Glossary

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G-infinity algebraWrite comment View comments

Mentioned in http://arxiv.org/abs/0710.4510

Mentioned here: http://www.ncatlab.org/nlab/show/homotopy+BV-algebra

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See the AMS book on several complex variables (Taylor???) for a good introduction to the classical GAGA.

Toen and Vaquie proves something about when a complex variety is algebraizable, in terms of a finiteness condition (saturatedness) on an associated dg-catégorie whose homotopy cat is the derived cat of perfect complexes on the variety. File: Toen web publ caca.pdf.

Some slider from a lecture on GAGA: File AG/Various/Lecture - presentation

http://mathoverflow.net/questions/10724/relationship-between-algebraic-and-holomorphic-differential-forms

http://mathoverflow.net/questions/17937/algebraic-de-rham-cohomology-vs-analytic-de-rham-cohomology

arXiv:1101.5123 Generalizing the GAGA Principle from arXiv Front: math.AG by Jack Hall This paper generalizes the fundamental GAGA results of Serre cite{MR0082175} in three ways---to the non-separated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of $\mathcal{M}_{g,n}$ possessing modular interpretations are algebraizable.

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Galois categoriesWrite comment View comments

Title: A Tannakian Context for Galois. http://front.math.ucdavis.edu/1110.6411 Authors: Eduardo J. Dubuc, Martin Szyld. Abstract: Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.

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Galois moduleWrite comment View comments

Bondarko has a few papers about Galois modules over local fields.

Burns is also interested in Galois modules I think

Many things by Snaith et al, for example the book Galois module structure, Fields Inst Monographs 2.

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Galois representationsWrite comment View comments

See reference list for the study group on Langlands, I think in an email from Tobias Berger

Galois rep folder under N TH

Taylor articles

Possibly of some interest: book by Snaith: Topological methods in Galois rep theory, late 80s

Strauch reference list

http://mathoverflow.net/questions/77278/introductory-text-on-galois-representations

http://mathoverflow.net/questions/2791/understanding-gal-bar-q-q

Title: Modular forms of weight one: Galois representations and dimension Authors: Denis Trotabas The present notes are the expanded and polished version of three lectures given in Stanford, concerning the analytic and arithmetic properties of weight one modular forms. The author tried to write them in a style accessible to non-analytically oriented number theoritists: in particular, some effort is made to be precise on statements involving uniformity in the parameters. On the other hand, another purpose was to provide an introduction, together with a set of references, consciously kept small, to the realm of Galois representations, for non-algebraists -- like the author. The proofs are sketched, at best, but we tried to motivate the results, and to relate them to interesting conjectures. http://arxiv.org/abs/0906.4579

[arXiv:1207.6724] Variations on a theorem of Tate fra arXiv Front: math.NT av Stefan Patrikis Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $Gal(\bar{F}/F) \to PGLn(C)$ lift to $GLn(C)$. We take special interest in the interaction of this result with algebraicity (on the automorphic side) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois "Tannakian formalisms"; monodromy (independence-of-$\ell$) questions for abstract Galois representations.

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Galois theoryWrite comment View comments

Hoobler, Raymond T. Purely inseparable Galois theory. Ring theory (Proc. Oklahoma Conf., Univ. Oklahoma, Norman, Okla., 1973), pp. 207–240. Lecture Notes in Pure and Appl. Math., Vol. 7, Dekker, New York, 1974.

See Rognes for ring spectra

Book: Galois theories. Abstract stuff

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Gauge theoryWrite comment View comments

http://ncatlab.org/nlab/show/gauge+theory

http://mathoverflow.net/questions/11427/looking-for-reference-on-gauge-fields-as-connections

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GeisserWrite comment View comments

Web page

MathSciNet

arXiv

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GelfandWrite comment View comments

Collected papers, 3 vols

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Gelfand-Naimark theoremWrite comment View comments

http://www.ncatlab.org/nlab/show/Gelfand+spectrum

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Generalised JacobianWrite comment View comments

Sem Bourbaki Exp 93.

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Generalized ringWrite comment View comments

nlab entry

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Generalized schemeWrite comment View comments

nlab

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Generating hypothesisWrite comment View comments

Title: The Equivariant Generating Hypothesis Authors: Anna Marie Bohmann We state the generating hypothesis in the homotopy category of G-spectra for a compact Lie group G, and prove that if G is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational S^1-equivariant spectra.

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Generic pointsWrite comment View comments

http://mathoverflow.net/questions/2162/what-are-the-most-important-instances-of-the-yoga-of-generic-points

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GenusWrite comment View comments

See maybe stuff by Carl McTague?

