How Grothendieck Simplified Algebraic Geometry Photo courtesy of the IHES. Colin McLarty
256 Notices of the AMS Volume 63, Number 3 The idea of scheme is childishly simple—so simple, The Beginnings of Cohomology so humble, no one before me dreamt of stooping Surfaces with holes are not just an amusing so low.…It grew by itself from the sole demands pastime but of quite fundamental importance for of simplicity and internal coherence. the theory of equations. (Atiyah [4, p. 293]) [A. Grothendieck, Récoltes et Semailles (R&S), pp. P32, P28] The Cauchy Integral Theorem says integrating a holo- morphic form 휔 over the complete boundary of any Algebraic geometry has never been really simple. It region of a Riemann surface gives 0. To see its signifi- was not simple before or after David Hilbert recast it in cance look at two closed curves 퐶 on Riemann surfaces his algebra, nor when André Weil brought it into number theory. Grothendieck made key ideas simpler. His schemes that are not complete boundaries. Each surrounds a hole, give a bare minimal definition of space just glimpsed as and each has ∫퐶 휔 ≠ 0 for some holomorphic 휔. early as Emmy Noether. His derived functor cohomology Cutting the torus along the dotted curve 퐶1 around the pares insights going back to Bernhard Riemann down to center hole of the torus gives a tube, and the single curve an agile form suited to étale cohomology. To be clear, étale 퐶1 can only bound one end. cohomology was no simplification of anything. It was a ...... radically new idea, made feasible by these simplifications...... ⋆ .... Grothendieck got this heritage at one remove from ...... the original sources, largely from Jean-Pierre Serre in ...... 퐶1 퐶2 ...... ⋆ shared pursuit of the Weil conjectures. Both Weil and ...... Serre drew deeply and directly on the entire heritage. The original ideas lie that close to Grothendieck’s swift The punctured sphere on the right has stars depicting reformulations. punctures, i.e. holes. The regions on either side of 퐶2 are unbounded at the punctures. Generality As the Superficial Aspect Riemann used this to calculate integrals. Any curves 퐶 Grothendieck’s famous penchant for generality is not and 퐶′ surrounding just the same holes the same number enough to explain his results or his influence. Raoul of times have ∫퐶 휔 = ∫퐶′ 휔 for all holomorphic 휔. That Bott put it better fifty-four years ago describing the is because 퐶 and the reversal of 퐶′ form the complete Grothendieck-Riemann-Roch theorem. boundary of a kind of collar avoiding those holes. So Riemann-Roch has been a mainstay of analysis for ∫퐶 휔 − ∫퐶′ 휔 = 0. one hundred fifty years, showing how the topology of a Modern cohomology sees holes as obstructions to Riemann surface affects analysis on it. Mathematicians solving equations. Given 휔 and a path 푃∶ [0, 1] → 푆 it from Richard Dedekind to Weil generalized it to curves would be great to calculate the integral ∫ 휔 by finding a over any field in place of the complex numbers. This 푃 function 푓 with d푓 = 휔, so ∫ 휔 = 푓(푃 ) − 푓(푃 ). Clearly, makes theorems of arithmetic follow from topological 푃 1 0 there is not always such a function, since that would and analytic reasoning over the field 픽 of integers 푝 imply ∫ 휔 = 0 for every closed curve 푃. But Cauchy, modulo a prime 푝. Friedrich Hirzebruch generalized the 푃 complex version to work in all dimensions. Riemann, and others saw that if 푈 ⊂ 푆 surrounds no Grothendieck proved it for all dimensions over all holes there are functions 푓푈 with d푓푈 = 휔 all over 푈. fields, which was already a feat, and he went further in Holes are obstructions to patching local solutions 푓푈 into a signature way. Beyond single varieties he proved it for one solution of d푓 = 휔 all over 푆. suitably continuous families of varieties. Thus: This concept has been generalized to algebra and Grothendieck has generalized the theorem to the number theory: point where not only is it more generally applica- Indeed one now instinctively assumes that all ble than Hirzebruch’s version but it depends on obstructions are best described in terms of a simpler and more natural proof. (Bott [6]) cohomology groups. [32, p. 103] This was the first concrete triumph for his new cohomol- ogy and nascent scheme theory. Recognizing that many Cohomology Groups mathematicians distrust generality, he later wrote: With homologies, terms compose according to I prefer to accent “unity” rather than “general- the rules of ordinary addition. [24, pp. 449-50] ity.” But for me these are two aspects of one Poincaré defined addition for curves so that 퐶 + 퐶′ is quest. Unity represents the profound aspect, and the union of 퐶 and 퐶′, while −퐶 corresponds to reversing generality the superficial. [16, p. PU 25] the direction of 퐶. Thus for every form 휔: Colin McLarty is Truman P. Hardy Professor of Philosophy and professor of mathematics, Case Western Reserve University. His ∫ 휔 = ∫ 휔 + ∫ 휔 and ∫ 휔 = − ∫ 휔. email address is [email protected]. 퐶+퐶′ 퐶 퐶′ −퐶 퐶
For permission to reprint this article, please contact: reprint- When curves 퐶1, … , 퐶푘 form the complete boundary [email protected]. of some region, then Poincaré writes 훴푘퐶푘 ∼ 0 and says DOI: http://dx.doi.org/10.1090/noti1338 their sum is homologous to 0. In these terms the Cauchy
March 2016 Notices of the AMS 257 Integral Theorem says concisely: So topologists wrote H푖(푆) for the 푖th cohomology group of space 푆 and left the coefficient group implicit in the
