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Grothendieck and Algebraic Geometry Asia Pacific Mathematics Newsletter 1 GrothendieckGrothendieck and and Algebraic Algebraic Geometry Geometry LucLuc Illusie Illusie and and Michel Michel Raynaud Raynaud Grothendieck’s thesis and subsequent publica- it was Grothendieck who showed all its wealth tions in the early 1950’s dealt with functional and flexibility. Starting with a category , to each C analysis. This was remarkable work, which is object X of one can associate a contravariant func- C attracting new attention today.a Still, his most tor on with values in the category of sets, h : C X important contributions are in algebraic geometry, Sets, sending the object T to Hom (T, X). By a C→ C a field which occupied him entirely from the late classical lemma of Yoneda, the functor X h is → X 1950’s on, in particular during the whole time he fully faithful. To preserve the geometric language, was a professor at the Institut des Hautes Etudes´ Grothendieck called hX(T) the set of points of X Scientifiques (IHES)´ (1959–1970). with values in T. Thus, an object X is known when Algebraic geometry studies objects defined by we know its points with values in every object T. polynomial equationsb and interprets them in a Grothendieck applied this to algebraic geometry. geometric language. A major problem faced by This was revolutionary as, until then, only field algebraic geometers was to define a good frame- valued points had been considered. work and develop local to global techniques. As an example, suppose we have a system of In the early 1950’s complex analytic geometry equations showed the way with the use of sheaf theory. f1(x1, ..., xn) = = fN(x1, ..., xn) = 0, (1) Thus, a complex analytic space is a ringed space, ··· with its underlying space and sheaf of holomor- where the fi’s are polynomials with coefficients in Z. Let be the opposite category of the category phic functions (Oka, H Cartan). Coherent sheaves A of modules over this sheaf of rings play an impor- of rings, and let F be the (contravariant) functor tant role.c In 1954 Serre transposed this viewpoint sending a ring A to the set F(A) of solutions (x ), x A, of (1). This functor is nothing but the to algebraic geometry for varieties defined over i i ∈ functor h for X the object of corresponding to an algebraically closed field. He employed the X A Zariski topology, a topology with few open sub- the quotient of Z[x1, ..., xn] by the ideal generated sets, whose definition is entirely algebraic (with by the fi’s: the functor F is represented by X, an no topology on the base field), but which is well affine scheme. Points of X with values in C are adapted, for example, to the description of a pro- points of a complex algebraic variety — that one jective space as a union of affine spaces, and gives can possibly study by analytic methods — while rise to a cohomology theory which enabled him, points with values in Z, Q, or in a finite field for example, to compare certain algebraic and are solutions of a diophantine problem. Thus the analytic invariants of complex projective varieties. functor F relates arithmetic and geometry. Inspired by this, Grothendieck introduced If the fi’s have coefficients in a ring B instead of Z, the analogous functor F on the category schemes as ringed spaces obtained by gluing (for A the Zariski topology) spectra of general com- opposite to that of B-algebras, sending a B-algebra mutative rings. Furthermore, he described these A to the set F(A) of solutions of (1) with values objects from a functorial viewpoint. The language in A, is similarly represented by the spectrum of categories already existed, having appeared in X of a B-algebra (quotient of B[x1, ..., xn] by the the framework of homological algebra, following ideal generated by the fi’s), a scheme over the the publication of Cartan–Eilenberg’s book (Ho- spectrum of B. In this way a relative viewpoint mological Algebra, Princeton Univ. Press, 1956). But appears, for which the language of schemes is perfectly suited. The essential tool is base change, aSee [G Pisier, Grothendieck’s Theorem, Past and Present, Bull. Amer. Math. Soc. (N.S.) 49(2) (2012) 237–323]. a generalisation of the notion of extension of bHowever, we seldom saw Grothendieck write an explicit scalars: given a scheme X over S and a base equation on the blackboard; he did it only for basic, crucial change morphism S S, we get a new scheme cases. → c Cf. Cartan’s famous theorems A and B. X over S, namely, the fiber product of X and S January 2015, Volume 5 No 1 1 Asia Pacific Mathematics Newsletter 2 over S. In particular, X defines a family of schemes covering, satisfying a small number of properties, Xs parameterised by the points s of S. The above