Hirzebruch: Top methods in AG. In folder AG/Various. Covers the Todd genus, and talks about RR.

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Geometric categoryWrite comment View comments

See discussion in the concept note.

http://mathoverflow.net/questions/33477/did-durovs-work-give-an-example-of-noncommutative-schemes

http://mathoverflow.net/questions/58428/basic-questions-about-stacks

See Motivic stuff blog post.

There are categories which can be viewed as geometric objects. One example is Balmer's tensor triangulated geometry, treated also in an arxiv preprint by Stevenson and Dell'Ambrogio. Another is Orlov-style geometry with derived categories of sheaves. Also: Tabuada's noncommutative motives - is this based on DG categories?

http://ncatlab.org/nlab/show/Cartan+geometry

See Operad for a more on the PROs etc hierarchy.

Toen has a notion of geometric setting in the under Spec Z paper.

Toen notion of geometric context, see his cours in the Toen web folder, definition in cours2.pdf

Various cats of motives.

Fulton and MacPherson: Categorical framework for the study of singular spaces. In Geometry-Various folder. This seems to be the original source for bivariant theories in general, with quite a lot of material.

Generalized schemes; see Durov thesis link under Field with one element

http://ncatlab.org/nlab/show/Gabriel-Rosenberg+theorem

Teleman and Simpson: De Rham's theorem for Mathematical formula -stacks.

http://mathoverflow.net/questions/56833/riemannian-manifolds-etc-as-locally-ringed-spaces

http://golem.ph.utexas.edu/category/2010/06/vladimir_arnold_12_june_1937_3.html mentions Arnold's ideas on various notions of geometry

Question: Can everything be subsumed by simplicial sheaves on a site?

http://ncatlab.org/nlab/show/geometry+(for+structured+(infinity,1)-toposes) (interesting)

arXiv:1103.2139 Localization of ringed spaces from arXiv Front: math.AG by W. D. Gillam Let $X$ be a ringed space together with the data $M$ of a set $Mx$ of prime ideals of $\O{X,x}$ for each point $x \in X$. We introduce the localization of $(X,M)$, which is a locally ringed space $Y$ and a map of ringed spaces $Y \to X$ enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general $\Spec$ functor. We conclude with a discussion of relative schemes.

http://mathoverflow.net/questions/84641/theme-of-isbell-duality

Demazure: Lectures on p-divisible groups. LNM302. Abstract intro to schemes, group schemes, formal schemes and more. Perhaps the reference I was really looking for is Demazure and Gabriel: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs.

Topological concrete category at nlab, with discussion of some terminology

nice topological space nlab

http://ncatlab.org/nlab/show/stratifold

http://nlab.mathforge.org/nlab/show/orbifold

http://ncatlab.org/nlab/show/scheme

http://nlab.mathforge.org/nlab/show/formal+scheme

arXiv:0907.3925 Compactly Generated Stacks: A Cartesian-Closed Theory of Topological Stacks from arXiv Front: math.AG by David Carchedi A convenient 2-category of topological stacks is constructed which is both complete and Cartesian closed. This 2-category, called the 2-category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of 2-categories between compactly generated stacks and those classical topological stacks which admit locally compact atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are homotopy equivalent.

arXiv:1101.2796 A-schemes and Zariski-Riemann spaces from arXiv Front: math.AG by Satoshi Takagi In this paper, we will investigate further properties of A-schemes. The category of A-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of A-schemes. We compare it with the conventional Zariski-Riemann space, and characterize the latter by a left adjoint.

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Geometric class field theoryWrite comment View comments

http://mathoverflow.net/questions/73054/a-reference-for-geometric-class-field-theory

http://mathoverflow.net/questions/54895/geometric-abelian-class-field-theory

http://mathoverflow.net/questions/100977/how-does-one-understand-geometric-cft-in-terms-of-modularity

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Geometric invariant theoryWrite comment View comments

Book by Mumford, Fogarty and Kirwan. This and much more in folder AG/Invariant theory

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Geometric LanglandsWrite comment View comments

Check out the MO answers of Ben-Zvi, in particular discussions of how physics should tell us what to look for when generalizing to higher dimensions.

http://www.ncatlab.org/nlab/show/geometric+Langlands

http://mathoverflow.net/questions/4180/consequences-of-geometric-langlands/

Frenkel and Gross: A rigid irregular connection on the projective line http://arxiv.org/abs/0901.2163 Some kind of analogue of a family of l-adic reps.

http://mathoverflow.net/questions/56571/a-precise-statement-of-the-categorical-version-of-geometric-langlands-conjecture

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Geometric realizationWrite comment View comments

nLab

http://mathoverflow.net/questions/28380/a-canonical-and-categorical-construction-for-geometric-realization

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Geometrically unibranchWrite comment View comments

Wikipedia has an article on the def for rings.

