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Algebraic Geometry from an a -Homotopic Viewpoint

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Algebraic Geometry from an a -Homotopic Viewpoint Algebraic geometry from an 1 A -homotopic viewpoint Aravind Asok February 4, 2019 2 Contents 1 Topological preliminaries 9 1 1 2 Affine varieties, A -invariance and naive A -homotopies 17 3 Projective modules, gluing and locally free modules 33 1 4 Picard groups, normality and A -invariance 55 1 5 Grothendieck groups, regularity and A -invariance 79 6 Local triviality and smooth fiber bundles 103 7 Schemes, sheaves and Grothendieck topologies 129 1 8 Vector bundles and A -invariance 155 1 9 Symmetric bilinear forms and endomorphisms of P 175 10 Punctured affine space and naive homotopy classes 189 11 Milnor K-theory 191 12 Milnor–Witt K-theory 195 A Sets, Categories and Functors 199 3 CONTENTS 4 Outline/Motivation 1 This course is supposed to serve as an introduction to A -algebraic topology, which is a term coined by Fabien Morel [Mor06b, Mor12]. Officially, the prerequisites for this class were the introduc- tory topology sequence (homology, covering spaces, fundamental groups, manifolds, elementary differential topology, de Rham cohomology) and the algebra sequence (groups, rings, fields, lin- ear algebra, a touch of commutative algebra and some elementary non-commutative ring theory). The true prerequisite is willingness to learn on the fly. Given that background, this course was constructed around the following conceit: it is possible and interesting to learn something about 1 A -homotopy theory. 1 Thus, the natural question to ask is: why should one care about A -homotopy theory (especially if you have not yet learned homotopy theory or algebraic geometry!)? This course is essentially my answer to this question: I want to provide a “modern” approach to the study of projective modules over commutative rings. The theory of projective modules is by now a very classical subject: the formal notion of projec- tive module goes back to the work of Cartan–Eilenberg in the foundations of homological algebra [CE99, Chapter I.2], but examples of projective modules arose much earlier (e.g., the theory of in- vertible fractional ideals). The notion of projective module becomes indispensible in cohomology, e.g., group cohomology may be computed using projective resolutions. One may look at the collec- tion of projective modules over a ring as a certain invariant of the ring itself (“representations” of the ring). The results of Serre showed that the language of algebraic geometry might provide a good language to study projective modules over commutative unital rings [Ser55, x50 p. 242]. More precisely, Serre showed that finitely generated projective modules over commutative unital rings are precisely the same things as finite rank algebraic vector bundles over affine algebraic varieties. Serre furthermore showed that this dictionary was useful for providing a better understanding about projective modules because it allowed one to exploit an analogy between algebraic geometry and algebraic topology: projective modules over rings are analogous to vector bundles over topological spaces. From this point of view, projective modules take on additional significance. For example, in differential topology, one may turn a non-linear problem (e.g., existence of an immersion of one manifold into another) into a linear problem by looking at associated bundles (a corresponding injection of tangent bundles in the case of immersions). In good situations, a solution to the linear problem can actually be promoted to a solution of the non-linear problem (in our parenthetical exmaple, this is an incarnation of the Hirsch-Smale theory of immersions). Based on this analogy, Serre observed that if R is a Noetherian ring of dimension d, one could “simplify” projective modules P of rank r > d greater than the dimension: any such module could be written as a sum of a projective module of rank r0 ≤ d and a free module of rank r0 − r [Ser58b]. After the work of Bass [Bas64], which furthermore amalgamated Grothendieck’s ideas regarding K-theory with Serre’s results, J.F. Adams wrote: “This leads to the following programme: take definitions, constructions and theorems from bundle-theory; express them as particular cases of definitions, constructions and statements about finitely-generated projective modules over a general ring; and finally, try to prove the statements under suitable assumptions”. 