Geometry.Net - Pure_And_Applied_Math: Homological Algebra

Commutative Diagram -- From MathWorld
Many other mathematical concepts and properties, especially in algebraic topology,homological algebra, and category theory, can be formulated in terms of
http://mathworld.wolfram.com/CommutativeDiagram.html

Extractions: Commutative diagrams are usually composed by commutative triangles and commutative squares. Commutative triangles and squares can also be combined to form plane figures or space arrangements. A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets. A looped arrow indicates a map from a set to itself. The above commutative diagram expresses the fact that g is the inverse map to f , since it is a pictorial translation of the map equalities and This can also be represented using two separate diagrams. Many other mathematical concepts and properties, especially in algebraic topology homological algebra , and category theory , can be formulated in terms of commutative diagrams.
KLUWER Academic Publishers | Category Theory, Homological Algebra
Home » Browse by Subject » Mathematics » Foundations, Sets andCategories » Category Theory, homological algebra. Sort listing
http://www.wkap.nl/home/topics/J/4/4/
KLUWER Academic Publishers | Non-Abelian Homological Algebra And Its Application
Books » NonAbelian homological algebra and Its Applications. Non-Abelianhomological algebra and Its Applications. Add to cart. by
http://www.wkap.nl/prod/b/0-7923-4718-8

Extractions: This book exposes methods of non-abelian homological algebra, such as the theory of satellites in abstract categories with respect to presheaves of categories and the theory of non-abelian derived functors of group valued functors. Applications to K-theory, bivariant K-theory and non-abelian homology of groups are given. The cohomology of algebraic theories and monoids are also investigated. The work is based on the recent work of the researchers at the A. Razmadze Mathematical Institute in Tbilisi, Georgia.
Elementary Operations
homological algebra. Some basic facts from homological algebra 9, 3 are neededto understand the purpose of the algorithms presented in this section.
http://www.bangor.ac.uk/~mas019/symb/node16.html

Extractions: Next: More Advanced Operations Up: Needed Functionality Previous: Input and Output The usual operations such as Expr exprPlus (Expr, Expr); Expr exprMinus (Expr, Expr); Expr exprTimes (Expr, Expr); Expr exprDiv (Expr, Expr); Expr exprNum (Expr); Expr exprDen (Expr); are available. The first four are obvious. To understand the last two, some terminology is needed. In ExprLib, every mathematical expression is a rational function in what are called parameters . A parameter is a constant, a symbolic constant (e.g. PI representing , etc.), a symbol (such as x, y, z, ... ), or the application of a function symbol (operator) to one or more expressions (e.g cos(3x + y) or myOp (x, 3.7 y) , etc.). Thus, every expression has a numerator and a denominator. The last two functions return those respectively.
Math 406 Homological Algebra
University of Illinois at UrbanaChampaign Math 406 homological algebraSpring 1999. Professor J. Rotman 1000 am, MWF 443 Altgeld Hall.
http://www.math.uiuc.edu/Classes/math406.html

Extractions: The course will construct the functors Ext and Tor on categories of R-modules, where R is a ring. We will begin with projective, injective, and flat resolutions and derived functors. Various dimensions of modules and of rings will be discussed, and applications will focus on rings and cohomology of groups (once some version of homological algebra is learned, applications to other areas, e.g. Lie algebras or sheaves, are routine). The course will end with a discussion of spectral sequences (via exact couples) and more applications to rings and groups. Although the origins of homological algebra are in Algebraic Topology, we will not assume familiarity with homology and cohomology of spaces. Prerequisite: Math 402 References:
Homological Algebra
homological algebra. Certain results in homological algebra regardingspectral sequences have been repeatedly used in various contexts.
http://www.imsc.ernet.in/~kapil/work/node14.html