Selected publications

Riemann-Roch theorems for higher algebraic K-theory (1981)
Universal cycle classes (1983)
Deligne homology and Abel-Jacobi maps (1984)
Homological descent for the K-theory of coherent sheaves (1984)
Intersection theory on stacks and algebraic Q-varieties (1984)
Gersten's conjecture for the K-theory of with torsion coefficients of a discrete valuation ring (1986)
K-theory and intersection theory revisited (1987)

by Gillet and Soulé:

Intersection theory using Adams operations (1987)
Descent, motives and K-theory (1996)
Filtrations on higher algebraic K-theory (1999)

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Globular objectWrite comment View comments

nLab

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Godement resolutionWrite comment View comments

Let Mathematical formula be a sheaf of abelian groups on a site with enough points. The Godement resolution of is defined as follows:

Mathematical formula

This resolution is functorial under morphisms of sites.

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GoncharovWrite comment View comments

Web page

MathSciNet

arXiv

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Goodwillie calculusWrite comment View comments

See introductory notes by Dundas

Kuhn: Goodwillie towers and chromatic homotopy: an overview

See also the original papers of Goodwillie.

Chromotopy has at least one or two blog basic blog posts.

http://ncatlab.org/nlab/show/Goodwillie+calculus

arXiv:1005.1698 Introduction to the manifold calculus of Goodwillie-Weiss from arXiv Front: math.AT by Brian A. Munson We present an introduction to the manifold calculus of functors, due to Goodwillie and Weiss. Our perspective focuses on the role the derivatives of a functor F play in this theory, and the analogies with ordinary calculus. We survey the construction of polynomial functors, the classification of homogeneous functors, and results regarding convergence of the Taylor tower. We sprinkle examples throughout, and pay special attention to spaces of smooth embeddings.

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Grading of cohomology theoriesWrite comment View comments

Why are theories in AG bigraded?? Two spheres in motivic homotopy theory? Something to do with grading of K-theory? Here is an MO question discussing the Brauer grading of K-theory and Bott periodicity, in topology: http://mathoverflow.net/questions/87345/brauer-groups-and-k-theory

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GriffithsWrite comment View comments

See his Selected Works volumes, 4 volumes.

MathSciNet

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Groebner basesWrite comment View comments

Gröbner bases: Introduction to Gröbner bases by Cox, in some book, no el copy. Also the 2 books by Cox et al (el)

preprint on a more abstract notion

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Gromov-Witten invariantsWrite comment View comments

Toen: On motives for DM stacks. Discusses Chow rings and Chow motives, 2 different defs. Motivation: Gromov-Witten invariants.

http://nlab.mathforge.org/nlab/show/Gromov-Witten+invariants

http://mathoverflow.net/questions/1312/gromov-witten-theory-and-compactifications-of-the-moduli-of-curves

A seminar

Some notes are in folder AG/Various/Gromov-Witten inv

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Grothendick constructionWrite comment View comments

Tamaki on the Grothendieck construction and enriched cats. http://front.math.ucdavis.edu/0907.0061

See nlab

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GrothendieckWrite comment View comments

The Grothendieck circle

FGA EGA SGA english index

A short Grothendieck biography: Part I and Part 2

Most of his publications including EGA, are online at NUMDAM. For SGA, see here. For SGA1 and SGA2, see also the LaTeXed versions at arXiv. For SGA3, see Gille and Polo

Lots of scans of Grothendieck writings, including Tohoku, Pursuing stacks, various letters on motives, and Recoltes et Semailles.

See also Joyal: A letter to A. Grothendieck (1984)

A. Grothendieck, Techniques de descente et Th´eor`emes d’existence en G´eom´etrie Alg´ebrique, expos´es dans le S´eminaire Bourbaki entre 1959 et 1962.

http://ncatlab.org/nlab/show/EGA

Publications:

The cohomology theory of abstract algebraic varieties pdf

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Grothendieck categoryWrite comment View comments

An abelian U-category is called Grothendieck if it admits a generator and U-small inductive limits, and U-small filtrant inductive limits are exact.

