Explicit linear minimal free resolutions over a natural class of Rees algebras (Q1423844)

Let \(A\) be a standard graded algebra over a field \(K\) and let \(R= A[{\mathfrak m}t]\) be its Rees algebra, equipped with the usual bigrading given by \(\deg(gt^j)= (\deg(g)- j,j)\). The ring \(A\) is called a Koszul algebra if \(\text{Tor}^A_{i,j}(K, K)= 0\) for \(i\neq j\). Given a finitely generated graded \(A\)-module \(N\) having a linear minimal free resolution, the authors construct an explicit linear free resolution of \(N\otimes_A R\) over \(R\). As a consequence, they obtain an explicit and constructive proof of a result of \textit{R. Fröberg} and \textit{L. T. Hoa} [see Commun. Algebra 20, No. 11, 3369--3380 (1992; Zbl 0768.13005)] which says that \(R\) is a Koszul algebra if and only if \(A\) is a Koszul algebra and that \(A\) has a linear resolution as an \(R\)-algebra.