Kim on fundamental groups in number theory the ncategory cafe. When going over to a general scheme, this geometric picture is lost, and one must rely completely on algebraic constructions. An nite etale cover is a morphism that is nite and etale. Absolute galois groups, the simplest case, are very difficult to calculate. An improvement on the fundamental group and the total fundamental groupoid relevant to the van kampen theorem for computing the fundamental group or groupoid is to use. Etale fundamental group examples october 15, 2014 in this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the zariski topology. For a pointed algebraic variety x, x over c there is a canonical isomorphism between the etale fundamental group. Definition and basic properties, properties of morphisms of schemes, general techniques and constructions. In relation to fundamental groups, you could say that abelian class field theory is the study of the galois action on. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. The etale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Namely, we study the topological properties of by studying its covers, the maps which are local isomorphisms.

We shall not be able to avoid using spectral sequences see pp 307309 of my book on etale cohomology for a brief summary of spectral sequences and chapter 5 of weibels book for a complete treatment. Galois groups and fundamental groups cambridge studies in. Galois groups and fundamental groups math berkeley. Friedlander presents many of his own applications of this theory to algebraic topology, finite chevalley groups, and algebraic geometry. He also discusses the fundamental group of a scheme in this chapter. The next two chapters concern the basic theory of etale sheaves and elementary etale cohomology, and are followed by an application of the cohomology to the study of the brauer group. For a more detailed resource on this topic, i suggest the book sza09, aptly titled galois. This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by m. The etale fundamental group dexter chua 1 introduction 1 2 etale morphisms 3 3 etale covers 6 4 the etale fundamental group 7 5 galois theory 8 appendix a faithfully at morphisms 10 1introduction the fundamental theorem of galois theory says theorem galois theory. Galois representations in etale fundamental groups and the. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of. But the author does show effectively that etale morphisms for schemes are essentially local isomorphisms in a sense.

We say that a scheme xis normal if all local rings o x. Tamas szamuely, galois groups and fundamental groups. Theres a nice book by janelidze and borceaux called galois theories that covers categorical galois theory generalise galois theory to a purely categorical setting, and then respecialise to various categories that covers some of this the topological equivalent of the classical galois connection is the equivalence between the the structure of covering spaces and the subgroup. It enables one to translate intricate geometric problems into representationtheoretic ones, easier to handle. Etale homotopy edition 1 by michael artin, barry mazur. Here the etale fundamental group is just the absolute galois group. X a\cap x, thus giving a set of base points which can be chosen. The etale fundamental group of a scheme is one of the most elaborated tool in arithmetic geometry. Thus, the etale fundamental group of the spectrum of a complete discrete valuation ring is isomorphic to the galois group of its residue eld. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. Numerous and frequentlyupdated resource results are available from this search. Functors, isomorphic and equivalent categories, representable functors, some constructions in the light of representable functors, schemes. For a more detailed resource on this topic, i suggest the book sza09, aptly titled \galois groups and fundamental groups.

Here are some texts which might be of interest for understanding the geometry of finite etale covers. The reader can find the basic facts about locals in the book. Berlin, heidelberg, new york, springer, 1969 ocolc655087247. Ive difficulty in understanding the difference between the two cases. Etale covers are closed under pullback and composition, and satisfy fpqc descent. Fundamental group homotopy homotopy group fibrations. Clark october 25, 2006 1 introduction fundamental groups and galois groups. The usual proof of it as given for instance in hatchers book h is tedious. What is the best textbook or book for studying etale cohomology. If kis the complex numbers, then xmay be regarded as a onedimensional compact complex manifold x. Etale cohomology is an important branch in arithmetic geometry. Ill complement the list of well known books on the subject by some freely available documents, which i find userfriendly. These are the theories of galois groups and eld extensions and of fundamental groups and covering spaces. Definition and the basic properties, cohomology of curves, cohomological dimension, purity.

Some of the examples will involve schemetheoretic concepts that we have not covered such as normality, smoothness, dimension etc. The purpose of this paper is to explain how padic hodge theory for the unipotent fundamental group provides examples of extensions of crystalline representations which are semistable but not crystalline, and where the monodromy operator has a. As a result, representations of the etale fundamental groupespecially the ones arising from etale cohomologyare ubiquitous. For a scheme x that is of finite type over c, the complex numbers, there is a close relation between the etale fundamental group of x and the usual, topological, fundamental group of xc, the complex analytic space attached to x. Etale morphisms, etale fundamental group, the local ring for the etale topology, sheaves for the etale topology, direct and inverse images of sheaves, cohomology. Etale cohomology pms33, volume 33 princeton university press. Let kbe a eld and g gal kk be the absolute galois group. A trivial cover of xis one that is a nite disjoint union of copies of x. Pdf stratified bundles and \etale fundamental group new. A glossary of the categories in which we shall work, and fibre resolutions. Etale cohomology pms33, volume 33 princeton mathematical. Verdier the whole theory is closely modelled on the usual theory of sheaves and their derived functor cohomology on a topological space. In comparison with my book, the emphasis is on heuristic arguments rather than.

This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ladic cohomology. Its no reason that the etale fundamental group is hard or even impossible to calculate. Etale cohomology theory nankai tracts in mathematics. On the whole, the book is useful for mathematicians and graduate students looking for one place where they can find information about the etale fundamental group and the related nori fundamental group scheme. A ne, and in particular etale morphisms are separated. Free algebraic geometry books download ebooks online. We write fet x for the category of etale covers of x. Mar 15, 2011 etale cohomology is an important branch in arithmetic geometry. New edition available hereetale cohomology is an important branch in arithmetic geometry. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Oct 11, 2016 this book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by m. Pdf stratified bundles and \etale fundamental group. This years seminar will focus on etale cohomology, the goal being to understand laumons proof of the main theorem of delignes weil ii paper that gave a powerful and vast generalization of the riemann hypothesis over finite fields.

The book is well written and contains much information about the etale fundamental group. I am looking for exact references for the comparison theorem for the etale fundamental group. There is an injection of the brauer group into the cohomological brauer group, and this chapter outlines what was known at the time of publication when this is a. A glossary of the categories in which we shall work, and fibre resolutions proobjects in the homotopy category completions cohomological criteria for. Topologythe fundamental group wikibooks, open books for. Let f be a nonconstant polynomial with coefficients in the complex numbers c \displaystyle \mathbb c. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. Let xkbe a smooth, complete connected curve over an algebraically closed. References for etale cohomology and related topics fall 2011.

1510 73 1161 511 662 1534 536 660 820 1471 435 428 373 1271 766 1045 230 696 1380 962 1461 429 293 230 1256 813 340 254 54 1096 408 908 361 505