Thm: A Grothendieck category has enough injectives and admits an injective cogenerator.

http://ncatlab.org/nlab/show/Grothendieck+category

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Grothendieck dualityWrite comment View comments

Some work of Neeman

LNM1960: Lipman and Hashimoto: Foundations on Grothendieck duality for diagrams of schemes

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Grothendieck EsquisseWrite comment View comments

http://mathoverflow.net/questions/24900/a-sketch-of-esquisse-dun-programme

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Grothendieck existenceWrite comment View comments

http://ncatlab.org/nlab/show/Grothendieck+existence

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Grothendieck groupWrite comment View comments

http://www.ncatlab.org/nlab/show/Grothendieck+group good entry

References: - Hartshorne appendix A - Manin: Lectures on the K-functor - Borel and Serre: Le thm de R-R (1958)

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Grothendieck pairingWrite comment View comments

arXiv:0909.4425 Reduction of Abelian Varieties and Grothendieck's Pairing from arXiv Front: math.AG by Klaus Loerke We prove that abelian varieties of small dimension over discrete valuated, stricty henselian ground fields with perfect residue class field obtain semistable reduction after a tamely ramified extension of the ground field. Using this result we obtain perfectness results for Grothendieck's pairing.

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Grothendieck ringWrite comment View comments

For the Grothendieck ring of higher Artin stacks, see Toen: Anneaux de Grothendieck etc, file Toen web prepr K-champ.pdf. For special Artin stacks, he shows by comparing this ring to the ordinary Grothendieck ring of varieties, that invariants such as Hodge numbers and Euler characteristics (l-adic and motivic) extend uniquely to special Artin stacks. In particular, get Lefschetz trace formula for such stacks.

http://ncatlab.org/nlab/show/Grothendieck+ring

http://mathoverflow.net/questions/74433/definition-of-a-grothendieck-ring

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Grothendieck section conjectureWrite comment View comments

Minyong Kim July 09 presentation is an excellent intro to the section conj (this is maybe the same as his Cambridge lecture).

See Toen AIM talk on Homotopy types of algebraic varieties. In the last section he discusses a nonabelian Abel-Jacobi map, but it looks different from the map usually appearing the section conjecture. What is the relation? Toen says that his conjecture can be viewed as a generalization to higher-dimensional varieties.

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Grothendieck spectral sequenceWrite comment View comments

http://mathoverflow.net/questions/24143/composing-left-and-right-derived-functors

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Grothendieck topologyWrite comment View comments

One reason for introducing topologies is that one wants a coequalizer Mathematical formula This doesn't exist in Aff, and the thing you get in presheaves does not behave as it should. Therefore need to take the presheaf coequalizer and sheafify it. Problem: It is not obvious which topology to take - the canonical topology is hard to get your hands on.

http://mathoverflow.net/questions/71893/cohomological-dimension-for-coarser-finer-topologies

de Jong on comparing topologies

There are lots of insights on Grothendieck topologies in general, and h and qfh in particular, in Voevodsky's thesis. I think Homology of schemes I is essentially an improved version of the thesis.

As pointed out by Peter, and also indicated by Voevodsky, one can think of a Grothendieck topology as something corresponding to fixing some colimits which should be preserved under the Yoneda embedding. Some explanation: Take a colimit diagram in the small cat, embed it under Yoneda, might get a new colimit, and a map from the new to the old. Localize by making these maps isomorphisms. Peter thinks: Every left exact localization should correspond to a Groth top.

See Anel factorization systems, and Lurie's discussion on localization, for two alternative ways of thinking of Groth topologies.

Remark: As seen for example in Toen, some topologies can be defined formally, while others need some real AG input.

Voevodsky: Homotopy theory of simplicial sheaves in cd topologies. Discusses two approaches to model structures on simplicial sheaves, Jardine-Joyal being very general, and Brown-Gersten more managable with better finiteness properties (finitely generated), but working only on certain sites associated to "cd-structures". Notions of complete, bounded and regular cd-structure.

See also the various pages on different types of sheaf cohomology.

See also Sheaf theory and similar entries under S.

References:

Shatz article, for cohomological dimension.

nLab, see also sieve

http://nlab.mathforge.org/nlab/show/Grothendieck+pretopology

http://ncatlab.org/nlab/show/Verdier+site

http://mathoverflow.net/questions/9571/canonical-topology-on-the-category-of-schemes

See Toen Essen talk section 4.3 and later, for the right way of defining Grothendieck tops on model categories, taking into account the model structure.

For abstract defs of flat, smooth and etale morphisms, see Toen Barcelona lectures.