similar to those satisfied by open coverings in functor F then becomes the functor sending a topological spaces. The conceptual jump is that S-scheme T to the set of S-morphisms from T to the arrows U U are not necessarily inclusions.f i → X. A number of useful properties of X/S (such Grothendieck developed the corresponding no- as smoothness or properness) can nicely be read tions of sheaf and cohomology. The basic example g on the functor hX. The two above mentioned is the ´etaletopology. A seminar run by M Artin properties are stable under base change, as is at Harvard in the spring of 1962 started its sys- flatness, a property that plays a central role in tematic study. Given a scheme X, the category to algebraic geometry, as Grothendieck showed. In be considered is that of etale´ maps U X, and → 1968, thanks to M Artin’s approximation theorem, covering families are families (U U) such that i → it became possible to characterise functors that U is the union of the images of the Uis. The defi- are representable by algebraic spaces (objects very nition of an etale´ morphism of schemes is purely close to schemes) by a list of properties of the algebraic, but one should keep in mind that if X is functor, each of them often being relatively easy a complex algebraic variety, a morphism Y X → to check. But already in 1960, using only the is etale´ if and only if the morphism Yan Xan → notion of flatness, Grothendieck had constructed, between the associated analytic spaces is a local in a very natural way, Hilbert and Picard schemes isomorphism. A finite Galois extension is another as representing certain functors, at once super- typical example of an etale´ morphism. seding — by far — all that had been written on For torsion coefficients, such as Z/nZ, one the subject before. obtains a good cohomology theory Hi(X, Z/nZ), Nilpotent elementsd in the local rings of at least for n prime to the residue characteristics schemes appear naturally (for example in fiber of the local rings of X. Taking integers n of the products), and they play a key role in questions form r for a fixed prime number , and pass- of infinitesimal deformations. Using them system- ing to the limit, one obtains cohomologies with r atically, Grothendieck constructed a very general values in Z = lim Z/ Z, and its fraction field differential calculus on schemes, encompassing Q. If X is a complex←−− algebraic variety, one has arithmetic and geometry. comparison isomorphisms (due to M Artin) be- In 1949 Weil formulated his conjectures on tween the etale´ cohomology groups Hi(X, Z/rZ) varieties over finite fields. They suggested that and the Betti cohomology groups Hi(Xan, Z/rZ),h it would be desirable to have at one’s disposal thus providing a purely algebraic interpretation of a cohomology with discrete coefficients satisfying the latter. Now, if X is an algebraic variety over an analogue of the Lefschetz fixed point formula. an arbitrary field k (but of characteristic ) (a In classical algebraic topology, cohomology with k-scheme of finite type in Grothendieck’s lan- discrete coefficients, such as Z, is reached by cut- guage), k an algebraic closure of k, and Xk de- ting a complicated object into elementary pieces, duced from X by extension of scalars, the groups i such as simplices, and studying how they overlap. H (Xk, Q) are finite dimensional Q-vector spaces, In algebraic geometry, the Zariski topology is too and they are equipped with a continuous action of coarse to allow such a process. To bypass this the Galois group Gal(k/k). It is especially through obstacle, Grothendieck created a conceptual rev- these representations that algebraic geometry in- olution in topology by presenting new notions of terests arithmeticians. When k is a finite field Fq, in gluing (a general theory of descent,e conceived al- which case Gal(k/k) is generated by the Frobenius ready in 1959), giving rise to new spaces: sites and substitution a aq, the Weil conjectures, which → topoi, defined by what we now call Grothendieck are now proven, give a lot of information about topologies. A Grothendieck topology on a category these representations. Etale´ cohomology enabled is the datum of a particular class of morphisms Grothendieck to prove the first three of these and families of morphisms (Ui U)i I, called → ∈ fMore precisely, monomorphisms, in categorical language. dAn element x of a ring is called nilpotent if there exists an gThe choice of the word ´etale is due to Grothendieck. integer n 1 such that xn = 0. hBut not between Hi(X, Z) and Hi(Xan, Z): by passing to e ≥ i The word “descent” had been introduced by Weil in the case the limit one gets an isomorphism between H (X, Z) and i an of Galois extensions.
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