2004/09/30 by Alexandre Tchernev, Tchernev, Alexandre
Computer Science · Mathematics · #13D02 (Primary) 13A02 #52B40 #52C35 (Secondary) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras #math.AC #math.CO #msc:13A02 #msc:13D02 #msc:52B40 #msc:52C35
paper · pdf · doi:10.48550/arxiv.math/0409591
53 pages, added a reference, added examples, corrected typos
openalex publication_date 2004/09/30 · arxiv created 2004/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a field, let R=K[x1,..., xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let F: E --> G be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T(F, S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation f over K of a matroid M a canonical finite complex of finite dimensional K-vector spaces T(f) that is a resolution of Ker(f). We also show that the length of T(f) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map f.