Toen and Vaquie: Under Spec Z. Some notes: Idea: Relative alg geom. Think of commutative monoids in a symm monoidal cat C as models for affine schemes relative to C. If there is a reasonable symmetric monoidal functor from C to Z-modules, get a base change functor, and a notion of scheme under Spec(Z). Homotopical version of this requires C to have a model structure. Now have flat and Zariski topology. Can make sense of schemes: a functor with a Zariski covering. Stuff about toric varieties and GL. Brave new AG over the sphere spectrum, and the spectrum with one element. Digression: Flat and Zariski ok. For etale (and maybe hence Nisnevich), Peter Arndt said there might be three ways of doing it: by a lifting property, by factorization systems (Anel), or by mimicking something Deitmar does for monoids, see Peter's thesis in progress, and see also the notion of formally etale.

A Grothendieck topology is a standard tool for constructing cohomology theories in algebraic geometry. See Wikipedia page for basic definitions: sieves, covering families, subcanonical topology, presheaf and sheaf with values in a category with products. More serious references include Tamme: Etale cohomology, Artin lecture notes (el), and SGA4.

Grothendieck topologies are used to define various kinds of Sheaf cohomology

A morphism of topologies

The command "\\" may only appear inside a "\begin ... \end" block

T \\\\\\\\to T\\\\\\\'

is a functor on the underlying categories which take covering families to covering families and \\\\"commutes with fiber products\\\\".

Here is something about the primitive topology by Walker.

de Jong on Comparing topologies

Examples

http://mathoverflow.net/questions/74549/a-bestiary-of-topologies-on-sch

The separated étale site

Objects are required to be separated, étale, and of finite type, rather than just the last two. Cohomology is canonically IMic to étale cohomology, but and advantage is that if Mathematical formula is separated, Noetherian and regular, and

The command "\\" may only appear inside a "\begin ... \end" block

V \\\\\\\\to X

is an object, then Mathematical formula

is also separated, Noetherian and regular. Ref: Jardine, Generalized étale cohomology, p. 278.

Lisse-etale

http://mathoverflow.net/questions/53563/why-is-lack-of-functoriality-of-the-lisse-etale-topology-specific-to-the-lisse-et

The big and small fppf site

Wikipedia

The big and small fpqc site

Wikipedia

http://mathoverflow.net/questions/39211/open-faithfully-flat-morphisms-are-fpqc

The qfh topology

The h topology

Geisser\\\\'s eh topology

Kahn in K-theory handbook says something about base change thm between étale and cdh topology, see page 384 bottom.

The cdh topology

We want to talk about singular schemes which admit resolutions by smooth schemes. For this purpose we introduce the cdh topology on Mathematical formula (schemes of finite type over ). This is the minimal Grothendieck topology for which Nisnevich coverings are coverings, and also proper surjective morphisms of the following type:

The command "\\" may only appear inside a "\begin ... \end" block

W \\\\\\\\coprod U_1 \\\\\\\\to U

where

The command "\\" may only appear inside a "\begin ... \end" block

U_1 \\\\\\\\to U

is a closed embedding and

The command "\\" may only appear inside a "\begin ... \end" block

p^{-1}(U-U_1) \\\\\\\\to (U-U_1)

is an isomorphism.

Examples in FGA?

The canonical topology

On any category (with products, I guess), we can define the canonical topology, by taking as coverings the collection of all families

The command "\\" may only appear inside a "\begin ... \end" block

\\\\\\\\{U_i \\\\\\\\to U \\\\\\\\}

of universal effective epimorphisms. On this topology, every presheaf of sets is a sheaf, and it is the finest topology with this property.

The perfect site

See Milne: Arithmetic duality theorems

The smooth site

Milne again.

The positive topology

See Schmidt

The canonical topology

Def by all representables being sheaves?

The alteration topology with variants

See Gabber\\\\'s abstract from IHES talk 2009

The cohomological descent topology

See Hodge III

Syntomic topology

See Syntomic cohomology

The Zink site

See MR1803955

More Voevodsky stuff

Thesis, p 35: The p-topology and f-topology, coverings given by proper (resp finite) surjective families of morphisms.

Glossary

Selected publications

Examples

The big and small Zariski site

The big and small Nisnevich site

The big and small étale site

The separated étale site

Lisse-etale

The big and small fppf site

The big and small fpqc site

The qfh topology

The h topology

Geisser\\\\'s eh topology

The cdh topology

Examples in FGA?

The canonical topology

The perfect site

The smooth site

The positive topology

The canonical topology

The alteration topology with variants

The cohomological descent topology

Syntomic topology

The Zink site

More Voevodsky stuff