If 훴푘퐶푘 ∼ 0, then ∫ 휔 = 0 context. 훴푘퐶푘 In contrast group theorists wrote H푖(퐺, 퐴) for the for all holomorphic forms 휔. 푖th cohomology of 퐺 with coefficients in 퐴, because Poincaré generalized this idea of homology to higher many kinds of coefficients were used and they were as dimensions as the basis of his analysis situs, today called interesting as the group 퐺. For example, the famed Hilbert topology of manifolds. Theorem 90 became Notably, Poincaré published two proofs of Poincaré H1(Gal(퐿/푘), 퐿×) ≅ {0}. duality using different definitions. His first statement of it was false. His proof mixed wild non sequiturs The Galois group Gal(퐿/푘) of a Galois field extension 퐿/푘 and astonishing insights. For topological manifolds 푀 has trivial 1-dimensional cohomology with coefficients in × of any dimension 푛 and any 0 ≤ 푖 ≤ 푛, there is a the multiplicative group 퐿 of all nonzero 푥 ∈ 퐿. Olga tight relation between the 푖-dimensional submanifolds Taussky[33, p. 807] illustrates Theorem 90 by using it on of 푀 and the (푛 − 푖)-dimensional. This relation is hardly the Gaussian numbers ℚ[푖] to show every Pythagorean expressible without using homology, and Poincaré had to triple of integers has the form revise his first definition to get it right. Even the second 푚2 − 푛2, 2푚푛, 푚2 + 푛2. version relied on overly optimistic assumptions about triangulated manifolds. Trivial cohomology means there is no obstruction to Topologists spent decades clarifying his definitions solving certain problems, so Theorem 90 shows that and theorems and getting new results in the process. some problems on the field 퐿 have solutions. Algebraic relations of H1(Gal(퐿/푘), 퐿×) to other cohomology groups They defined homology groups H푖(푆) for every space 푆 imply solutions to other problems. Of course Theorem 90 and every dimension 푖 ∈ ℕ. In each H1 the group addition is Poincaré’s addition of curves modulo the homology was invented to solve lots of problems decades before relations 훴푘퐶푘 = 0. They also defined related cohomology group cohomology appeared. Cohomology organized and 푖 extended these uses so well that Emil Artin and John Tate groups H (푆) such that Poincaré duality says H푖(푀) is isomorphic to H푛−푖(푀) for every compact orientable made it basic to class field theory. 푛-dimensional manifold 푀. Also in the 1940s topologists adopted sheaves of Poincaré’s two definitions of homology split into many coefficients. A sheaf of Abelian groups ℱ on a space using simplices or open covers or differential forms or 푆 assigns Abelian groups ℱ(푈) to open subsets 푈 ⊆ 푆 metrics, bringing us to the year 1939: and homomorphisms ℱ(푈) → ℱ(푉) to subset inclusions Algebraic topology is growing and solving prob- 푉 ⊆ 푈. So the sheaf of holomorphic functions 풪푀 assigns lems, but nontopologists are very skeptical. At the additive group 풪푀(푈) of holomorphic functions on 푈 to each open subset 푈 ⊆ 푀 of a complex manifold. Harvard, Tucker or perhaps Steenrod gave an 푖 expert lecture on cell complexes and their homol- Cohomology groups like H (푀, 풪푀) began to organize ogy, after which one distinguished member of the complex analysis. audience was heard to remark that this subject Leaders in these fields saw cohomology as a unified had reached such algebraic complication that it idea, but the technical definitions varied widely. In the was not likely to go any further. (MacLane [21, Séminaire Henri Cartan speakers Cartan, Eilenberg, and p. 133] Serre organized it all around resolutions. A resolution of an Abelian group 퐴 (or module or sheaf) is an exact sequence of homomorphisms, meaning the image of each Variable Coefficients and Exact Sequences homomorphism is the kernel of the next: / / / / In his Kansas article (1955) and Tôhoku article {0} 퐴 퐼0 퐼1 …. (1957) Grothendieck showed that given any cate- gory of sheaves a notion of cohomology groups It quickly follows that many sequences of cohomology results. (Deligne [10, p. 16]) groups are also exact. That proof rests on the Snake Lemma immortalized by Hollywood in a scene widely Algebraic complication went much further. Methods available online: “A clear proof is given by Jill Clayburgh in topology converged with methods in Galois theory at the beginning of the movie It’s My Turn” [35, p. 11]. and led to defining cohomology for groups as well as Fitting a cohomology group H푖(푋, ℱ) into the right for topological spaces. In the process, what had been a exact sequence might show H푖(푋, ℱ) ≅ {0}, so the ob- technicality to Poincaré became central to cohomology, 푖 namely, the choice of coefficients. Certainly he and others structions measured by H (푋, ℱ) do not exist. Or it may prove some isomorphism, H푖(푋, ℱ) ≅ H푘(푌, 풢). Then the used integers, rational numbers or reals or integers 푖 modulo 2 as coefficients: obstructions measured by H (푋, ℱ) correspond exactly to those measured by H푘(푌, 풢). 푎1퐶1 + ⋯ + 푎푚퐶푚 푎푖 ∈ ℤ or ℚ or ℝ or ℤ/2ℤ. Group cohomology uses resolution by injective mod- But only these few closely related kinds of coefficients ules 퐼푖. A module 퐼 over a ring 푅 is injective if for were used, chosen for convenience for a given calculation. every 푅-module inclusion 푗∶ 푁 ↣ 푀 and homomorphism
258 Notices of the AMS Volume 63, Number 3 푓∶ 푁→퐼 there is some 푔∶ 푀→퐼 with 푓 = 푔푗. The diagram They were wrong on both counts. These axioms also sim- is simple: plified proofs of already-known theorems. Most especially / 푗 / they subsumed many useful spectral sequences (not all) 푁 N 푀 NN under the Grothendieck spectral sequence so simple as to NNN NN 푔 be Exercise A.3.50 of (Serre [11, p. 683]). NNN 푓 NN' Early editions of Serge Lang’s Algebra gave the Abelian 퐼 category axioms with a famous exercise: “Take any book This works because every 푅-module 퐴 over any ring 푅 on homological algebra, and prove all the theorems embeds in some injective 푅-module. No one believed without looking at the proofs given in that book” [20, p. anything this simple would work for sheaves. Sheaf coho- 105]. He dropped that when homological algebra books mology was defined only for sufficiently regular spaces all began using axiomatic proofs themselves, even if their using various, more complicated topological substitutes theorems are stated only for modules. David Eisenbud, for injectives. Grothendieck found an unprecedented for example, says his proofs for modules “generalize with proof that sheaves on all topological spaces have injec- just a little effort to [any] nice Abelian category” [11, tive embeddings. The same proof later worked for sheaves p. 620]. on any Grothendieck topology. Injective resolutions in any Abelian category give derived functor cohomology of that category. This was obviously general beyond any proportion to the then- Tôhoku known cases. Grothendieck was sure it was the right