On finite algebras projectively represented by graded complete intersections
type: Item
instance of: scholarly article
title: On finite algebras projectively represented by graded complete intersections
zbMATH Open document ID: 0822.13005
DOI: 10.1007/BF02571663
author: Günter Scheja
author: Uwe Storch
published in: Mathematische Zeitschrift
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174492
review text: Let \(A\) be a commutative noetherian ring and \(C = A[T_ 0, \dots, T_ n]/(F_ 1, \dots, F_ n)\) a graded complete intersection with \(\deg (F_ i) = \delta_ i\) of relative dimension one. Put \(X = \text{Proj} (C)\), \(L_ k = H^ 0 (X, {\mathcal O}_ X (k))\), and \(B = L_ 0 = H^ 0 (X, {\mathcal O}_ X)\) (the algebra projectively represented by \(C)\). Using the theory of generalized Grothendieck residue symbol, the authors define a bilinear function \(L_ k \times L_{\sigma - 1 -k} \to A\) with \(\sigma = \sum (\delta_ i - 1)\), which is a perfect duality. In particular, \(L_{\sigma - 1} \cong E = \Hom_ A (B,A)\), the relative canonical module of \(B\), and \(C_{\sigma - 1} \cong E^ 0 = \{\alpha \in E \mid \alpha (1) = 0\}\), the reduced relative canonical module of \(B\), and the \(B\)-modules \(L_ k\) are stably free over \(A\). As applications of this duality, the authors give various results about the canonical projective representations of \(B\), explicit formulas for the discriminants of \(A\)-linear forms on \(B\) in terms of the resultants, and so on.
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 549761
zbMATH Keywords: graded complete intersection
zbMATH Keywords: residue symbol
zbMATH Keywords: duality
reviewed by: Akira Ooishi
MaRDI profile type: Publication
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OpenAlex ID: W2040889479
description: scientific article; zbMATH DE number 549761