Yuri Manin Introduction into theory of schemes Translated from the Russian and edited by Dimitry Leites Abdus Salam School of Mathematical Sciences Lahore, Pakistan . Summary. This book provides one with a concise but extremely lucid exposition of basics of algebraic geometry seasoned by illuminating examples. The preprints of this book proved useful to students majoring in mathematics and modern mathematical physics, as well as professionals in these ¯elds. °c Yu. Manin, 2009 °c D. Leites (translation from the Russian, editing), 2009 Contents Editor's preface :::::::::::::::::::::::::::::::::::::::::::::::: 5 Author's preface ::::::::::::::::::::::::::::::::::::::::::::::: 6 1 A±ne schemes ::::::::::::::::::::::::::::::::::::::::::::: 7 1.1 Equations and rings . 7 1.2 Geometric language: Points . 12 1.3 Geometric language, continued . 16 1.4 The Zariski topology on Spec A ........................... 19 1.5 The a±ne schemes (a preliminary de¯nition) . 29 1.6 Topological properties of morphisms and Spm . 33 1.7 The closed subschemes and the primary decomposition . 42 1.8 Hilbert's Nullstellensatz (Theorem on zeroes) . 49 1.9 The ¯ber products . 52 1.10 The vector bundles and projective modules . 55 1.11 The normal bundle and regular embeddings . 63 1.12 The di®erentials . 66 1.13 Digression: Serre's problem and Seshadri's theorem . 71 1.14 Digression: ³-function of a ring . 74 1.15 The a±ne group schemes . 79 1.16 Appendix: The language of categories. Representing functors . 87 1.17 Solutions of selected problems of Chapter 1 . 96 2 Sheaves, schemes, and projective spaces ::::::::::::::::::: 97 2.1 Basics on sheaves . 97 2.2 The structure sheaf on Spec A ............................. 103 2.3 The ringed spaces. Schemes . 107 2.4 The projective spectra . 112 2.5 Algebraic invariants of graded rings . 116 2.6 The presheaves and sheaves of modules . 125 2.7 The invertible sheaves and the Picard group . 131 4 Contents 2.8 The Cech· cohomology . 137 2.9 Cohomology of the projective space . 145 2.10 Serre's theorem . 150 2.11 Sheaves on Proj R and graded modules . 153 2.12 Applications to the theory of Hilbert polynomials . 156 2.13 The Grothendieck group: First notions . 161 2.14 Resolutions and smoothness . 166 References ::::::::::::::::::::::::::::::::::::::::::::::::::::: 172 Index :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 177 Prefaces 5 Editor's preface It is for more than 40 years now that I wanted to make these lectures | my ¯rst love | available to the reader: The preprints of these lectures [Ma1, Ma2] of cir- culation of mere 500+200 copies became bibliographic rarities almost immediately. Meanwhile the elements of algebraic geometry became everyday language of work- ing theoretical physicists and the need in a concise manual only increased. Various (nice) text-books are usually too thick for anybody who does not want to become a professional algebraic geometer, which makes Manin's lectures even more appealing. The methods described in these lectures became working tools of theoretical physicists whose subject ranges from high energy physics to solid body physics (see [Del] and [Ef], respectively) | in all questions where supersymmetry naturally arises. In mathematics, supersymmetry in in-build in everything related to homotopy and exterior products of di®erential forms, hence to (co)homology. In certain places, supersymmetric point of view is inevitable, as in the study of integrability of certain equations of mathematical physics ([MaG]). Therefore, preparing proceedings of my Seminar on Supermanifolds (see [SoS]) I urgently needed a concise and clear introduction into basics of algebraic geometry. In 1986 Manin wrote me a letter allowing me to include a draft of this trans- lation as a Chapter in [SoS], and it was preprinted in Reports of Department of Mathematics of Stockholm University. My 1972 de¯nition of superschemes and su- permanifolds [L0] was based on these lectures; they are the briefest and clearest source of the background needed for studying those aspects of supersymmetries that can not be reduced to linear algebra. Written at the same time as Macdonald's lectures [M] and Mumford's lectures [M1, M2, M3] Manin's lectures are more lucid and easier to come to grasps. Later on there appeared several books illustrating the topic from di®erent positions, e.g., [AM], [Sh0], [E1, E2, E3], [H, Kz] and I particularly recommend [Reid] for the ¯rst reading complementary to this book. For preliminaries on algebra, see [vdW, Lang]; on sheaves, see [God, KaS]; on topology, see [K, FFG, RF, Bb3]; on number theory, see [Sh1, Sh2]. £ In this book, N := f1; 2;::: g and Z+ := f0; 1; 2;::: g; Fm := Z=m; A is the group of invertible elements of the ring A. The term \identity" is only applied to relations and maps, the element 1 (sometimes denoted by e) such that 1a = a = a1 is called unit (sometimes unity). Other notation is standard. I advise the reader to digress from the main text and skim the section on cate- gories as soon as the word \functor" or \category" appears for the ¯rst time. The responsibility for footnotes is mine. D. Leites, April 2, 2009. 