5 CONTENTS 1 My point of view in this class is that the Morel-Voevodsy A -homotopy theory provides arguably the ultimate realization of this program. Pontryagin and Steenrod observed that one could use techniques of homotopy theory to study vector bundles on spaces having the homotopy type of CW complexes. Indeed, the basic goal of the class will be to establish the analog in algebraic geometry of this result, at least for sufficiently nice (i.e., non-singular) affine varieties. Looking beyond this, just as the Weil conjectures provides a beautiful link between the arithmetic problem of counting the number of solutions of a system of equations over a finite field and a “topologically inspired” etale´ cohomology theory of algebraic 1 varieties, A -homotopy theory allows one to construct a link between the algebraic theory of projec- tive modules over commutative rings, and an “algebro-geometric” analog of the homotopy groups of spheres! After very quickly recalling some of the topological constructions that provide sources of in- spiration (and which we will attempt to mirror), I will begin a brief study of the theory of affine algebraic varieties. While this will not suffice for our eventual applications, affine varieties are, arguably, intuitively appealing, and it seemed better not to require too much algebro-geometric so- phistication at first. Then, I will introduce a “naive” version of homotopy for algebraic varieties and, following the topological story, describe various “homotopy invariants” in algebraic geometry. Along the way, I will introduce a number of important invariants of algebraic varieties: projective modules, Picard groups, and K-theory. The ultimate goal is to prove Lindel’s theorem that shows that the functor “isomorphism classes of projective modules” is homotopy invariant, in a suitable algebro-geometric sense, on suitably nice (i.e., non-singular, affine) algebraic varieties. Along the way, I will try to 1 build things up in a way that motivates some of the tools used in the study of A -homotopy theory over a field. There are many texts that talk about cohomology theories in algebraic geometry and these notes are not intended to be another such text. Rather, there is a hope, supported by recent results, that 1 A -homotopy theory can give us information not just about cohomology of algebraic varieties, but actually about their geometry. We have attempted to illustrate this by focusing on projective modules and vector bundles on algebraic varieties. What’s next? To proceed from the “naive” theory to the “true” theory, requires more sophistication: one needs to know some homotopy theory of simplicial sets, Grothendieck topologies, model categories etc. The syllabus listed the following plan, which I would argue is the “next step” beyond what I now want to cover in the class. The subsequent background is written with this plan in mind. • Week 1. Some abstract algebraic geometry: the Nisnevich topology and basic properties. • Week 2. Simplicial sets and simplicial (pre)sheaves. 1 • Week 3. Model categories in brief; the simplicial and A -homotopy categories 1 • Week 4. Basic properties of the A -homotopy category (e.g., homotopy purity) • Week 5. Fibrancy, cd-structures and descent • Week 6. Classifying spaces: simplicial homotopy classification of torsors 1 • Week 7. A -homotopy classification results 1 • Week 8. Eilenberg-MacLane spaces and strong and strict A -invariance CONTENTS 6 • Week 9. Postnikov towers 1 • Week 10. Homotopy sheaves and A -connectivity 1 • Week 11. The unstable A -connectivity property and applications • Week 12. Loop spaces and relative connectivity 1 • Week 13. Gersten resolutions and strong/strict A -invariance 1 1 • Week 14. A -homology and A -homotopy sheaves 1 • Week 15. A -quasifibrations and some computations of homotopy sheaves Background. While there isn’t a specific textbook for the class, I will use a number of different sources for some of the background material. There is formally quite a lot of background for the subject of the class and I don’t expect anyone to have digested all the prerequisites in any sort of linear fashion. Instead, there will be a lot of “on-the-fly” learning and going backwards to fill in details as necessary. • To get started, I will expect that people know some basic things about commutative ring theory. A good introductory textbook is [AM69], but [Mat89] is more comprehensive. For a discussion that is more algebro-geometric, you can look at [Eis95]. We will also need more detailed results about modules, for which you consule [Lam99]. • We will study affine varieties and eventually discuss sheaf cohomology on topological spaces. Beyond what I mention in the class, useful references for the theory of algebraic varieties will include [Har77, Chapters 1-2]. Useful background for the notions of sheaf cohomology we will need on topological spaces in general, and on schemes in particular, can be found in [God73] or [Har77, Chapter 3]. Implicit here is a basic understanding of some ideas from homological algebra [Wei94]. Furthermore, from the standpoint of references, I think there is now no better definitive source than Johan de Jong’s Stacks Project [Sta15]. • I will also expect some familiarity with basic concepts of algebraic and differential topology, e.g., topological spaces, smooth manifolds and maps, CW complexes, singular homology, covering spaces, vector bundles, and homotopy groups as can be found in [Spa81] or [Hat02]. The point of view exposed in [AGP02] will also be useful. • Finally, the course will, from the beginning, use category-theoretic terminology. Beyond the usual notions of categories, functors, and natural transformations, I will expect some famil- iarity with various kinds of universal properties, limits (and colimits) and adjoint functors and their properties, as can be picked up in [ML98] or [Kel05].
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