Araya, Tokuji; Yoshino, Yuji Remarks on a depth formula, a grade inequality and a conjecture of Auslander. (English) Zbl 0906.13002 Commun. Algebra 26, No. 11, 3793-3806 (1998). 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 \(i > p\). Reviewer: Takesi Kawasaki (Tokyo) Cited in 1 ReviewCited in 46 Documents MSC: 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13H15 Multiplicity theory and related topics 13D05 Homological dimension and commutative rings Keywords:depth of Tor; CI dimension; Auslander conjecture; Noetherian local ring Citations:Zbl 0104.26202; Zbl 0803.13008; Zbl 0891.13007 PDFBibTeX XMLCite \textit{T. Araya} and \textit{Y. Yoshino}, Commun. Algebra 26, No. 11, 3793--3806 (1998; Zbl 0906.13002) Full Text: DOI References: [1] Auslander M., J. Math 5 pp 631– (1961) [2] DOI: 10.1006/jabr.1993.1076 · Zbl 0778.13007 · doi:10.1006/jabr.1993.1076 [3] Avramov L.L., Publ. Math. I.H.E.S. [4] Bruns W., Cambridge studies in advanced Math 39 (1993) [5] Hochster M., Regional Conf. Ser. in Math 24 (1975) [6] DOI: 10.1007/BF01459794 · Zbl 0803.13008 · doi:10.1007/BF01459794 [7] Huneke C., rigidity and local cohomology [8] Lichtenbaum S., J. Math pp 220– (1966) [9] Murthy M.P., J. Math 7 pp 558– (1963) [10] Roberts P., C. R. Acad. Sc. Paris Sér 304 pp 177– (1987) [11] DOI: 10.1006/jabr.1996.6821 · Zbl 0928.13003 · doi:10.1006/jabr.1996.6821 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.