Consider the set of all sheaves on a given topolog- generality: For a cohomological solution to any prob- ical space or, if you like, the prodigious arsenal lem, notably the Weil conjectures, find the right Abelian of all “meter sticks” that measure the space. We category. consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so The Weil Conjectures to speak “right in front of your nose”; that is what This truly revolutionary idea thrilled the mathe- we call the structure of a “category.” [16, p. P38] maticians of the time, as I can testify at first hand. We will not fully define sheaves, let alone spectral [30, p. 525] sequences and other “drawings (called “diagrams”) full of The Weil Conjectures relat- arrows covering the blackboard” which “totally escaped” ing arithmetic to topology were Grothendieck at the time of the Séminaire Cartan [R&S, immediately recognized as a The p. 19]. We will see why Grothendieck wrote to Serre on huge achievement. Weil knew conjectures February 18, 1955: “I am rid of my horror of spectral that just conceiving them was sequences” [7, p. 7]. a great moment in his career. were too The Séminaire Cartan emphasized how few specifics The cases he proved were im- about groups or modules go into the basic theorems. pressive. The conjectures were beautiful not Those theorems only use diagrams of homomorphisms. too beautiful not to be true and For example, the sum 퐴 + 퐵 of Abelian groups 퐴, 퐵 can yet nearly impossible to state to be true and be defined, uniquely up to isomorphism, by the facts that fully. it has homomorphisms 푖퐴 ∶ 퐴 → 퐴 + 퐵 and 푖퐵 ∶ 퐵 → 퐴 + 퐵 Weil [37] presents the yet nearly and any two homomorphisms 푓∶ 퐴→퐶 and 푔∶ 퐵→퐶 give topology using the nineteenth- impossible to a unique 푢∶ 퐴 + 퐵→퐶 with 푓 = 푢푖퐴 and 푔 = 푢푖퐵: century terminology of Betti numbers. But he was an estab- 푖퐴 / o 푖퐵 state fully. 퐴 FF 퐴 + 퐵 y 퐵 lished expert on cohomology FF yy 푓 FF 푢 yy 푔 and in conversations: FF yy F" y| y At that time, Weil was explaining things in terms of 퐶 cohomology and Lefschetz’s fixed point formula The same diagram defines sums of modules or of sheaves [yet he] did not want to predict [this could actually of Abelian groups. work]. Indeed, in 1949–50, nobody thought that Grothendieck [13, p. 127] took the basic patterns used it could be possible. (Serre quoted in [22, p. 305].) by the Séminaire Cartan as his Abelian category axioms. Lefschetz used cohomology, relying on the continuity He added a further axiom, AB5, on infinite colimits. of manifolds, to count fixed points 푥 = 푓(푥) of continuous Theorem 2.2.2 says if an Abelian category satisfies AB5 functions 푓∶ 푀→푀 on manifolds. Weil’s conjectures deal plus a set-theoretic axiom, then every object in that with spaces defined over finite fields. No known version category embeds in an injective object. These axioms of those was continuous. Neither Weil nor anyone knew taken from module categories obviously hold as well for what might work. Grothendieck says: sheaves of Abelian groups on any topological space, so Serre explained the Weil conjectures to me in the conclusion applies. cohomological terms around 1955 and only in People who thought this was just a technical result on these terms could they possibly “hook” me. No sheaves found the tools disproportionate to the product. one had any idea how to define such a cohomology
March 2016 Notices of the AMS 259 and I am not sure anyone but Serre and I, not subvarieties 푉′ ⊆ 푉 of a variety 푉 purely in terms even Weil if that is possible, was deeply convinced of fields of rational functions. Raynaud [25] gives an such a thing must exist. [R&S, p. 840] excellent overview. We list three key topics. (1) Weil has generic points. Indeed, a variety defined Exactly What Scheme Theory Simplified by polynomials over a field 푘 has infinitely many Kronecker was, in fact, attempting to describe and generic points with coordinates transcendental to initiate a new branch of mathematics, which over 푘, all conjugate to each other by Galois would contain both number-theory and algebraic actions over 푘. geometry as special cases. (Weil [38, p. 90]) (2) Weil defines abstract algebraic varieties by data Riemann’s treatment of complex curves left much to telling how to patch together varieties defined by geometric intuition. So Dedekind and Weber [8, p. 181] equations. But these do not exist as single spaces. proved a Riemann-Roch theorem from “a simple yet rig- They only exist as sets of concrete varieties plus orous and fully general viewpoint,” over any algebraically patching data. closed field 푘 containing the rational numbers. They note (3) Weil could not define a variety over the integers, 푘 can be the field of algebraic numbers. They saw this though he could systematically relate varieties bears on arithmetic as well as on analysis and saw all too over ℚ to others over the fields 픽푝. well that their result is “very difficult in exposition and expression” [8, p. 235]. Serre Varieties and Coherent Sheaves Meromorphic functions on any compact Riemann sur- Then Serre [27] temporarily put generic points and non- face 푆 form a field 푀(푆) of transcendence degree 1 over closed fields aside to describe the first really penetrating the complex numbers ℂ. Each point 푝 ∈ 푆 determines a cohomology of algebraic varieties: function 푒푝 from 푀(푆) to ℂ + {∞}: namely 푒푝(푓) = 푓(푝) This rests on the use of the famous Zariski topol- when 푓 is defined at 푝, and 푒푝(푓) = ∞ when 푓 has a pole ogy, in which the closed sets are the algebraic at 푝. Then, if we ignore sums ∞ + ∞: sub-varieties. The remarkable fact that this coarse topology could actually be put to genuine mathe- 푒푝(푓 + 푔) = 푒푝(푓) + 푒푝(푔), matical use was first demonstrated by Serre and 푒푝(푓 ⋅ 푔) = 푒푝(푓) ⋅ 푒푝(푔), it has produced a revolution in language and 1 1 techniques. (Atiyah [3, p. 66]) 푒푝( ) = . 