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.