Glossary
Letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Other
- A-infinity algebraWrite comment View comments
http://mathoverflow.net/questions/57589/heuristic-behind-a-infty-algebras
See relevant nLab entries, also A-infinity space, A-infinity operad, A-infinity ring.
http://mathoverflow.net/questions/51817/question-about-a-infty-maps
arXiv:1205.6058 Homotopy unital Ainfinity-algebras fra arXiv Front: math.AT av Volodymyr Lyubashenko It is well known that the differential graded operad of Ainfinity-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital Ainfinity-algebras. We prove that the operad bimodule of Ainfinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.
<]]>- A-infinity categoryWrite comment View comments
[CDATA[See nLab.
Book draft by Seidel on Fukaya categories and Picard-Lefschetz theory, in Symplectic folder. First chapter covers A-infty categories.
<]]>- abc conjectureWrite comment View comments
Rumour, summer 2012 - has the abc conjecture been proven by Mochizuki???
See this review of a Soulé article and the (paper) references it contains for stuff on the Arakelov theory approach to the abc conj via the Bogomolov inequality, and possible solution to the problem of absent absolute differentials. I have the article itself. The notion of successive minima might also be relevant - he mentioned some unpublished preprint with errors from the 90s or so.
http://mathoverflow.net/questions/78061/an-effective-shafarevich-and-the-belyi-degree-of-a-curve
Brief notes in Silverberg: Open questions. In Elliptic curves folder
Various posts on http://www.noncommutative.org/ and maybe other blogs of Lieven. E.g. http://www.neverendingbooks.org/index.php/meanwhile-at-angs.html and http://www.noncommutative.org/index.php/the-abc-conjecture.html
<]]>- Abelian categoryWrite comment View comments
An introduction by Daniel Murfet.
Borceaux vol 2.
http://ncatlab.org/nlab/show/additive+and+abelian+categories
P. Freyd, Abelian categories, Harper, 1966.
[Ga] P. Gabriel, Des categoryégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. ONline version exists
Grothendieck's Tohoku paper
An abelian category is an additive category in which every map:
- admits a kernel and a cokernel (Ab1)
- is strict (Ab2)
Recall that a kernel of a morphism
is the fiber product 
. Also recall that a morphism is strict if the natural map from its coimage to its image is an isomorphism.Some properties of abelian cats:
- mono + epi implies iso
- finite limits and colimits exist
- any functor category from a small to an abelian category is abelian
- the opposite of an abelian category is abelian
An abelian category is called semisimple if all short exact sequences split.
A full subcategory of an abelian category is called
- thick if it is closed under kernels, cokernels and extensions.
- generating if "its objects can surject to anything"
- cogenerating if "its objects can "eat" anything", i.e. can receive an injection from any object.
Recall that Hom is left exact in both variables. An object is said to be injective if it is exact as a contravariant functor. An abelian category has enough injectives if the injectives are cogenerating. An object is projective if it is exact as a covariant functor. Enough projectives means that the projectives are generating.
Properties of injective objects: "Any morphism from a subobject to I can be extended to the whole object". (Probably enough for generators). Three facts about short exact sequences: In any short exact sequence, if A, B inj, then C inj. Also, if A inj, then the sequence splits. In a split short exact sequence, B is inj iff A & C are inj.
Freyd-Mitchell thm: Let C be a small abelian category. Then there exists a ring R and an exact fully faithful functor from C to R-mod. (See Kashiwara-Schapira chapter 9)
Some further "axioms" on abelian cats that are sometimes useful to consider (these are not satisfied for all abelian cats):
- All direct sums exist (Ab3)
- (assuming Ab3) "A compatible family of morphisms on an increasingly filtered family of subobjects induce a unique morphism on the sum subobject". (Ab5)
- Also Ab4. See Tohoku
Thm: Let
be abelian, satisfying Ab3. Then direct limits exist in , and the functor 
is additive and right exact. (What is the significance of this result??)
A category is said to be
-linear if the Hom sets are -modules, composition is -linear, and if the category admits finite sums. A functor which respects the -module structure will also respect direct sums, up to a natural isomorphism.<]]>- Abelian varietyWrite comment View comments
Folder: AG/Theta functions (many nice things)
References: Mumford, Milne?
Notes from Tim's course (don't have all pages)
Swinnerton-Dyer: Book on analytic theory, in Elliptic curves folder
http://mathoverflow.net/questions/51955/references-for-abelian-schemes on equations for Jacobians
<]]>- Abstract algebraWrite comment View comments
Berrick and Keating: An introduction to rings and modules with K-theory in view. CUP 2000. (K-th folder)
<]]>- Abstract Brown representabilityWrite comment View comments
See Kashiwara-Schapira.
