Restuccia, Gaetana On the symmetric algebra for a module of projective dimension two. (English) Zbl 0781.13007 An. Univ. Bucur., Mat. 40, No. 1-2, 83-91 (1991). Let \(R\) be a Noetherian ring, \(E\) a finitely generated \(R\)-module, \(\text{Sym}_ R(E)\) the symmetric algebra of \(E\) on \(R\), \(\Psi\) the inclusion map \(R^ 2\to R^ m\) in the resolution \(0\to R^ 2\to R^ m\to R^ n\to E\to 0\), and \(I_ 2=I_ 2(\Psi)\) the ideal generated by \(2\times 2\) minors of a matrix representation of \(\Psi\). The author considers the question: What information can be obtained about \(I_ 2\) when \(\text{Sym}_ R(E)\) is a Cohen-Macaulay domain?The author shows that if the ideal of relations of \(E\) is prime, the \(\text{coker} Q\) of the map \(\bigwedge^ 2\Psi\) is reflexive, and if other conditions on the first syzygy module of \(E\) are satisfied, then \(\text{ht}(I_ 2)\) is odd. Finally, if \(R\) is Gorenstein and \(E\) satisfies the “sliding depth condition”, \(SD_{e+1}\) where \(e=\text{rank}(E)\), the author obtains a variety of information about the Koszul homology of \(I_ 2\). Reviewer: T.W.Hungerford (Cleveland) Cited in 3 Documents MSC: 13D02 Syzygies, resolutions, complexes and commutative rings 13D05 Homological dimension and commutative rings 13C14 Cohen-Macaulay modules 13E05 Commutative Noetherian rings and modules Keywords:Noetherian ring; symmetric algebra; Cohen-Macaulay domain; first syzygy module; Koszul homology PDFBibTeX XMLCite \textit{G. Restuccia}, An. Univ. Bucur., Mat. 40, No. 1--2, 83--91 (1991; Zbl 0781.13007)