Promenade 16
Motives-Or The Inner Heart
The "topos" theme came from that of "schemes" in the year of their appearance; yet it has greatly surpassed the mother notion in its extent. It is the topos, not schemes, which is the "bed", or that "deep river", in which the marriage of geometry, topology and arithmetic, mathematical logic, the theory of categories, and that of continous and discontinuous or "discrete" structures, is celebrated. If the theme of schemes is at the heart of the new geometry, the theme of the topos envelopes it as a kind of residence. It is my grandest conception, devised in order to grasp with precision, in the same language rich in resonances of geometry, an "essence" common to the most disparate situations, coming from every region of the universe of mathematical objects.
Yet the topos has not known the good fortune of the schemes. I discuss this subject in several places in Récoltes et Semailles, and this is not the place at which to dwell upon the strange adventures which have befallen this concept. However, two of the principal themes of the new geometry have derived from that of the topos, two "cohomological theories" have been conceived, one after the other, with the same purpose of providing an approach to the Weil conjectures: the étale ( or l-adic) theme , and the crystalline theme.
The first was given concrete form in my hands as the tool of l-adic cohomology, which has been shown to be one of the most powerful mathematical tools of this century.
As for the crystalline theme, (which had been reduced since my departure to a virtually quasi-occult standing) , it has finally been revitalized ( under the pressure of necessity), in the footlights and under a borrowed name, in circumstances even more bizarre than those which have surrounded the topos.
As predicted, it was the tool of l-adic cohomology which was needed to solve the Weil conjectures. I did most of the work, before the remainder was accomplished, in a magistral fashion, 3 years after my departure, by Pierre Deligne, the most brilliant of all my "cohomological" students.
Around 1968 I came up with a stronger version, (more geometric above all), of the Weil conjectures. These are still "stained" ( if one may use that expression) with an "arithmetical" quality which appears to be irreducible. All the same, the spirit of these conjectures is to grasp and express the "arithmetical" ( or discrete) through the mediation of the "geometric" ( or the "continuous".)(*)
(*For the mathematical reader) The Weil conjectures are subject to hypotheses of an essentially arithmetical nature, principally because the varieties involved must be defined over finite fields. From the point of view of the cohomological formalism, this results in a privileged status being ascribed to the
Frobenius endomorphism allied with such situations. In my approach, the crucial properties (analogous to 'generalized index theorems') are present in the various
algebraic correspondances, without making any arithmetic hypotheses about some previously assigned field.
In this sense the version of these conjectures which I've extracted from them appears to my mind to be more "faithful" to the "Weil philosophy" than those of Weil himself! - a philosophy that has never been written down and rarely expressed, yet which probably has been the primary motivating force in the extraordinary growth and development of geometry over the course of the last 4 decades.(*)
(*)Since my departure in 1970 however, a reactionary tendency has set in, finding its concrete expression in a state of relative stagnation, which I speak of on several occasions in the pages of Récoltes et Semailles.
My reformulation consisted, essentially, in extracting a sort of "quintessence" of what is truly valuable in the framework of what are called "abstract" algebraic varieties, in classical "Hodge theory", and in the study of "ordinary" algebraic varieties.(*)
(*)Here the word 'ordinary' signifies:"defined over complex fields". Hodge theory ( for "harmonic integrals") was the most powerful of the known cohomological theories in the context of complex algebraic varieties.
I've named this entirely geometric form of these celebrated conjectures the " standard conjectures".
To my way of thinking, this was, after the development of l-adic cohomology, a new step in the direction of these conjectures. Yet, at the same time and above all, it was also one of the principal possible approaches towards what still appears to me to be the most profound of all the themes I've introduced into mathematics (*) , that of motives, ( themselves originating in the "l-adic cohomology theme")
(*)This was the deepest theme at least during my period of mathematical activity between 1950 and 1969, that is to say up to the very moment of my departure from the mathematical scene. I deem the themes of anabelian algebraic geometry and that of Galois-Teichmuller theory, which have developed since 1977, to be of comparable depth.
