homological algebra


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homological algebra

[¦hä·mə¦läj·ə·kəl ′al·jə·brə]
(mathematics)
The study of the structure of modules, particularly by means of exact sequences; it has application to the study of a topological space via its homology groups.
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The subject area is an innovative blend of group theory, homological algebra, topology, geometry, number theory and computer science.
Maxim Kontsevich , Institut des Hautes Etudes Scientifiques, for work making a deep impact in a vast variety of mathematical disciplines, including algebraic geometry, deformation theory, symplectic topology, homological algebra and dynamical systems.
He assumes students are familiar with homological algebra, algebraic topology based on different forms, and de Rham cohomology.
We present here a synopsis of the results together with applications of this beautiful interplay between combinatorial topology and homological algebra.
This two-volume research monograph on the general Lagrangian Floer theory and the accompanying homological algebra of filtered $A_\infty$-algebras provides the most important step towards a rigorous foundation of the Fukaya category in general context.
Some familiarity with basic homological algebra is needed in the final chapter.
In Section 5 we use homological algebra to study dashings; our main result is the enumeration of odd dashings for any chromotopology.
Readers are assumed to have a basic knowledge of commutative algebra, homological algebra, and category theory.
An emphasis on homological algebra allows basic notions on complexes to be presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject.
Based on a June 2005 summer school, this series of 24 lectures introduces the algebraic sets, sheaf theory, and homological algebra leading to the definition and alternative characterizations of local cohomology.
Other topics include topological homological algebra, topological algebraic geometry, sheaf theory, and L-theory.