Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebras over absolutely flat commutative rings


Author: Joseph A. Wehlen
Journal: Trans. Amer. Math. Soc. 196 (1974), 149-160
MSC: Primary 16A16; Secondary 16A60
MathSciNet review: 0345996
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of $ {A_x}$ for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and $ {A^e}$.

It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.


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

  • [1] M. Auslander, On the dimension of modules and algebras. III: Global dimension, Nagoya Math. J. 9 (1955), 67-77. MR 0074406 (17:579a)
  • [2] N. Bourbaki, Eléments de mathématique. Fasc. XXVII. Algèbre commutative. Chap. 1: Modules plats. Chap. 2: Localisation, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [3] W. C. Brown, A splitting theorem for algebras over commutative von Neumann regular rings, Proc. Amer. Math. Soc. 36 (1972), 369-374. MR 0314887 (47:3436)
  • [4] S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J. 18 (1961), 13-25. MR 23 #A919. MR 0123594 (23:A919)
  • [5] J. Dauns and K. H. Hofmann, Representation of rings by sections, Mem. Amer. Math. Soc. No. 83 (1968). MR 40 #752. MR 0247487 (40:752)
  • [6] S. Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310-319. MR 16, 442. MR 0065544 (16:442c)
  • [7] S. Eilenberg, A. Rosenberg, and D. Zelinsky, On the dimension of modules and algebras. VIII: Dimension of tensor product, Nagoya Math. J. 12 (1957), 71-93. MR 20 #5229. MR 0098774 (20:5229)
  • [8] F. DeMeyer and E. C. Ingraham, Separable algebras over commutative rings, Lecture Notes in Math., vol. 181, Springer-Verlag, Berlin and New York, 1971. MR 43 #6199. MR 0280479 (43:6199)
  • [9] S. Endo and Y. Watanabe, On separable algebras over a commutative ring, Osaka J. Math. 4 (1967), 233-242. MR 37 #2796. MR 0227211 (37:2796)
  • [10] N. Jacobson, Structure of rings, 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964 MR 36 #5158. MR 0222106 (36:5158)
  • [11] J. Jans and T. Nakayama, Dimension of modules and algebras. VII: Algebras with finite-dimensional residue-algebras, Nagoya Math. J. 11 (1957), 65-76. MR 19, 250. MR 0086824 (19:250a)
  • [12] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [13] R. S. Pierce, Modules over commutative regular rings, Mem. Amer. Math. Soc. No. 70 (1967). MR 36 #151. MR 0217056 (36:151)
  • [14] J. A. Wehlen, Algebras of finite cohomological dimension, Nagoya Math. J. 43 (1971), 127-135. MR 46 #212. MR 0301054 (46:212)
  • [15] -, Cohomological and global dimension of algebras, Proc. Amer. Math. Soc. 32 (1972), 75-80. MR 45 #320. MR 0291226 (45:320)
  • [16] -, Triangular matrix algebras over Hensel rings, Proc. Amer. Math. Soc. 37 (1973), 69-74. MR 0308196 (46:7311)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A16, 16A60

Retrieve articles in all journals with MSC: 16A16, 16A60


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0345996-0
PII: S 0002-9947(1974)0345996-0
Keywords: Hochschild dimension, weak global dimension, idempotents, separable algebra, biregular ring, absolutely flat ring, triangular matrix ring, maximal algebra
Article copyright: © Copyright 1974 American Mathematical Society