This theme is like the heart, or soul, that which is most hidden, most completely shielded from view within the "schematic" theme, which is itself at the very heart of the new vision. And several key phenomena retrieved from the standard conjectures (**) can also be seen as constituting a sort of ultimate quintessence of the motivic theme, like the "vital breath" of this most subtle of all themes, of this "heart within the heart" of the new geometry.
(**) (For the algebraic geometer). Sooner or later there must be a revision of these conjectures. Fir more detailed commentary, go to "The tower of scaffoldings" (R&S IV footnote #178, p. 1215-1216), and the note at the bottom of page 769, in "Conviction and knowledge" (R&S III, footnote#162)
Roughly speaking, this is what's involved. We've come to understand , for a given prime number p, the importance of knowing how to construct "cohomological theories" ( particularly in light of the Weil conjectures) for the "algebraic varieties of characteristic p" . Now, the celebrated "cohomological l-adic tool" supplies one with just such a theory, and indeed, an
infinitude of different cohomological theories , that is to say, one associated with each prime number different from p. Clearly there is a "missing" theory, namely that in which l and p are equal. In order to provide for this case I conceived of yet another cohomological theory ( to which I've already alluded), entitled "crystalline cohomology". Furthermore, in the case in which p is infinite, there are yet 3 more cohomological theories (***)
(***) (For the benefit of the mathematical reader) These theories correspond, respectively, to
Betti cohomology( by means of transcendentals, and with the help of an embedding of the base field into the field of the complex numbers),
Hodge cohomology, and
de Rham cohomology as interpreted by myself. The latter two date back to the 50's ( that of Betti to the 19th century).
Furthermore there is nothing to prevent the appearance, sooner or later, of yet more cohomological theories, with totally analogous formal properties. In contradistinction to what one finds in ordinary topology, one finds oneself in the presence of a disconcerting abundance of differing cohomological theories. One had the impression that, in a sense that should be taken rather flexibly, all of these theories "boiled down" to the same one, that they "gave the same results". (****)
(****)(For the benefit of the mathematical reader). For example, if f is an endomorphism of the algebraic variety X, inducing an endomorphism of the cohomology space Hi(X), then the fact that the "characteristic polynomial" of the latter must have integrer coefficients does not depend on the kind of cohomology employed ( for example, l-adic for some arbitrary l) . Likewise for algebraic correspondances in general, which X is presumed proper and smooth. The sad truth, ( and this gives one an idea of the deporable state in which the cohomological theory of algebraic varieties of characterstic p finds itself since my departure), is that there is no demonstration of this fact, as of this writing, even in the simplest case in which X is a smooth projective surface , and i = 2. Indeed, to my knowledge, nobody since my departure has deigned to interest himself in this crucial question, which is typical of all those which are subsidiary to the standard conjecture. The doctrine
a-la-mode is that the only endomorphism worthy of anyone's attention is the Frobenius endomorphism, ( which could have been treated by Deligne by the method of boundaries ...)
It was through my intention to give expression to this "kinship" between differing cohomological theories that I arrived at the notion of associating an algebraic variety with a "motive". My intention in using this term is to suggest the notion of the "common motive" ( or of the "common rationale") subsidiary to the great diversity of cohomological invariants associated with the variety, owing to the enormous collection of cohomologies possible
apriori .The differing cohomological theories would then be merely so many differing thematic developments, ( each in the "tempo" , the "key", and "mode" ("major" or "minor") appropriate to it) , of an identical "basic motive" (called the
"motivic cohmological theory" ), which would also be at the same time the most fundamental, the ultimate "refinement" of all the differing thematic incarnations ( that is to say, of all the possible cohomological theories).
Thus the motive associated with an algebraic variety would constitute the ultimate invariant, the invariant par excellence from the cohomological standpoint among so many musical "incarnations", or differing "realizations". All of the essential properties of the cohomology of the variety could already be read off ( or be "extended to") on the corresponding motive, with the result that the properties and familiar structures of particular cohomological invariants, ( l-adic, crystalline for example) would be merely the faithful reflection of the properties and structures intrinsic to the motive(*) .
