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# On Set Theoretically and Cohomologically Complete Intersection Ideals

Published:2014-04-28
Printed: Sep 2014
• Majid Eghbali,
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran-Iran
 Format: LaTeX MathJax PDF

## Abstract

Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities $\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq \operatorname{ara}(\mathfrak a) \leq l(\mathfrak a) \leq \mu(\mathfrak a)$ are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.
 Keywords: set-theoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of ideals
 MSC Classifications: 13D45 - Local cohomology [See also 14B15] 13C14 - Cohen-Macaulay modules [See also 13H10]

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