푛 푓 푒푝(푓) Say a naive variety over any field 푘 is a subset 푉 ⊆ 푘 Dedekind and Weber define a general field of algebraic defined by finitely many polynomials 푝푖(푥1, … , 푥푛) over 푘: 푛 functions as any transcendence degree 1 extension 퐿/푘 of 푉 = { ⃗푥∈ 푘 | 푝1(⃗푥)= ⋯ = 푝ℎ(⃗푥)= 0 }. any algebraically closed field 푘. They define a point 푝 of 퐿 They form the closed sets of a topology on 푘푛 called to be any function 푒 from 퐿 to 푘 + {∞} satisfying those 푝 the Zariski topology. Even their infinite intersections are equations. Their Riemann-Roch theorem treats 퐿 as if it defined by finitely many polynomials, since the polyno- were 푀(푆) for some Riemann surface. mial ring 푘[푥 , … , 푥 ] is Noetherian. Also, each inherits Kronecker [19] achieved some “algebraic geometry over 1 푛 a Zariski topology where the closed sets are the subsets an absolutely algebraic ground-field” [38, p. 92]. These 푉′ ⊆ 푉 defined by further equations. fields are the finite extensions of ℚ or of finite fields 픽푝. These are very coarse topologies. The Zariski closed They are not algebraically closed. He aimed at “algebraic subsets of any field 푘 are the zero-sets of polynomials geometry over the integers” where one variety could be over 푘: that is, the finite subsets and all of 푘. defined over all these fields at once, but this was far too Each naive variety has a structure sheaf 풪푉 which difficult at the time [38, p. 95]. assigns to every Zariski open 푈 ⊆ 푉 the ring of regular Italian algebraic geometers relied on an idea of generic functions on 푈. Omitting important details: points of a complex variety 푉, which are ordinary complex points 푝 ∈ 푉 with no apparent special properties [26]. For 푓(⃗푥) 풪푉(푈) = { such that when ⃗푥∈ 푈 then 푔(⃗푥)≠ 0 }. example, they are not points of singularity. Noether and 푔(⃗푥) Bartel van der Waerden gave abstract generic points which A Serre variety is a topological space 푇 plus a sheaf actually have only those properties common to all points 풪푇 which is locally isomorphic to the structure sheaf of 푉. Van der Waerden [34] made these rigorous but not of a naive variety. Compare the sheaf of holomorphic so usable as Weil would want. Oscar Zariski, trained in functions 풪푀 of a complex manifold. The sheaf apparatus Italy, worked with Noether in Princeton, and later with lets Serre actually paste varieties together on compatible Weil, to give algebraic geometry a rigorous algebraic basis patches, just as patches of differentiable manifolds are [23, p. 56]. pasted together. Weil could not do this with his abstract Weil’s bravura Foundations of Algebraic Geometry [36] varieties. combined all these methods into the most complicated Certain sheaves related to the structure sheaves 풪푇 foundation for algebraic geometry ever. To handle vari- are called coherent. Serre makes them the coefficient eties of all dimensions over arbitrary fields 푘, he uses sheaves of a cohomology theory widely used today with algebraically closed field extensions 퐿/푘 of infinite tran- schemes. The close tie of coherent sheaves to structure scendence degree. He defines not only points but also sheaves makes this cohomology unsuitable for the Weil
260 Notices of the AMS Volume 63, Number 3 conjectures. When variety (or scheme) 푉 is defined over a modulo a prime 푝, such as 푥 = 2 and 푦 = 6 in the finite finite field 픽푝, its coherent cohomology is defined modulo field 픽13, are not complex numbers. And solutions modulo 푝 and can count fixed points of maps 푉→푉 only modulo one prime are different from those modulo another. All 푝. Still: these solutions are organized in the single scheme The principal, and perhaps only, external inspira- Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). tion for the sudden vigorous launch of scheme theory in 1958 was Serre’s (1955) article known The coordinate functions are simply integer polynomi- 2 2 by the acronym FAC. [R&S, p. P28] als modulo 푥 + 푦 − 1. The nonzero prime ideals are not simple at all. They correspond to solutions of this equa- Schemes tion in all the absolutely algebraic fields by which Weil explicated Kronecker’s goal, including all finite fields. In- The point, grosso modo, was to rid algebraic geom- deed, the closest Grothendieck comes to defining schemes etry of parasitic hypotheses encumbering it: base in Récoltes et Semailles is to call a scheme a “magic fan” fields, irreducibility, finiteness conditions. (Serre (éventail magique) folding together varieties defined over [29, p. 201]) all these fields (p. P32). This is algebraic geometry over Schemes overtly simplify algebraic geometry. Where the integers. earlier geometers used complicated extensions of alge- Now consider the ideal (푥2 + 푦2 − 1) consisting of braically closed fields, scheme theorists use any ring. all polynomial multiples of 푥2 + 푦2 − 1 in ℤ[푥, 푦]. It is Polynomial equations are replaced by ring elements. prime, so it is a point of Spec(ℤ[푥, 푦]). And schemes Generic points become prime ideals. The more intri- are not Hausdorff spaces: their points are generally not cate concepts come back in when needed, which is fairly closed in the Zariski topology. The closure of this point is often, but not always and not from the start. Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). This ideal is the generic point In fact this perspective goes back to unpublished work of the closed subscheme by Noether, van der Waerden, and Wolfgang Krull. Prior to Grothendieck: Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)) ↣ Spec(ℤ[푥, 푦]). The person who was closest to scheme-thinking The irreducible closed subschemes of any scheme are, (in the affine case) was Krull (around 1930). He roughly speaking, given by equations in the coordinate used systematically the localization process, and ring, and each has exactly one generic point. proved most of the nontrivial theorems in Com- In the ring ℤ[푥, 푦]/(푥2 + 푦2 − 1) the ideal (푥2 + 푦2 − 1) mutative Algebra. (Serre, email of 21/06/2004, appears as the zero ideal, since in this ring 푥2 + 푦2 − Serre’s parentheses) 1 = 0. So the zero ideal is the generic point for the Grothendieck made it work. He made every ring 푅 whole scheme Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). What happens the coordinate ring of a scheme Spec(푅) called the at this generic point also happens almost everywhere spectrum of 푅. The points are the prime ideals of 푅, on Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). Generic points like this and the scheme has a structure sheaf 풪푅 on the Zariski achieve what earlier algebraic geometers sought from topology for those points, like the structure sheaf on their attempts. a Serre variety. It follows that the continuous structure Schemes vindicate more classical intuitions as well. preserving maps from Spec(푅) to another affine scheme Ancient Greek geometers debated whether a tangent line Spec(퐴) correspond exactly to the ring homomorphisms meets a curve in something more than a point. Scheme in the other direction: theory says yes: a tangency is an infinitesimal segment 푓 / Spec(푓) / around a point. 