6 Prefaces Author's preface In 1966{68, at the Department of Mechanics and Mathematics of Moscow State University, I read a two-year-long course in algebraic geometry. The transcripts of the lectures of the ¯rst year were preprinted [Ma1, Ma2] (they constitute this book), that of the second year was published in the Russian Mathematical Surveys [Ma3]. These publications bear the remnants of the lecture style with its pros and cons. Our goal is to teach the reader to practice the geometric language of commutative algebra. The necessity to separately present algebraic material and later \apply" it to algebraic geometry constantly discouraged geometers, see a moving account by O. Zariski and P. Samuel in their preface to [ZS]. The appearance of Grothendieck's scheme theory opened a lucky possibility not to draw any line whatever between \geometry" and \algebra": They appear now as complimentary aspects of a whole, like varieties and the spaces of functions on these spaces in other geometric theories. From this point of view, the commutative algebra coincides with (more pre- cisely, is functorially dual to) the theory of local geometric objects | affine schemes. This book is devoted to deciphering the above claim. I tried to consecutively explain what type of geometric images should be related with, say, primary de- composition, modules, and nilpotents. In A. Weil's words, the spacial intuition is \invaluable if its restrictedness is taken into account".I strived to take into account both terms of this neat formulation. Certainly, geometric accent considerably influenced the choice of material. In particular, this chapter should prepare the ground for introducing global objects. Therefore, the section on vector bundles gives, on \naive" level, constructions be- longing, essentially, to the sheaf theory. Finally, I wanted to introduce the categoric notions as soon as possible; they are not so important in local questions but play ever increasing role in what follows. I advise the reader to skim through the section \Language of categories" and return to it as needed. 1) I am absolutely incapable to edit my old texts; if I start doing it, an irresistible desire to throw everything away and rewrite completely grips me. But to do some- thing new is more interesting. Therefore I wish to heartily thank D. Leites who saved my time. The following list of sources is by no means exhaustive. It may help the reader to come to grasps with the working aspects of the theory: [Sh0], [Bb1] (general courses); [M1], [M2], [Ma3], [MaG], [S1], [S2] (more special questions). The approach of this book can be extended, to an extent, to non-commutative geometry [Kas]. My approach was gradually developing in the direction along which the same guiding principle | construct a matrix with commuting elements satisfying only the absolutely necessary commutation relations | turned out applicable in ever wider context of n o n - c o m m u t a t i v e geometries, see [MaT], see also [GK], [BM] (extension to operads and further). Yu. Manin, March 22, 2009. 1 See also [McL, GM]. Chapter 1 A±ne schemes 1.1. Equations and rings Study of algebraic equations is an ancient mathematical science. In novel times, vogue and convenience dictate us to turn to rings. 1.1.1. Systems of equations. Let I, J be some sets of indices, let T = fTjgj2J be indeterminates and F = fFi 2 K[T ]gi2I a set of polyno- mials. A system X of equations for unknowns T is the triple (the ring K, the unknowns T , the functions F ), more exactly and conventionally expressed as Fi(T ) = 0; where i 2 I: (1.1) Why the ground ring or the ring of constants K enters the de¯nition is clear: The coe±cients of the Fi belong to a ¯xed ring K. We say that the system X is de¯ned over K. What should we take for a solution of (1.1)? To say, \a solution of (1.1) is a set t = (tj)j2J of elements of K such that Fi(t) = 0 for all i" is too restrictive: We wish to consider, say, complex roots of equations with real coe±cients. The radical resolution of this predicament is to consider solutions in any ring, and, \for simplicity", in all of the rings simultaneously. To consider solutions of X belonging to a ring L, we should be able to substitute the elements of L into Fi, the polynomials with coe±cients from K, i.e., we should be able to multiply the elements from L by the elements from K and add the results, that is why L must be a K-algebra. Recall that the set L is said to be a K-algebra if L is endowed with the structures of a ring and a K-module, interrelated by the following properties: 1.