Generalized analytic independence
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- by Jacob Barshay PDF
- Proc. Amer. Math. Soc. 58 (1976), 32-36 Request permission
Abstract:
If ${\mathbf {a}}$ is a proper ideal of a commutative ring with unity $A$, a set of elements ${a_1}, \ldots ,{a_n} \in A$ is called ${\mathbf {a}}$-independent if every form in $A[{X_1}, \ldots ,{X_n}]$ vanishing at ${a_1}, \ldots ,{a_n}$ has all its coefficients in ${\mathbf {a}}$. $\sup {\mathbf {a}}$ is defined as the maximum number of ${\mathbf {a}}$-independent elements in ${\mathbf {a}}$. It is shown that grade ${\mathbf {a}} \leq \sup {\mathbf {a}} \leq {\text {height }}{\mathbf {a}}$. Examples are given to show that $\sup {\mathbf {a}}$ need take neither of the limiting values and strong evidence is given for the conjecture that it can assume any intermediate value. Cohen-Macaulay rings are characterized by the equality of sup and grade for all ideals (or just all prime ideals). It is proven that equality of sup and height for all powers of prime ideals implies that the ring is ${S_1}$ (the Serre condition). Finally, independence is related to the structure of certain Rees algebras.References
- Jacob Barshay, Graded algebras of powers of ideals generated by $A$-sequences, J. Algebra 25 (1973), 90â99. MR 332748, DOI 10.1016/0021-8693(73)90076-8
- Erwin BÜger, Minimalitätsbedingungen in der Theorie der Reduktionen von Idealen, Schr. Math. Inst. Univ. Mßnster 40 (1968), viii+57 (German). MR 238840
- Artibano Micali, Sur les algèbres universelles, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 33â87 (French). MR 177009
- Giuseppe Valla, Elementi indipendenti rispetto ad un ideale, Rend. Sem. Mat. Univ. Padova 44 (1970), 339â354 (Italian). MR 297750
- Giuseppe Valla, Su certi isomorfismi di algebre e lâindipendenza rispetto ad un ideale, Matematiche (Catania) 26 (1971), 35â45 (1972) (Italian). MR 316444
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 58 (1976), 32-36
- MSC: Primary 13C15; Secondary 14M10
- DOI: https://doi.org/10.1090/S0002-9939-1976-0406997-4
- MathSciNet review: 0406997