Constructing schemes with prescribed cohomology in arbitrary codimension
Let \(k\) be a field, \(R = k[X_1, \dots, X_n]\) a polynomial ring, \(\mathfrak m = (X_1, \dots, X_n)\) and \(M_2\), \dots, \(M_{d-1}\) graded \(R\)-modules of finite length where \(2 < d \leq n-2\). The authors give a homogeneous ideal \(I\) of \(R\) such that \(\dim R/I = d\) and H\(_{\mathfrak m}^i(R/I) \cong M_i\) (up to shift) for \(2 \leq i \leq d-1\). If \(n-d = 2\), then Evans and Griffith already gave such a ring by using Bourbaki sequence. However the authors use the theory of orientable modules and give the ring when \(n - d \geq 3\).
scientific article; zbMATH DE number 1517304