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), 135165
MSC:
Primary 16A58; Secondary 16A61, 18E25, 55N25, 55U10
MathSciNet review:
965749
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Abstract: To each presheaf (over a poset) of associative algebras we associate an algebra . We define a full exact embedding of the category of (presheaf) bimodules in that of 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.
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Murray
Gerstenhaber and Samuel
D. Schack, Simplicial cohomology is Hochschild cohomology, J.
Pure Appl. Algebra 30 (1983), no. 2, 143–156.
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 [G1]
 M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267288. MR 0161898 (28:5102)
 [G2]
 , On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59103. 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), 150. MR 704600 (85d:16021)
 [GS2]
 , Simplicial cohomology is Hochschild cohomology, J. Pure Appl. Algebra (2) 30 (1983), 143156. MR 722369 (85f:18008)
 [GS3]
 , On the cohomology of an algebra morphism, J. Algebra (1) 95 (1985), 245262. MR 797666 (87a:18021)
 [GS4]
 , Relative Hochschild cohomology, rigid algebras, and the Bockstein, J. Pure Appl. Algebra (1) 43 (1986), 5374. MR 862872 (88a:16045)
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 , The cohomology of presheaves of algebras II: the barycentric subdivision of a small category (to appear).
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 [Gr]
 A. Grothendieck, Sur quelques points d'algebre homologique, Tôhoku Math. J. 9 (1957), 119221. MR 0102537 (21:1328)
 [H]
 G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 5867. MR 0011076 (6:114f)
 [M]
 S. Mac Lane, Homology, SpringerVerlag, 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), 290320. MR 0022071 (9:154a)
 [VZ]
 O. Villamayor and D. Zelinsky, Galois theory for rings with finitely many idempotents, Nagoya Math. J. 27 (1966), 721731. MR 0206055 (34:5880)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719880965749X
PII:
S 00029947(1988)0965749X
Keywords:
Associative algebra,
bimodule,
Yoneda cohomology,
Hochschild cohomology,
simplicial cohomology,
deformation,
triangulable space
Article copyright:
© Copyright 1988 American Mathematical Society
