Let $A=K[a_1,...,a_n]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1–26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A={⨁}_{p\in \mathbb{N}}{A}_p$ is an $\mathbb{N}$-graded algebra with the degree-0 homogeneous part $A_0=K$; (2) $A$ is an $\mathbb{N}$-filtered algebra with the filtration ${\left\{F_{p}A\right\}}_{p\in \mathbb{N}}$ determined by a positive-degree...

Let $K$ be a field and $S=K[x_1,...,x_m,y_1,...,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” $S$-modules with respect to $Q=(y_1,...,y_n)$. Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to $Q$ are considered.