퐴 푅 Spec(푅) Spec(퐴) . The contact of the parabola 푦 = 푥2 with the 푥-axis The points can be quite intricate: “When one has to 푦 = 0 in ℝ2 is plainly given by 푥2 = 0. As a variety it would construct a scheme one generally does not begin with the just be the one point space {0}, but it gives a nontrivial set of points” [10, p. 12]. scheme Spec(ℝ[푥]/(푥2)). The coordinate functions are For example, the ring ℝ[푥] of real polynomials in real polynomials modulo 푥2 or, in other words, real linear one variable is the natural coordinate ring for the real polynomials 푎 + 푏푥. line, so the spectrum Spec(ℝ[푥]) is the scheme of the Intuitively Spec(ℝ[푥]/(푥2)) is an infinitesimal line seg- real line. Each nonzero prime ideal is generated by a ment containing 0 but no other point. This segment is monic irreducible real polynomial. Those polynomials big enough that a function 푎 + 푏푥 on it has a slope 푏 are 푥 − 푎 for 푎 ∈ ℝ and 푥2 − 2푏푥 + 푐 for 푏, 푐 ∈ ℝ with but is too small to admit a second derivative. Intuitively 푏2 < 푐. The first kind correspond to ordinary points 푥 = 푎 a scheme map 푣 from Spec(ℝ[푥]/(푥2) to any scheme 푆 of the real line. The second kind correspond to pairs is an infinitesimal line segment in 푆, i.e. a tangent vector of conjugate complex roots 푏 ± √푏2 − 푐. The scheme with base point 푣(0) ∈ 푆. Spec(ℝ[푥]) automatically includes both real and complex Grothendieck’s signature method, called the relative points, with the nuance that a single complex point is a viewpoint, also reflects classical ideas. Earlier geometers conjugate pair of complex roots. would speak of, for example, 푥2 + 푡 ⋅ 푦2 = 1 as a quadratic A polynomial equation like 푥2 + 푦2 = 1 has many kinds equation in 푥, 푦 with parameter 푡. So it defines a conic of solutions. One could think of rational and algebraic section 퐸푡 which is an ellipse or a hyperbola or a pair of solutions as kinds of complex solutions. But solutions lines depending on the parameter. More deeply, this is
March 2016 Notices of the AMS 261 a cubic equation in 푥, 푦, 푡 defining a surface 퐸 bundling information and express operations which earlier geome- together all the curves 퐸푡. Over the real numbers this ters had only begun to explore. Grothendieck and Jean gives a map of varieties Dieudonné took this as a major advantage of scheme / theory: 퐸={⟨푥, 푦, 푡⟩∈ℝ3|푥2 +푡⋅푦2 = 1} ℝ ⟨푥, 푦, 푡⟩↦푡 . The idea of “variation” of base ring which we intro- Each curve 퐸 is the fiber of this map over its parameter 푡 duce gets easy mathematical expression thanks to 푡 ∈ ℝ. Classical geometers used the continuity of the the functorial language whose absence no doubt family of curves 퐸 bundled into surface 퐸 but generally 푡 explains the timidity of earlier attempts. [17, p. 6] left the cubic surface implicit as they spoke of the variable quadratic curve 퐸푡. Grothendieck used rigorous means to treat a scheme Étale Cohomology map 푓∶ 푋→푆 as a single scheme simpler than either one of In the Séminaire Chevalley of April 21, 1958, Serre pre- 1 푋 and 푆. He calls 푓 a relative scheme and treats it roughly sented new 1-dimensional cohomology groups H̃ (X, G) 1 suitable for the Weil conjectures: “At the end of the as the single fiber 푋푝 ⊂ 푋 over some indeterminate 푝 ∈ 푆. Grothendieck had this viewpoint even before he had oral presentation Grothendieck said this would give the schemes: Weil cohomology in all dimensions! I found this very Certainly we’re now so used to putting some optimistic” [31, p. 255]. That September Serre wrote: problem into relative form that we forget how One may ask if it is possible to define higher revolutionary it was at the time. Hirzebruch’s cohomology groups H̃ 푞(X, G)…in all dimensions. proof of Riemann-Roch is very complicated, while Grothendieck (unpublished) has shown it is, and the proof of the relative version, Grothendieck- it seems that when G is finite these furnish Riemann-Roch, is so easy, with the problem “the true cohomology” needed to prove the Weil shifted to the case of an immersion. This was conjectures. On this see the introduction to [14]. fantastic. [18, p. 1114] [28, p. 12] What Hirzebruch proved for complex varieties Grothen- Grothendieck later described that unpublished work dieck proved for suitable maps 푓∶ 푋→푆 of varieties over of 1958, saying, “The two key ideas crucial in launching any field 푘. Among other advantages this allows reducing and developing the new geometry were those of scheme the proof to the case of maps 푓, called immersions, with and of topos. They appeared almost simultaneously and simple fibers. The method relies on base change transforming a in close symbiosis.” Specifically he framed “the notion relative scheme 푓∶ 푋→푆 on one base 푆 to some 푓′ ∶ 푋′ → 푆′ of site, the technical, provisional version of the crucial on some related base 푆′. Fibers themselves are an example. notion of topos” [R&S, pp. P31 and P23n]. But before Given 푓∶ 푋→푆 each point 푝 ∈ 푆 is defined over some field pursuing this idea into higher-dimensional cohomology 푘, and 푝 ∈ 푆 amounts to a scheme map 푝∶ Spec(푘) → 푆. he used Serre’s idea to define the fundamental group of The fiber 푋푝 is intuitively the part of 푋 lying over 푝 and a variety or scheme in a close analogy with Galois theory. is precisely the relative scheme 푋푝 → Spec(푘) given by Notice that Zariski topology registers punctures much pullback: more directly than it registers holes like those through / 푋푝 푋 the center of the torus or inside the tube.