Extractions: Next: Motives Up: Current Mathematical Research Previous: Rationality and Unirationality Certain results in homological algebra regarding spectral sequences have been repeatedly used in various contexts. The usual reference has been to unpublished work of Deligne for a proof. The general applicability of these results has been a matter of some interest due to the interest in ``motives'' (see below). The author has generalised and clarified the existence and properties of the constructions used; roughly speaking this is a theory of spectral sequences of spectral sequences. Recent discussions with B. Kahn (CNRS) have led to a further generalisation of the results to unbounded complexes. A paper based on this work has appeared in the Journal of Algebra. A talk based on this work was also presented at the ICM Satellite conference at Essen.
HOMOLOGICAL ALGEBRA
NextPrev Right Left Up Index Root HOMOLOGICALALGEBRA. Chapters. BASIC ALGEBRAS. CHAIN COMPLEXES.
http://magma.maths.usyd.edu.au/magma/htmlhelp/part13.htm
Homological Algebra
next up previous Next Basic Algebras Up V2.9 Features Previous Invariantsof the Symmetric homological algebra. Basic Algebras; Chain Complexes.
http://magma.maths.usyd.edu.au/magma/Features/node199.html
Cartan, H. And Eilenberg, S.: Homological Algebra (PMS-19).
of the book homological algebra (PMS19) by Cartan, H. and Eilenberg,S., published by Princeton University Press. homological algebra (PMS-19).
http://pup.princeton.edu/titles/250.html

Extractions: Shopping Cart Table of Contents When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.
UWM Math: Homological Algebra
homological algebra. Algebra Group modelled after). All members ofthe algebra group have done work involving homological algebra.
http://www.uwm.edu/Dept/Math/Research/Algebra/homological/homological.html
UWM Math: Algebra Group
Algebra Group. People, research areas, seminars.Category Science Math Algebra Research Groups Every semester, we also offer an advanced course in algebra; some recent examplesinclude Ring Theory, homological algebra, and special topics courses
http://www.uwm.edu/Dept/Math/Research/Algebra/algebra.html

Extractions: The general address for anonymous ftp is ftp://ftp.uwm.edu and the Math directory is pub/Math Below are some of the main research areas among the group, along with the researchers involved in these areas. (Obviously these are often rough approximations.) To find out more about each topic, click on the name of the topic.
Introduction To Commutative And Homological Algebra
Introduction to Commutative and homological algebra. Commutative and homological SIGelfand, Yu. I. Manin, Methods of homological algebra, I.
http://www.mccme.ru/mathinmoscow/courses/homolalg.htm

Extractions: Commutative and homological algebra studies algebraic structures, say, modules over commutative rings, in terms of their generators and relations. It provides the most powerful algebraic tools for applications in algebraic and differential geometry, number theory, algebraic topology, etc. This course gives a quick introduction to these techniques. It is designed as the starting point for those who intend to study algebraic geometry and related topics. Prerequisites: basic linear and multilinear algebra (tensor products, multilinear maps), some experience in geometry and topology is desirable but not essential. Polynomial ideals and algebraic varieties. Noetherian rings. Hilbert base theorem. Integer ring extensions. Gauss-Kronecker-Dedekind lemma. Finitely generated algebras over a field. Noether's normalization theorem. Hilbert's Nullstellensatz. Geometry of ring homorphisms: category of affine schemes. Category of modules. Generators and relations. Exact sequences. Grothendieck group.
Homological Algebra (Spring 2002)
?.?. (A.Gorodentsev). homological algebra. Notes in English.Gzipped postscript (may be viewd directly with some versions of ghostview).
http://www.mccme.ru/ium/s02/homalg.html
An Introduction To Homological Algebra
``An introduction to homological algebra''. by Charles Weibel Published1994 by Cambridge Univ. Press (450pp.). Corrections to 1994
http://www.math.rutgers.edu/~weibel/Hbook-corrections.html
Analysis And Homological Algebra
Analysis and homological algebra. joint work with M. Olbrich. Givena real semisimple Lie group we have the category of its Harish
http://www.uni-math.gwdg.de/bunke/project2.html

