Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set

Authors: Murray Gerstenhaber and Samuel D. Schack
Journal: Trans. Amer. Math. Soc. 310 (1988), 135-165
MSC: Primary 16A58; Secondary 16A61, 18E25, 55N25, 55U10
MathSciNet review: 965749
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: To each presheaf (over a poset) of associative algebras $ \mathbb{A}$ we associate an algebra $ \mathbb{A}!$. We define a full exact embedding of the category of (presheaf) $ \mathbb{A}$-bimodules in that of $ \mathbb{A}!$-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.

References [Enhancements On Off] (What's this?)

  • [CE] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • [G1] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267-288. MR 0161898 (28:5102)
  • [G2] -, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. MR 0171807 (30:2034)
  • [GS1] M. Gerstenhaber and S. D. Schack, On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc. 279 (1983), 1-50. MR 704600 (85d:16021)
  • [GS2] -, Simplicial cohomology is Hochschild cohomology, J. Pure Appl. Algebra (2) 30 (1983), 143-156. MR 722369 (85f:18008)
  • [GS3] -, On the cohomology of an algebra morphism, J. Algebra (1) 95 (1985), 245-262. MR 797666 (87a:18021)
  • [GS4] -, Relative Hochschild cohomology, rigid algebras, and the Bockstein, J. Pure Appl. Algebra (1) 43 (1986), 53-74. MR 862872 (88a:16045)
  • [GS5] -, The cohomology of presheaves of algebras II: the barycentric subdivision of a small category (to appear).
  • [GS6] -, The cohomology of presheaves of algebras III: Embedding theorems (to appear).
  • [Gr] A. Grothendieck, Sur quelques points d'algebre homologique, Tôhoku Math. J. 9 (1957), 119-221. MR 0102537 (21:1328)
  • [H] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58-67. MR 0011076 (6:114f)
  • [M] S. Mac Lane, Homology, Springer-Verlag, Berlin and New York, 1967. MR 1344215 (96d:18001)
  • [S] N. E. Steenrod, Products of cocycles and extensions of mappings, Ann. of Math. (2) 48 (1947), 290-320. MR 0022071 (9:154a)
  • [VZ] O. Villamayor and D. Zelinsky, Galois theory for rings with finitely many idempotents, Nagoya Math. J. 27 (1966), 721-731. MR 0206055 (34:5880)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A58, 16A61, 18E25, 55N25, 55U10

Retrieve articles in all journals with MSC: 16A58, 16A61, 18E25, 55N25, 55U10

Additional Information

Keywords: Associative algebra, bimodule, Yoneda cohomology, Hochschild cohomology, simplicial cohomology, deformation, triangulable space
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society