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On the symmetric algebra for a module of projective dimension two. (English) Zbl 0781.13007

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 \(I_ 2\).

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13D05 Homological dimension and commutative rings
13C14 Cohen-Macaulay modules
13E05 Commutative Noetherian rings and modules
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