Computation of Minimal Filtered Free Resolutions over $\mathbb{N}$-Filtered Solvable Polynomial Algebras

TL;DR

This paper develops a method to compute minimal filtered free resolutions of modules over weighted solvable polynomial algebras, establishing their uniqueness and providing an algorithmic approach using non-commutative Gr"obner bases.

Contribution

It introduces minimal F-bases and standard bases for modules over filtered solvable polynomial algebras, proving their properties and algorithmic computability.

Findings

Minimal F-bases and standard bases have invariant size and degree.

Minimal filtered free resolutions are unique up to isomorphism.

Algorithm for computing resolutions using Gr"obner basis theory is provided.

Abstract

Let $A=K[a_1,\ldots,a_n]$ be a weighted $\mathbb{N}$-filtered solvable polynomial algebra with filtration $FA=\{ F_pA\}_{p\in\mathbb{N}}$, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gr\"obner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26), and $FA$ is constructed with respect to a positive-degree function $d(~)$ on $A$. By introducing minimal F-bases and minimal standard bases respectively for left $A$-modules and their submodules with respect to good filtrations, minimal filtered free resolutions for finitely generated $A$-modules are introduced. It is shown that any two minimal F-bases, respectively any two minimal standard bases have the same number of elements and the same number of elements of the same filtered degree; that minimal filtered free resolutions are unique up to strict filtered isomorphism of chain complexes in the category of filtered $A$-modules; and that minimal finite filtered free resolutions can be algorithmically computed by employing Gr\"obner basis theory for modules over $A$ with respect to any graded left monomial ordering on free left $A$-modules.