푓 ⋆푃 / ⋆ Spec(푘) 푝 푆 Other examples of base change include extending a ⋆ scheme 푓∶ 푌 → Spec(ℝ) defined over the real numbers into one 푓′ ∶ 푌′ → Spec(ℂ) over the complex numbers by Zariski closed subsets are (locally) the zero-sets of pullback along the unique scheme map from Spec(ℂ) to polynomials, so a nonempty Zariski open subset of Spec(ℝ): the torus is the torus minus finitely many punctures // 푌′ 푌 (possibly none). Such a subset might or might not be punctured at some point 푃 itself, so the Zariski opens 푓′ 푓 themselves distinguish between having and not having / Spec(ℂ) Spec(ℝ) that puncture. But every nonempty Zariski open subset surrounds the hole through the torus center and the Other changes of base go along scheme maps 푆′ → 푆 one through the torus tube. These subsets by themselves between schemes 푆, 푆′ taken as parameter spaces for cannot distinguish between having and not having those serious geometric constructions. Each is just a pullback holes. Coherent cohomology registers those holes by in the sense of category theory, yet they encode intricate using coherent sheaves, which cannot work for the Weil 1In 1942 Oscar Zariski urged something like this to Weil [23, conjectures, as noted above. p. 70]. Weil took the idea much further without finally making it So Serre used many-sheeted covers. Consider two a working method [38, p. 91ff]. different 2-sheeted covers of one torus 푇. Let torus 푇′ be
262 Notices of the AMS Volume 63, Number 3 ′ twice as long as 푇, with the same tube diameter. Wrap 푇 set of étale maps 푈푖 → 푆 such that the union of all the twice around 푇 along the tube: images is the whole of 푆. Sites today are often called Grothendieck topologies, and this site may be called the ″ 푇 étale topology on 푆. There are two basic ways to solve a problem locally in the étale topology on 푆. You could solve it on each of a set of Zariski open subsets of 푆 whose union is 푆, or you could solve it in a separable algebraic extension wrap 2× | of the coordinate ring of 푆. The first gives an actual, ′ ↓ vertically 푇 global solution if the local solutions agree wherever they wrap 2× horizontally overlap. The second gives a global solution if the local ⟶ 푇 solution is Galois invariant—like first factoring a real polynomial over the complex numbers, then showing the ″ Let torus 푇 be as long as 푇 with twice the tube diameter. factors are actually real. Étale cohomology would measure Wrap it twice around the tube. The difference between obstructions to patching actual solutions together from these two covers, and both of them from 푇 itself, reflects combinations of such local solutions. the two holes in 푇. In 1961 Michael Artin proved the first higher- Riemann created Riemann surfaces as analogues to dimensional geometric theorem in étale cohomology [1, number fields. As ℚ[√2] is a degree 2 field extension of p. 359]. According to David Mumford this was that the rational numbers ℚ, so 푇′ → 푇 is a degree 2 cover of the plane with origin deleted has nontrivial H3; in the 푇. As ℚ[√2]/ℚ has a two element Galois group where context of étale cohomology that means the coordinate the nonidentity element interchanges √2 with −√2, so plane punctured at the origin, 푘2 − {0}, for any field of 푇′ → 푇 has a two-element symmetry group over 푇 where coordinates 푘. Weil’s conjectures suggest that, when 푘 the nonidentity symmetry interchanges the two sheets of is absolutely algebraic, this cohomology should largely 푇′ over 푇. agree with the classical cohomology of the complex case Serre consciously extended Riemann’s analogy to a ℂ2 − ⟨0, 0⟩. That space is topologically ℝ4 punctured at far-reaching identity. He gave a purely algebraic defini- its origin. It has the classical cohomology of the 3-sphere 3 tion of unramified covers 푆′ → 푆 which has the Riemann 푆3, and that is nontrivial in H . So Artin’s result needed surfaces above as special cases, as well as Galois field to hold in any Weil cohomology. Artin proved it does extensions, and much more. Naturally, in this generality hold in the derived functor cohomology of sheaves on some theorems and proofs are a bit technical, but over the étale site. Today this is étale cohomology. and over Serre’s unramified covers make intuitions taken In short, Artin showed the étale site yields not only from Riemann surfaces work for all these cases. Groth- some sheaf cohomology but a good usable one. Classical endieck used these to give the first useful theory of the theorems of cohomology survive with little enough change. fundamental group of a variety or a scheme, that is, the Grothendieck invited Artin to France to collaborate in the one-dimensional homotopy. He also worked with a slight seminar that created Théorie des topos et cohomologie generalization of unramified covers, called étale maps, étale [2]. The subject exploded, and we will go no further which include all algebraic Riemann covering surfaces. into it. Serre had not calculated cohomology of sheaves but Toposes are less popular than schemes or sites in ge- of isotrivial fiber spaces. Over a torus 푇 those are roughly ometry today. Deligne expresses his view with care: “The spaces mapped to 푇 which may twist around 푇 but tool of topos theory permitted the construction