(*)(For the benefit of the mathematical reader). Another way of viewing the category of motives over a field k, is to visualize it as a kind of "covering Abelian category" of the category of distinct schemes of finite type over k. Then the motive associated with a given schema X ("cohomological motive" of X which I notate as H*(mot)(X) ) thereby appears as a sort of "Abelianized avatar" of X. The essential point is that , even as an Abelian variety X is susceptible to "continuous variation" ( with a dependence of its' isomorphism class on "continuous parameters", or "modules") , the motive associated with X , or more generally, a "variable" motive, is also susceptible to continuous variation.This is an aspect of motivic cohomology which is in flagrant contrast to what one normally has with respect to all the classical cohomological invariants, (including the l-adic invariants), with the sole exception of the Hodge cohomology of complex algebraic varieties.
This should give one an idea of to what extent "motivic cohomology" is a more refined invariant, encapsulating in a far tighter manner the "arithmetical form" ( if I can risk such an expression) of X, than do the traditional invariants of pure topology. In my way of looking at motives, they consitute a kind of delicate and hidden "thread" linking the algebraic-geometric properties of an algebraic variety to the properties of an "arithmetic" nature incarnated in its motive. The latter may then be considered to be an object which, in its spirit, is geometric in nature, yet for which the "arithmetic" properties implicit in its geometry have been laid bare.
Thus, the motive presents itself as the deepest "form invariant" which one has been able to associate up to the present moment with an algebraic variety, setting aside its "motivic fundamental group". For me both invariants represent the "shadows" projected by a "motivic homotopy type" which remains to be discovered (and about which I say a few things in the footnote: "The tower of scaffoldings- or tools and vision" (R&S IV, #178, see scafolding 5 ( Motives), and in particular page 1214 )) .
It is the latter object which appears to me to be the most perfect incarnation of the elusive intuition of "arithmetic form" ( or "motivic"), of an arbitrary algebraic variety.
Here we find, expressed in the untechnical language of musical metaphor, the quintessence of an idea (both delicate and audacious at once), of virtually infantile simplicity. This idea was developed, on the fringes of more fundamental and urgent tasks, under the name of the "theory of motives", or of "philosophy ( or "yoga") of the "motives", through the years 19673-69. It's a theory of a fascinating structural richness, a large part of which remains purely conjectural. (*)
(*)I've explained my vision of motives to any who wished to learn about them all through the years, without taking the trouble to publish anything in black and white on this subject (not lacking in other tasks of importance). This enabled several of my students later on to pillage me all the more easily, and under the tender gaze of my circle of friends who were well aware of the situation. (See the following footnote)
IN R &S I often return to this topic of the "yoga of motives", of which I am particularly fond. There is no need to dwell here on what is discussed so throughly elsewhere. It suffices for me to say that the "standard conjectures" flow in a very natural way from the world of this yoga of motives. These conjectures furnish at the same time a primary means for effecting one of the possible formal constructions of the notion of the motive.
The standard conjectures appeared to me then, and still do today, as one of the two questions which are the most fundamental in Algebraic Geometry. Neither this question, nor the other one ( known as the "resolution of singularities") has been answered at the present time. However, whereas the second of them has a venerable history of a century, the other one, which I've had the honor of discovering, now tends to be classfied according to the dictates of fad-and-fashion ( over the years following my departure from the mathematical scene, (and similarly for the theme of motives) ) , as some kind of genial "grothendieckean" fol-de-rol. Once more I'm getting ahead of myself .... (*)
(*) In point of fact, this theme was exhumed ( one year after the crystalline theme) , but this time under its own name , ( and in a truncated form, and only in the single case of a base field of null characteristic) , without the name of its discoverer being so much as mentioned. It constiutes one example among so many others, of an idea and a theme which were buried at the time of my departure as some kind of "grothendieckean" fantasmagoria", only to be revived, one after another, by certain of my students over the course of the next 10 to 15 years, with shameless pride and ( need one spell it out?) never a mention of its originator.