Hovey suggests the following problem (page 199 of his book). Define "pre-triangulated category with sequential colimits". Show: Every triangulated category gives rise to such a thing. The homotopy category of a model category is such a thing. Define a cohomology functor on such a thing, and prove that all cohomology functors on the homotopy category of a pointed cofibrantly generated model category are representable.
Rosicky: Generalized Brown Representability in homotopy categories (online at TAC), with an erratum here
Raventos and Muro has some project on representability for functors a priori defined only on compact objects, I believe. This could be of interest for example in the case of CTs defined on geometric objects but not a priori on all generalised spaces.
<]]>- Abstract homotopy theoryWrite comment View comments
Cisinski on homotopy theory in toposes. See also other papers by Cisinski.
Cox: Homotopy limits and the homotopy type of functor categories (1976) MR0407022
Jardine on the ideas of Cisinski
Anderson: Axiomatic homotopy theory (1978)
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Symmetric powers in stable homotopy cats by Guletskii and Gorchinskiy
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- AdamsWrite comment View comments
The selected works of Frank Adams, 2 vols
<]]>- Adams operationsWrite comment View comments
See Feliu thesis
Florence Lecomte, Simplicial schemes and Adams operations
<]]>- Adams spectral sequenceWrite comment View comments
See Adams summary (early in part III).
We consider the stable group
for some finite complexes 
and 
.The reduced cohomology of
with coefficients in 
is a module over the Steenrod algebra 
.The Adams spectral sequence goes from
to the above stable group. (What does this mean???)
There is actually a much better and more general formulation of all this. See Chapter 15 in Adams Part III.
<]]>- Additive categoryWrite comment View comments
A category is additive if it satisfies four conditions, including existence of zero object, finite products and coproducts, and "coprod IMic to prod". One proves that a category is additive iff Hom sets are abelian groups, composition is bilinear, and it admits finite products.
http://ncatlab.org/nlab/show/additive+and+abelian+categories
<]]>- Adelic geometryWrite comment View comments
There is a lot of stuff written by Parshin, see for example his Invitation to higher local fields on arxiv. This also contains references to papers by people like Huber and Beilinson, some of which may be difficult to find.
arXiv:0909.1568 Igusa integrals and volume asymptotics in analytic and adelic geometry from arXiv Front: math.NT by Antoine Chambert-Loir, Yuri Tschinkel We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
arXiv:0912.1577 Harmonic analysis on local fields and adelic spaces II from arXiv Front: math.NT by D. V. Osipov, A. N. Parshin This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.
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- Adic spacesWrite comment View comments
See also rigid geometry, analytic space.
Wahle good refs for adic and analytic spaces.
<]]>- Adjoint functorWrite comment View comments
"A left adjoint is right exact"
In general, a left adjoint preserves colimits, and a right adjoint preserves limits.
http://mathoverflow.net/questions/6376/why-forgetful-functors-usually-have-left-adjoint
Examples of adjoints: (left adjoint, right adjoint)
- (geom. realization, sing)
- Lots of examples in Weibel (via index)
- "Informally, a free functor is a left adjoint to a forgetful functor"
<]]>- AdviceWrite comment View comments
Manin: I've had maybe 50 students. 3 or 4 of them failed, all because they insisted on reading. Do mathematics, look at recent arxiv preprints and track back the results you need. Only read what is of immediate use to your research.
Method of poetry: immerse yourself in quality papers
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- Algebra over a monoidal categoryWrite comment View comments
Let
be a Monoidal category. A -algebra structure on a category 
is a monoidal structure on together with a monoidal functor 
. A -algebra functor is a monoidal functor "commuting with 
up to natural isomorphism". The -algebras form a 2-category, and there is a forgetful 2-functor to -modules.Example: Let
be 
-mod ( a commutative ring), and consider a map of -algs, 
. Get a -algebra functor from 
to 
by tensoring with 
over the base ring 
.If
is symmetric monoidal, then a symmetric -algebra structure on a category is a symmetric monoidal structure on together with a symmetric monoidal functor 
. A symmetric -algebra functor is a symmetric monoidal functor that is also a -algebra functor. We can also define a central -algebra structure.Example: The category of modules over a commutative Hopf algebra (over a field) is a central algebra over the category of vector spaces. It is symmetric iff the Hopf algebra is cocommutative.