Extractions: Given a real semisimple Lie group we have the category of its Harish Chandra modules and the canonical globalization functors which produce group representations from Harish-Chandra modules. Further given a discrete (arithmetic, or S-arithmetic) subgroup we can consider the cohomology of the subgroup with coefficients in the globalizations of Harish-Chandra modules. We are interested in structural results such as finitness, Poincare duality, computation in terms of automorphic forms etc. Our main technique is to employ invariant differential operators on symmetric spaces in order to construct acyclic resolutions of the representations.
Homological Algebra - Wikipedia
homological algebra. homological algebra is that branch of mathematics whichstudies the methods of homology and cohomology in a general setting.
http://www.wikipedia.org/wiki/Homological_algebra

Extractions: Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help From Wikipedia, the free encyclopedia. Homological algebra is that branch of mathematics which studies the methods of homology and cohomology in a general setting. These concepts originated in algebraic topology Cohomology theories have been described for topological spaces sheaves , and groups . The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology.
Title Details - Cambridge University Press
Home Catalogue An Introduction to homological algebra. Related Areas Pure Mathematics.An Introduction to homological algebra. Charles A. Weibel. £60.00.
http://books.cambridge.org/0521435005.htm
Sheafhom, Programs For Homological Algebra And Algebraic Topology
Sheafhom. Sheafhom is a set of programs I'm developing for work inhomological algebra and algebraic topology. It allows you to use
http://www.math.okstate.edu/~mmcconn/shh0498.html

Extractions: Sheafhom is a set of programs I'm developing for work in homological algebra and algebraic topology. It allows you to use finite-dimensional rational vector spaces and their morphisms (i.e. linear maps). The system supports tensor products, direct sums, and wedge products, as well as partially-ordered sets. A data structure called a sheaf represents sheaves (like the intersection homology sheaves) on certain topological spaces. This includes the spaces given by regular cell complexes and simplicial complexes. The main application so far is to toric varieties. The machine can find the rational intersection cohomology ( IH ) of n -dimensional toric varieties, for any n and in any perversity. This includes the ordinary cohomology H ^* and homology H _* as special cases. The varieties should be defined over the complex numbers. They should be normal, but they do not need to be smooth or projective. The IH is computed by a direct approach that constructs explicit cycles on the space, modeling Deligne's construction of intersection homology. Slight modifications to the code allow it to compute IH with integer coefficients, though this has not been put on the Web page yet.
Sheafhom, Programs For Homological Algebra And Algebraic Topology
Sheafhom (first Web version, May 11, 1996). Sheafhom is a set of programsI'm developing for homological algebra and algebraic topology.
http://www.math.okstate.edu/~mmcconn/shh0596.html

Extractions: Sheafhom is a set of programs I'm developing for homological algebra and algebraic topology. It allows you to work with finite-dimensional rational vector spaces, chain complexes, and their morphisms (i.e. linear maps). The system supports tensor products, direct sums, and wedge products. The main application so far is to toric varieties: the machine can find the rational homology of (complete, normal, possibly singular) toric varieties in any dimension, using an algorithm of Burt Totaro (see the source code for the reference). Applications I hope to develop in the future include Finding intersection (co)homology of toric varieties. Finding maps on homology between toric varieties. Finding the canonical intersection product on (intersection) (co)homology. n ,R) and Gamma is an arithmetic subgroup, separating out the part of the cohomology that does not come from the Borel-Serre boundary, using a recent spectral sequence of Ash and McConnell. Computing Hecke operators on locally symmetric spaces, using techniques of Ash, of MacPherson and McConnell, or of Paul Gunnells.
Math 677. Homological Algebra
Math 677. homological algebra terms offered, credit hours, prerequisites, andcourse description. Math 677. homological algebra. Offered W even years.
http://www.math.byu.edu/Programs/677.html

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