Cohomology of algebraic theories

Jibladze, Mamuka; Pirashvili, Teimuraz

doi:10.1016/0021-8693(91)90093-n

Cited by 100 publications

(62 citation statements)

References 8 publications

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“…We define this cohomology theory for graded rings using the normalized cohomology of categories [BW85]. Its ungraded version is equivalent to Mac Lane's original definition [JP91]. This theory is, for various reasons, an appropriate replacement of the Hochschild cohomology used in [BKS04].…”

Section: Introductionmentioning

confidence: 99%

Universal Toda brackets of ring spectra

Sagave

2007

Trans. Amer. Math. Soc.

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Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum.For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.

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Section: Introductionmentioning

confidence: 99%

Universal Toda brackets of ring spectra

Sagave

2007

Trans. Amer. Math. Soc.

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“…In section 3 we further reduce these models to appropriate (no longer stabilized) cohomology of small categories. In section 4 we reinterpret these results in terms of Mac Lane homology following the ideas of [9].…”

Section: Where Hh Is the Hochschild Homology Complex For Q * (R)mentioning

confidence: 99%

“…This can be thought of as a generalization of the bar construction for abelian groups, and hence this stabilization is a direct transliteration of the Dold-Puppe stable derived functors ( [4]) to the setting of exact categories. Our method of computation for the stabilized functors is first to relate them to a stabilized version of the cohomology of small categories in the sense of [2] and then to relate this to Mac Lane homology much in the manner of [9].…”

Section: Introductionmentioning

confidence: 99%

On the computation of stabilized tensor functors and the relative algebraic 𝐾-theory of dual numbers

McCarthy

2001

Trans. Amer. Math. Soc.

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“…In 1991, an innocuous paper by Jibladze and Pirashvili [JP91] proved that the Mac Lane homology of a ring A (and a module M ) is Tor F * (A⊗, M ⊗) in the functor category F = F (A) of functors from the category of fin. gen. free A-modules to the category of A-modules.…”

Section: Mac Lane Cohomology and Topological Hochschild Homologymentioning

confidence: 99%

History of Homological Algebra

Weibel

1999

History of Topology

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Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on "homology numbers," and the rigorous development of the notion of homology numbers by Poincaré in 1895. A 1925 observation of Emmy Noether [N25] shifted the attention to the "homology groups" of a space, and algebraic techniques were developed for computational purposes in the 1930's. Yet homology remained a part of the realm of topology until about 1945.During the period 1940-1955, these topologically-motivated techniques for computing homology were applied to define and explore the homology and cohomology of several algebraic systems: Tor and Ext for abelian groups, homology and cohomology of groups and Lie algebras, and the cohomology of associative algebras. In addition, Leray introduced sheaves, sheaf cohomology and spectral sequences.At this point Cartan and Eilenberg's book [CE] crystallized and redirected the field completely. Their systematic use of derived functors, defined via projective and injective resolutions of modules, united all the previously disparate homology theories. It was a true revolution in mathematics, and as such it was also a new beginning. The search for a general setting for derived functors led to the notion of abelian categories, and the search for nontrivial examples of projective modules led to the rise of algebraic K-theory. Homological algebra was here to stay.Several new fields of study grew out of the Cartan-Eilenberg Revolution. The importance of regular local rings in algebra grew out of results obtained by homological methods in the late 1950's. The study of injective resolutions led to Grothendieck's theory of sheaf cohomology, the discovery of Gorenstein rings and Local Duality in both ring theory and algebraic geometry. In turn, cohomological methods played a key role in Grothendieck's rewriting of the foundations of algebraic geometry, including the development of derived categories. Number theory was infused with new results from Galois cohomology, which in turn led to the development ofétale cohomology and the eventual solution of the Weil Conjectures by Deligne.Simplicial methods were introduced in the 1950's by Dold, Kan, Moore and Puppe. They led to the rise of homotopical algebra and nonabelian derived functors in the 1960's. Among its many applications, perhaps André-Quillen homology for commutative rings and higher algebraic K-theory are the most noteworthy. Simplicial methods also played a more recent role in the development of Hochschild homology, topological Hochschild homology and cyclic homology.This completes a quick overview of the history we shall discuss in this article. Now let us turn to the beginnings of the subject.

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Cohomology of algebraic theories

Cited by 100 publications

References 8 publications

Universal Toda brackets of ring spectra

Universal Toda brackets of ring spectra

On the computation of stabilized tensor functors and the relative algebraic 𝐾-theory of dual numbers

History of Homological Algebra

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