On finite algebras projectively represented by graded complete intersections
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.
scientific article; zbMATH DE number 549761