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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [CE]
Henri
Cartan and Samuel
Eilenberg, Homological algebra, Princeton University Press,
Princeton, N. J., 1956. MR 0077480
(17,1040e)
 [G1]
Murray
Gerstenhaber, The cohomology structure of an associative ring,
Ann. of Math. (2) 78 (1963), 267–288. MR 0161898
(28 #5102)
 [G2]
Murray
Gerstenhaber, 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), no. 1, 1–50. MR 704600
(85d:16021), http://dx.doi.org/10.1090/S00029947198307046005
 [GS2]
Murray
Gerstenhaber and Samuel
D. Schack, Simplicial cohomology is Hochschild cohomology, J.
Pure Appl. Algebra 30 (1983), no. 2, 143–156.
MR 722369
(85f:18008), http://dx.doi.org/10.1016/00224049(83)900518
 [GS3]
Murray
Gerstenhaber and Samuel
D. Schack, On the cohomology of an algebra morphism, J.
Algebra 95 (1985), no. 1, 245–262. MR 797666
(87a:18021), http://dx.doi.org/10.1016/00218693(85)901048
 [GS4]
Murray
Gerstenhaber and Samuel
D. Schack, Relative Hochschild cohomology, rigid algebras, and the
Bockstein, J. Pure Appl. Algebra 43 (1986),
no. 1, 53–74. MR 862872
(88a:16045), http://dx.doi.org/10.1016/00224049(86)900046
 [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]
Alexander
Grothendieck, Sur quelques points d’algèbre
homologique, Tôhoku Math. J. (2) 9 (1957),
119–221 (French). 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]
Saunders
Mac Lane, Homology, Classics in Mathematics, SpringerVerlag,
Berlin, 1995. Reprint of the 1975 edition. 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.
E. Villamayor and D.
Zelinsky, Galois theory for rings with finitely many
idempotents, Nagoya Math. J. 27 (1966),
721–731. MR 0206055
(34 #5880)
 [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), 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)
 [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), 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)
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
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
