## Algebras over absolutely flat commutative rings

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- by Joseph A. Wehlen
- Trans. Amer. Math. Soc.
**196**(1974), 149-160 - DOI: https://doi.org/10.1090/S0002-9947-1974-0345996-0
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## 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
N. Bourbaki, - 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

*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.

## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**196**(1974), 149-160 - MSC: Primary 16A16; Secondary 16A60
- DOI: https://doi.org/10.1090/S0002-9947-1974-0345996-0
- MathSciNet review: 0345996