<]]>- Algebraic categoryWrite comment View comments
nLab "essentially algebraic theory"
nLab on algebraic theory
http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory
See also Barr-Beck thm or something like that
Schwede has some notions for triangulated cats I think
Boerceaux vol 2 page 158: A cat equipped with a functor U to sets is called algebraic if (a) it has coequalizers and kernel pairs (b) U has a left adjoint F (c) U reflects isomorphisms (d) U preserves regular epimorphisms (e) UF preserves filtered colimits.
arXiv:1109.1598 Algebraic theories, span diagrams and commutative monoids in homotopy theory from arXiv Front: math.CT by James Cranch We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length.
We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie's concept of a commutative algebra object.<]]>- Algebraic curveWrite comment View comments
Loads of material on algebraic curves and Riemann surfaces in the folder under AG.
<]]>- Algebraic geometryWrite comment View comments
FGA EGA SGA english index
Add good introductory references. Should include:
Vakil: New revised book/course notes, see his blog.
http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course
http://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry
http://mathoverflow.net/questions/34110/algebraic-geometry-examples
http://mathoverflow.net/questions/78696/is-there-an-intuitive-reason-for-zariskis-main-theorem
http://mathoverflow.net/questions/111685/on-some-finiteness-properties-for-schemes
Mumford's Red Book
Cox, Little, O\'Shea: Ideals, varieties and algorithms
Dieudonne: History of AG
Putinar and Sullivant ed: Emerging applications of algebraic geometry
FGA, in folder AG/Various. Grothendieck topologies and stacks, Hilbert and Quot schemes, elementary deformation theory, Grothendieck's existence thm in formal geometry, the Picard scheme, summary of intersection theory.
Totaro recommends Lazarsfeld: Positivity
Many useful insights are listed here: http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course
http://mathoverflow.net/questions/46/what-is-the-universal-property-of-normalization
Bourbaki seminars possibly of some interest:
- Exp 37: Quelques varietes usuelles en G.A.
- exp 49: Hyperplane sections of normal varieties
- Exp 53: Picard vareity and Neron-Severi group
- Exp 71: Serre: Cohomology and complex variables
- Exp 72: Weil on Picard varieties and jacobians
- Exp 75: Equivalence relations on algebraic curves having multiple points
- Exp 66: Neron on arithmetic
- Exp 77: Serre: Cohomology and arithmetic
- Exp 78: Thom on subvarieties and homologu classes of differentiable varieties
- Possibly 84 and 85
- Exp 86: Samuel: Les fonctions holomorphes abstraites de Zariski
- Exp 88: Travaux de Hirzebruch sur la topologie des varietes
- Exp 89: Thom
- Exp 94: The Enriques-Severi Lemma
- Exp 95: Analytic sheaves
- Exp 99: Travaux de Zariski sur Hilbert 14.
Books in the folder AG/Introductory (only the most interesting listed)
- Abhyankar: AG for scientists and engineers. Very down-to-earth, actually quite nice! Goes up to RoS for surfaces and similar things.
- AG1 by Shokurov and Danilov: Basic theory of curves and Jacobians. Basic theory of varieties, some schemes at the end.
- Dieudonne: Algebraic Geometry. Notes in English surveying EGA I-III. (Course notes from Maryland)
- Dieudonne: Fondements de la GA Moderne. Continuation of the Maryland notes. Together these two sets of notes seem like a marvellous route into EGA.
- Dolgachev Topics in Classical AG: Polarity, Conics, Plane cubics, Determinental equations, Theta characteristics, Plane quartics, Planar Cremona transformations, Del Pezza surfaces, Cubic surfaces, Automorphisms of Del Pezzo surfaces, Geometry of Lines.
- Eisenbud-Harris: The geometry of schemes. Excellent book.
- Griffiths-Harris
- Harder: Lectures on AG (Vol 1 out of 3 planned): Basic homological algebra, sheaf theory, cohomology of sheaves, Riemanns surfaces and AVs
- Harris: First course. Classical, no schemes, many examples.
- Hartshorne
- Liu
- Parshin-Shafarevich AG III. Covers Hodge theory, periods, curves and Jacobians.
- Ravi Vakil book project
- Shafarevich ed AG II. Covers cohomology and algebraic surfaces.
- There is one big file with all the Russian AG books (I-V), ed Shafarevich
- Thomas Zeta functions. Contains basic function field arithmetic and some stuff on Weil conjectures.
<]]>- Algebraic geometry examplesWrite comment View comments
Title: Schubert varieties are log Fano over the integers Authors: Dave Anderson, Alan Stapledon http://front.math.ucdavis.edu/1203.6678 Categories: math.AG Algebraic Geometry Comments: 3 pages; to appear in Proc. Amer. Math. Soc MSC: 14M15, 14E30, 20G99 Abstract: Given a Schubert variety Xw, we exhibit a divisor \Delta, defined over the integers, such that the pair (Xw,\Delta) is log Fano in all characteristics.