of étale can be untwisted by lifting to some other torus 푇‴ → 푇 cohomology” [10, p. 15]. Yet, once constructed, this coho- wrapped some number of times around each hole of 푇. mology is “so close to classical intuition” that for most While Grothendieck [12] also used fiber spaces for one- purposes one needs only some ordinary topology plus dimensional cohomology, he found his Tôhoku methods “a little faith/un peu de foi” [9, p. 5]. Grothendieck would more promising for higher dimensions. He wanted some “advise the reader nonetheless to learn the topos lan- notion of sheaf matching Serre’s idea. guage which furnishes an extremely convenient unifying During 1958 Grothendieck saw that instead of defining principle” [5, p. VII]. sheaves by using open subsets 푈 ⊆ 푆 of some space 푆, he We close with Grothendieck’s view of how schemes and could use étale maps 푈→푆 to a scheme. He published this his cohomology and toposes all came together in étale idea by spring 1961 [15, §4.8, p. 298]. Instead of inclusions cohomology, which indeed in his hands and Deligne’s 푉 ⊆ 푈 ⊆ 푆, he could use commutative triangles over 푆: gave the means to prove the Weil conjectures: The crucial thing here, from the viewpoint of the 푓 / / Weil conjectures, is that the new notion of space 푉 >> 푈 푈 ×푆 푉 푉 >> is vast enough, that we can associate to each >> scheme a “generalized space” or “topos” (called > / the “étale topos” of the scheme in question). 푆 푈 푆 Certain “cohomology invariants” of this topos In place of intersections 푈∩푉 ⊆ 푆 he could use pullbacks (as “babyish” as can be!) seemed to have a good 푈 ×푆 푉 over 푆. Then an étale cover of a scheme 푆 is any chance of offering “what it takes” to give the
March 2016 Notices of the AMS 263 conjectures their full meaning, and (who knows!) 10. Pierre Deligne, Quelques idées maîtresses de l’œuvre de perhaps to give the means of proving them. [16, A. Grothendieck, in Matériaux pour l’Histoire des Mathéma- e p. P41] tiques au XX Siècle (Nice, 1996), Soc. Math. France, 1998, pp. 11–19. 11. David Eisenbud, Commutative Algebra, Springer-Verlag, New York, 2004. 12. Alexander Grothendieck, A General Theory of Fibre Spaces with Structure Sheaf, technical report, University of Kansas, 1955. 13. , Sur quelques points d’algèbre homologique, Tôhoku Mathematical Journal, 9:119–221, 1957. 14. , The cohomology theory of abstract algebraic va- rieties, in Proceedings of the International Congress of Mathematicians, 1958, Cambridge University Press, 1958, pp. 103–18. 15. , Revêtements Étales et Groupe Fondamental, Sémi- naire de géométrie algébrique du Bois-Marie, 1, Springer- Verlag, 1971. Generally cited as SGA 1. 16. , Récoltes et Semailles, Université des Sciences et Tech- niques du Languedoc, Montpellier, 1985–1987. Published in several successive volumes. 17. Alexander Grothendieck and Jean Dieudonné, Éléments de Géométrie Algébrique I, Springer-Verlag, 1971. 18. Luc Illusie, Alexander Beilinson, Spencer Bloch, Vladimir
Courtesy of Patricia Princehouse. Drinfeld et al., Reminiscences of Grothendieck and his Colin McLarty with a Great Pyrenees, from the region school, Notices of the Amer. Math. Soc., 57(9):1106–15, 2010. 19. Leopold Kronecker, Grundzüge einer arithmetischen theorie where Grothendieck spent much of his life. der algebraischen grössen, Crelle, Journal für die reine und angewandte Mathematik, XCII:1–122, 1882. 20. Serge Lang, Algebra, 3rd edition, Addison-Wesley, Reading, References MA, 1993. 1. Michael Artin, Interview, in Joel Segel, editor, Recountings: 21. Saunders Mac Lane, The work of Samuel Eilenberg in topol- Conversations with MIT Mathematicians, A K Peters/CRC ogy, in Alex Heller and Myles Tierney, editors, Algebra, Press, Wellesley, MA, 2009, pp. 351–74. Topology, and Category Theory: A Collection of Papers in 2. Michael Artin, Alexander Grothendieck, and Jean-Louis Honor of Samuel Eilenberg, Academic Press, New York, 1976, Verdier, Théorie des Topos et Cohomologie Etale des Sché- pp. 133–44. mas, Séminaire de géométrie algébrique du Bois-Marie, 4, 22. Colin McLarty, The rising sea: Grothendieck on simplicity Springer-Verlag, 1972, Three volumes, cited as SGA 4. and generality I, in Jeremy Gray and Karen Parshall, edi- 3. Michael Atiyah, The role of algebraic topology in mathemat- tors, Episodes in the History of Recent Algebra, American ics, Journal of the London Mathematical Society, 41:63–69, Mathematical Society, 2007, pp. 301–26, 1966. 23. Carol Parikh, The Unreal Life of Oscar Zariski, Springer- 4. , Bakerian lecture, 1975: Global geometry, Proceedings Verlag, New York, 2009. of the Royal Society of London Series A, 347(1650):291–99, 24. Henri Poincaré, Analysis situs, and five complements to it, 1976. 1895–1904, Collected in G. Darboux et al., eds., Oeuvres de 5. Pierre Berthelot, Alexander Grothendieck, and Luc Illusie, Henri Poincaré in 11 volumes, Paris: Gauthier-Villars, 1916– Théorie des intersections et théorème de Riemann-Roch, Num- 1956, Vol. VI, pp. 193–498. I quote the translation by John ber 225 in Séminaire de géométrie algébrique du Bois-Marie, Stillwell, Papers on Topology: Analysis Situs and Its Five 6, Springer-Verlag, 1971. Generally cited as SGA 6. Supplements, American Mathematical Society, 2010, p. 232. 