In relation to fundamental groups, you could say that abelian class field theory is the study of the galois action on. This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by m. For a pointed algebraic variety x, x over c there is a canonical isomorphism between the etale fundamental group. In comparison with my book, the emphasis is on heuristic arguments rather than. As a result, representations of the etale fundamental groupespecially the ones arising from etale cohomologyare ubiquitous. Clark october 25, 2006 1 introduction fundamental groups and galois groups. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the zariski topology. Ill complement the list of well known books on the subject by some freely available documents, which i find userfriendly.
Galois representations in etale fundamental groups and the. The etale fundamental group of a scheme is one of the most elaborated tool in arithmetic geometry. The book is well written and contains much information about the etale fundamental group. Etale cohomology theory nankai tracts in mathematics. Definition and basic properties, properties of morphisms of schemes, general techniques and constructions. 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. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of. Etale covers are closed under pullback and composition, and satisfy fpqc descent. X a\cap x, thus giving a set of base points which can be chosen. He also discusses the fundamental group of a scheme in this chapter. Berlin, heidelberg, new york, springer, 1969 ocolc655087247. Some of the examples will involve schemetheoretic concepts that we have not covered such as normality, smoothness, dimension etc.
Number theory learning seminar 20162017 the seminar will meet wednesdays 1. Definition and the basic properties, cohomology of curves, cohomological dimension, purity. The usual proof of it as given for instance in hatchers book h is tedious. An nite etale cover is a morphism that is nite and etale. Absolute galois groups, the simplest case, are very difficult to calculate. 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.
If kis the complex numbers, then xmay be regarded as a onedimensional compact complex manifold x. A trivial cover of xis one that is a nite disjoint union of copies of x. Etale cohomology theory lei fu bok 9789814307727 bokus. Galois groups and fundamental groups math berkeley.
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. 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. Here are some texts which might be of interest for understanding the geometry of finite etale covers. 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. What is the best textbook or book for studying etale cohomology. Verdier the whole theory is closely modelled on the usual theory of sheaves and their derived functor cohomology on a topological space.
A glossary of the categories in which we shall work, and fibre resolutions. We say that a scheme xis normal if all local rings o x. Let f be a nonconstant polynomial with coefficients in the complex numbers c \displaystyle \mathbb c. 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.
A glossary of the categories in which we shall work, and fibre resolutions proobjects in the homotopy category completions cohomological criteria for. 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. Kim on fundamental groups in number theory the ncategory cafe. 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. It enables one to translate intricate geometric problems into representationtheoretic ones, easier to handle. For a more detailed resource on this topic, i suggest the book sza09, aptly titled galois. Let kbe a eld and g gal kk be the absolute galois group. The etale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Etale homotopy edition 1 by michael artin, barry mazur. Etale morphisms, etale fundamental group, the local ring for the etale topology, sheaves for the etale topology, direct and inverse images of sheaves, cohomology.
The brauer group is defined in terms of equivalence classes of azumaya algebras, with the cohomological brauer group defined as the torsion part of the second etale cohomology group. Mar 15, 2011 etale cohomology is an important branch in arithmetic geometry. Namely, we study the topological properties of by studying its covers, the maps which are local isomorphisms. Thus, the etale fundamental group of the spectrum of a complete discrete valuation ring is isomorphic to the galois group of its residue eld. Ive difficulty in understanding the difference between the two cases. Here the etale fundamental group is just the absolute galois group. Numerous and frequentlyupdated resource results are available from this search. But the author does show effectively that etale morphisms for schemes are essentially local isomorphisms in a sense.
When going over to a general scheme, this geometric picture is lost, and one must rely completely on algebraic constructions. For a more detailed resource on this topic, i suggest the book sza09, aptly titled \galois groups and fundamental groups. These are the theories of galois groups and eld extensions and of fundamental groups and covering spaces. Friedlander presents many of his own applications of this theory to algebraic topology, finite chevalley groups, and algebraic geometry. New edition available hereetale cohomology is an important branch in arithmetic geometry. We write fet x for the category of etale covers of x. Fundamental group homotopy homotopy group fibrations. 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. A ne, and in particular etale morphisms are separated. Pdf stratified bundles and \etale fundamental group. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
References for etale cohomology and related topics fall 2011. Let xkbe a smooth, complete connected curve over an algebraically closed. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. 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. Tamas szamuely, galois groups and fundamental groups. Etale cohomology is an important branch in arithmetic geometry. Topologythe fundamental group wikibooks, open books for. Etale cohomology pms33, volume 33 princeton university press.
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. 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. Pdf stratified bundles and \etale fundamental group new. Galois groups and fundamental groups cambridge studies in. The reader can find the basic facts about locals in the book. Etale cohomology pms33, volume 33 princeton mathematical.
Its no reason that the etale fundamental group is hard or even impossible to calculate. Free algebraic geometry books download ebooks online. 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. I am looking for exact references for the comparison theorem for the etale fundamental group.
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