdam, 1987.

51. J.-P. Serre, Faisceaux alg?briques coh?rents, Ann. of Math. 61 (1955) 197-278.

52. -, Rapport au comit? Fields sur les travaux de A. Grothendieck, K-theory 3 (1989) 199-204.

W. A. Weil, Number of solutions of equations over finite fields, Bull. Amer. Math. Soc. 55 (1949) 497-508.

Instytut Matemaryczny PAN, Sniadeckich 8, 00-956 Warszawa, Poland

P. Pragacz @ impan. gov.pl

846 ? the mathematical association of America [Monthly 113