6. Raoul Bott, Review of A. Borel and J-P. Serre, Le théorème 25. Michel Raynaud, André Weil and the foundations of algebraic de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136. geometry, Notices of the American Mathematical Society, 46: MR0116022 (22 #6817). 864–867, 1999. 7. Pierre Colmez and Jean-Pierre Serre, editors, Correspon- 26. Norbert Schappacher, A historical sketch of B. L. van der dance Grothendieck-Serre, Société Mathématique de France, Waerden’s work in algebraic geometry: 1926–1946, in Jeremy 2001. Expanded to Grothendieck-Serre Correspondence: Bilin- Gray and Karen Parshall, editors, Episodes in the History gual Edition, American Mathematical Society, and Société of Modern Algebra (1800–1950), American Mathematical Mathématique de France, 2004. Society, 2011, pp. 245–84. 8. Richard Dedekind and Heinrich Weber, Theorie der alge- 27. Jean-Pierre Serre, Faisceaux algébriques cohérents, Annals of braischen funktionen einer veränderlichen, J. Reine Angew. Mathematics, 61:197–277, 1955. Math., 92:181–290, 1882. Translated and introduced by John 28. , Espaces fibrés algébriques, in Séminaire Chevalley, Stillwell as Theory of Algebraic Functions of One Variable, chapter exposé no. 1, Secrétariat Mathématique, Institut copublication of the AMS and the London Mathematical Henri Poincaré, 1958. Society, 2012. 29. , Rapport au comité Fields sur les travaux de 9. Pierre Deligne, editor, Cohomologie Étale, Séminaire de A. Grothendieck (1965), 퐾-Theory, 3:199–204, 1989. géométrie algébrique du Bois-Marie, Springer-Verlag, 1977. 30. , André Weil: 6 May 1906–6 August 1998, Biographical Generally cited as SGA 4 1/2. This is not strictly a report on Memoirs of Fellows of the Royal Society, 45:520–29, 1999. Grothendieck’s Seminar.
264 Notices of the AMS Volume 63, Number 3 A M S 31. , Exposés de Séminaires 1950–1999, Société Mathéma- tique de France, 2001. 32. Peter Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, 2001. 33. Olga Taussky, Sums of squares, American Mathematical Monthly, 77:805–30, 1970. 34. Bartel L. van der Waerden, Zur Nullstellentheorie der Polynomideale, Mathematische Annalen, 96:183–208, 1926. 35. Charles Weibel, An Introduction to Homological Algebra, Algebraic Geometry II Cambridge University Press, 1994. David Mumford, Brown University, Providence, RI, and 36. André Weil, Foundations of Algebraic Geometry, American Tadao Oda, Tohoku University, Japan Mathematical Society, 1946. Several generations of students of algebraic geometry have 37. , Number of solutions of equations in finite fields, learned the subject from David Mumford’s fabled “Red Book”, Bulletin of the American Mathematical Society, 55:487–95, which contains notes of his lectures at Harvard University. Their 1949. genesis and evolution are described by Mumford in the preface: 38. , Number-theory and algebraic geometry, in Proceed- Initially, notes to the course were mimeographed and bound and sold by the Harvard mathematics department with a red cover. ings of the International Congress of Mathematicians (1950: These old notes were picked up by Springer and are now sold as Cambridge, Mass.), American Mathematical Society, 1952, pp. The Red Book of Varieties and Schemes. However, every time I 90–100. taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes… This book contains what Mumford had then intended to be Volume II. It covers the material in the “Red Book” in more depth, with several topics added. Mumford has revised the notes in col- laboration with Tadao Oda. The book is a sequel to Algebraic Geometry I, published by Springer-Verlag in 1976. Hindustan Book Agency; 2015; 516 pages; Hardcover; ISBN: 978-93-80250-80-9; List US$76; AMS members US$60.80; Order code HIN/70
Operators on Hilbert Space V. S. Sunder, Institute of Mathematical Sciences, Chennai, India
This book’s principal goals are: (i) to present the spectral theorem as a statement on the existence of a unique continuous and mea- surable functional calculus, (ii) to present a proof without digress- ing into a course on the Gelfand theory of commutative Banach algebras, (iii) to introduce the reader to the basic facts concerning the various von Neumann-Schatten ideals, the compact oper- ators, the trace-class operators and all bounded operators, and finally, (iv) to serve as a primer on the theory of bounded linear operators on separable Hilbert space. Hindustan Book Agency; 2015; 110 pages; Softcover; ISBN: 978- 93-80250-74-8; List US$40; AMS members US$32; Order code HIN/69
Problems in the Theory of Modular Forms M. Ram Murty, Michael Dewar, and Hester Graves, Queen’s University, Kingston, Ontario, Canada
This book introduces the reader to the fascinating world of mod- ular forms through a problem-solving approach. As such, it can be used by undergraduate and graduate students for self-instruc- tion. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ-function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms, and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field. Hindustan Book Agency; 2015; 310 pages; Softcover; ISBN: 978- 93-80250-72-4; List US$58; AMS members US$46.40; Order code HIN/68
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March 2016 Notices of the AMS 265