<]]>- Algebraic groupWrite comment View comments
See Alg groups folder under AG. Many good sources, for example PSPUM-9.
Short article in some Clay volume???
<]]>- Algebraic spaceWrite comment View comments
Ref: Knutson LNM0203.
Algebraic stacks may be viewed as groupoid objects in the category of algebraic spaces. (Joshua: The intersection cohomology...)
http://mathoverflow.net/questions/3194/what-are-the-benefits-of-using-algebraic-spaces-over-schemes
See Toen course in cours folder under Toen, I think around chapters 2 to 4-2 in particular.
http://mathoverflow.net/questions/11226/commutative-rings-to-algebraic-spaces-in-one-jump
<]]>- Algebraic stackWrite comment View comments
See also Stack.
An excellent introduction to derived AG is the CRM 2008 notes, in Toen web unpublished folder. These notes also covers algebraic stacks, the idea of moduli spaces, a little about cotangent complexes, and examples of derived algebraic stacks.
http://mathoverflow.net/questions/73115/query-on-comment-in-deligne-mumford-1969
<]]>- Algebraic surfaceWrite comment View comments
Zariski lectures LNM0083
Folder: AG/Surfaces
http://front.math.ucdavis.edu/0912.4291 Lecture notes on surfaces in positive characteristic
http://mathoverflow.net/questions/40555/calculations-of-pic0-pic-ns-of-surfaces
<]]>- Algebraic topologyWrite comment View comments
General references:
- Books of Hatcher (online)
- The book of May (online)
- Selick: Introduction to homotopy theory
- Many other books in the Alg Top folder!
Also folder under GEOMETRY
A History of Algebraic and Differential Topology, 1900 - 1960 Series: Modern Birkhäuser Classics Dieudonné, Jean 1st ed. 1989. 2nd printing, 2009, XXIV, 648 p., Softcover ISBN: 978-0-8176-4906-7
nLab almost no info at all Aug 09
<]]>- AlgebrasWrite comment View comments
There are loads of different types of algebras around. Start to list here for future ref.
Some introduction is in MacLane: Homology. In Homol alg folder
Skowronski ed: Trends in Representation Theory of Algebras and Related Topics, EMS
Central simple algebras: See Grillet
Jordan algebras: See Bourbaki exp 31
Chiral algebras: See Beilinson book
Hecke algebras
Vertex operator algebras
http://ncatlab.org/nlab/show/von+Neumann+algebra
Book by Chevalley on giga: The construction and study of certain important algebras
<]]>- Almost complex varietyWrite comment View comments
I think Kodaira did important work on almost complex structures, but might be wrong.
Bourbaki exp 35.
<]]>- Almost etaleWrite comment View comments
See Faltings theory folder, under NUMBER TH. Document: almost-etale-stuff
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- AlterationsWrite comment View comments
Some really really nice things in Resolution of singularities (tribute to Zariski), ed Hauser, Lipman et al. Progress in Mathematics.
An improved version of the alteration thm comes from Gabber's work on purity, see maybe study group notes of Riou or maybe Illusie. Or ask Riou. It was something along the following lines: Take any quite general scheme, and a prime l. Then there exist an alteration of degree prime to l (this is not exactly right, so check proper ref).
http://mathoverflow.net/questions/27821/alterations-of-regular-varieties
The work of Gabber (see Illusie et al on arXiv) contains applications of alterations.
<]]>- Anabelian geometryWrite comment View comments
See lots of material from the Newton Institute programme, fall of 2009. In particular, the presentation of Kim.
Jardine weird quote, I think: Grothendieck fundamental groupoid, leads to anabelian geometry in simplicial (pre?)sheaves
<]]>- Analytic continuationWrite comment View comments
Madhav Nori on analytic continuation and the Poisson formula: http://front.math.ucdavis.edu/1204.3533
<]]>- Analytic geometryWrite comment View comments
Paugam: Global analytic geometry
<]]>- Analytic number theoryWrite comment View comments
Old list of refs:
Article by Shintani on zeta-functions and prehomogeneous vector spaces (related to Sato's work)
Automorphic Forms and L-Functions for the Group GL(n,R). Goldfeld, CUP
Recent Perspectives in Random Matrix Theory and Number Theory. Ed. Mezzadri
Spectral Theory of the Riemann Zeta-Function. Motohashi, CUP
Euler through time... Varadarajan
Dynamical, Spectral, and Arithmetic Zeta Functions/ Lapidus ed.
