Algebras over absolutely flat commutative rings

Wehlen, Joseph A.

doi:10.1090/S0002-9947-1974-0345996-0

Algebras over absolutely flat commutative rings
HTML articles powered by AMS MathViewer

by Joseph A. Wehlen PDF

Trans. Amer. Math. Soc. 196 (1974), 149-160 Request permission

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

Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR 74406, DOI 10.1017/S0027763000023291

Eléments de mathématique

Algèbre commutative

Modules plats

Localisation

36

William C. Brown, A splitting theorem for algebras over commutative von Neumann regular rings, Proc. Amer. Math. Soc. 36 (1972), 369–374. MR 314887, DOI 10.1090/S0002-9939-1972-0314887-7
Stephen U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J. 18 (1961), 13–25. MR 123594, DOI 10.1017/S0027763000002208
John Dauns and Karl Heinrich Hofmann, Representation of rings by sections, Memoirs of the American Mathematical Society, No. 83, American Mathematical Society, Providence, R.I., 1968. MR 0247487
Samuel Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310–319. MR 65544, DOI 10.1007/BF02566937
Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky, On the dimension of modules and algebras. VIII. Dimension of tensor products, Nagoya Math. J. 12 (1957), 71–93. MR 98774, DOI 10.1017/S0027763000021954
Frank DeMeyer and Edward Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971. MR 0280479, DOI 10.1007/BFb0061226
Shizuo Endo and Yutaka Watanabe, On separable algebras over a commutative ring, Osaka Math. J. 4 (1967), 233–242. MR 227211
Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
J. P. Jans and Tadasi Nakayama, On the dimension of modules and algebras. VII. Algebras with finite-dimensional residue-algebras, Nagoya Math. J. 11 (1957), 67–76. MR 86824, DOI 10.1017/S0027763000001951
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056
Joseph A. Wehlen, Algebras of finite cohomological dimension, Nagoya Math. J. 43 (1971), 127–135. MR 301054, DOI 10.1017/S0027763000014409
Joseph A. Wehlen, Cohomological dimension and global dimension of algebras, Proc. Amer. Math. Soc. 32 (1972), 75–80. MR 291226, DOI 10.1090/S0002-9939-1972-0291226-1
Joseph A. Wehlen, Triangular matrix algebras over Hensel rings, Proc. Amer. Math. Soc. 37 (1973), 69–74. MR 308196, DOI 10.1090/S0002-9939-1973-0308196-0