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Remarks on a depth formula, a grade inequality and a conjecture of Auslander. (English) Zbl 0906.13002

Let \(A\) be a Noetherian local ring and \(M\), \(N\) two finitely generated \(A\)-modules. Assume that \(\text{Tor}^A_q(M,N) \neq 0\) and \(\text{Tor}^A_i(M,N) = 0\) for all \(i > q\). If \(M\) and \(N\) have finite projective dimension and \(q = 0\), then we can easily see \[ \text{depth} M + \text{depth} N = \text{depth} R + \text{depth Tor}^A_q(M,N) - q \tag{1} \] by using the Auslander-Buchsbaum theorem. Several mathematicians are interested in improving (1); see for example M. Auslander, Ill. J. Math. 5, 631-647 (1961; Zbl 0104.26202); C. Huneke and R. Wiegand, Math. Ann. 299, No. 3, 449-476 (1994; Zbl 0803.13008) and P. Constapel, Commun. Algebra 24, No. 3, 833-846 (1996; Zbl 0891.13007). In the present paper, the authors show (1) by assuming a certain homological invariant of \(N\) – called the CI-dimension – is finite. They also study the integer \(p\) such that \(\text{Ext}_A^p(M,N) \neq 0\) and \(\text{Ext}_A^i(M,N) = 0\) for all \(i > p\).

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13H15 Multiplicity theory and related topics
13D05 Homological dimension and commutative rings
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References:

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