On Random Matrices, Zeta Functions and Dynamical Systems (Springer, 3-540-23189-7)
Katz and Sarnak: Random matrices, Frobenius eigenvalues, and monodromy.
<]]>- Analytic spaceWrite comment View comments
I think there are real-analytic and complex-analytic spaces.
LNM0025 Narasimhan: Intro to analytic spaces
<]]>- Analytic torsionWrite comment View comments
Burgos talk abstract, Paris, March 2011: The holomorphic analytic torsion classes for Kähler fibrations provide us with a refinement of the Grothendieck-Riemann-Roch theorem at the level of differential forms. They are a key ingredient in the arithmetic Grothendieck-Riemann-Roch theorem. In this talk we will give an axiomatic characterization of holomorphic analytic torsion classes, use it extend analytic torsion classes to arbitrary projective morphisms and extract some consequences. This is joint work with R. Litcanu and G. Freixas.
<]]>- Andre-Oort conjectureWrite comment View comments
See Yafaev and Klingler (arxiv 2012)
Maybe Martin is working on this??
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- AnelWrite comment View comments
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- Anodyne extensionWrite comment View comments
Consider the category of simplicial sets. We define the set
as the set of inclusions 
for 
. Define 
to be the set of inclusions 
for 
. A map 
is a cofibration iff it is in 
. A map is a (Kan) fibration iff it is in 
. A map is a weak equivalence iff its geometric realization is a weak equivalence of topological spaces. The maps in 
are called anodyne extensions. See Cofibrantly generated for the notation.Every anodyne extension is a trivial cofibration of simplicial sets.
I think the anodyne extensions include the maps
. Possibly these suffice for certain "testing".Another formulation: A class of simplicial set monomorphisms is called saturated if contains all IMs, is closed under pushouts, retracts, countable compositions and arbitrary disjoint unions. An anodyne extension is a member of the smallest saturated class which contains the standard inclusions of horns.
Kan fibrations have the RLP wrt all standard inclusions of horns, and hence wrt all anodyne extensions.
<]]>- ArakelovWrite comment View comments
MathSciNet only has four items. I believe there might be other things by Arakelov in Russian.
<]]>- Arakelov theoryWrite comment View comments
An introduction by Burgos-Gil
Lang: Introduction to Arakelov theory
http://www.ncatlab.org/nlab/show/Arakelov+geometry
http://mathoverflow.net/questions/78460/learning-arakelov-geometry
Manin-Panchiskin has some material
Faltings: Calculus on arithmetic surfaces
Gillet in Arcata: Intro to higher-dimensional Arakelov theory
Soulé, Christophe Hermitian vector bundles on arithmetic varieties. Algebraic geometry---Santa Cruz 1995, 383--419, Proc. Sympos. Pure Math., 62, Part 1. This looks excellent, I haven't read it but noted that on p 397 there are references to several attempts to develop adelic intersection theory.
I think Neukirch ANT works out things for Pic-hat of the ring of integers in a number field.
MR1087394 (92d:14016) Gillet, Henri(1-ILCC); Soulé, Christophe(F-IHES) Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. No. 72 (1990), 93--174 (1991). (Long review)
MR1724892 (2000m:14030) Künnemann, Klaus(D-KOLN); Maillot, Vincent(F-ENS-MI) Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire. (French)
A course syllabus for a course by Gillet
Vojta: Applications of arithmetic algebraic geometry to Diophantine approximations (1993)
One could maybe look at Bismut, at least the reviews
arXiv:0909.3680 On Volumes of Arithmetic Line Bundles II from arXiv Front: math.AG by Xinyi Yuan For a hermitian line bundle over an arithmetic variety, we construct a convex continuous function on the Okounkov body associated to the generic fibre of the line bundle. The integration of the continuous function gives the growth of the Euler characteristic of the hermitian line bundle. It is the global version of the recent work of Nystrom.
arXiv:1010.1599 Toward Dirichlet's unit theorem on arithmetic varieties from arXiv Front: math.NT by Atsushi Moriwaki In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet's unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.
arXiv:1102.2063 Hermitian structures on the derived category of coherent sheaves from arXiv Front: math.AG by José Ignacio Burgos Gil, Gerard Freixas i Montplet, Razvan Litcanu The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. To this end, we first define hermitian structures on the objects of the bounded derived category of coherent sheaves on a smooth complex variety. Secondly we extend the theory of Bott-Chern classes to these hermitian structures. Finally we introduce the category $\oSm_{\ast/\CC}$ whose morphisms are projective morphisms with a hermitian structure on the relative tangent complex.
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