Noninvertible retracts

Yanik, Joe

doi:10.1090/S0002-9939-1983-0677224-1

Noninvertible retracts

Author: Joe Yanik
Journal: Proc. Amer. Math. Soc. 87 (1983), 29-32
MSC: Primary 13F20; Secondary 13D15
DOI: https://doi.org/10.1090/S0002-9939-1983-0677224-1
MathSciNet review: 677224
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Abstract: We demonstrate that if is a projective -module that is not stably extended from an -module, then the symmetric algebra of over is a retract of a polynomial ring over , but is not an invertible -algebra. Hence, there are noninvertible retracts over a quite general class of rings.

[B] Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
[Bo] N. Bourbaki, Algebra. I, Hermann, Paris, 1970.
[C] E. H. Connell, A 𝐾-theory for the category of projective algebras, J. Pure Appl. Algebra 5 (1974), 281–292. MR 384803, https://doi.org/10.1016/0022-4049(74)90038-3
[E-H] Paul Eakin and William Heinzer, A cancellation problem for rings, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Springer, Berlin, 1973, pp. 61–77. Lecture Notes in Math., Vol. 311. MR 0349664
[G] Cornelius Greither, Seminormality, projective algebras, and invertible algebras, J. Algebra 70 (1981), no. 2, 316–338. MR 623811, https://doi.org/10.1016/0021-8693(81)90221-0
[M] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
[W] Charles A. Weibel, 𝐾-theory and analytic isomorphisms, Invent. Math. 61 (1980), no. 2, 177–197. MR 590161, https://doi.org/10.1007/BF01390120
[Y] Joe Yanik, Projective algebras, J. Pure Appl. Algebra 21 (1981), no. 3, 339–358. MR 617139, https://doi.org/10.1016/0022-4049(81)90022-0