Free resolutions fo rmultigraded modules: a generalization of Taylor's construction
Abstrak
Let $Q=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with the standard $N^n$-grading. Let $φ$ be a morphism of finite free $N^n$-graded $Q$-modules. We translate to this setting several notions and constructions that appear originally in the context of monomial ideals. First, using a modification of the Buchsbaum-Rim complex, we construct a canonical complex $T_\bullet(φ)$ of finite free $N^n$-graded $Q$-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel $M$ of $φ$ when $φ$ satisfies certain rank criteria. We also introduce the Scarf complex of $φ$, and a notion of ``generic'' morphism. Our main result is that the Scarf complex of $φ$ is a minimal free resolution of $M$ when $φ$ is minimal and generic. Finally, we introduce the LCM-lattice for $φ$ and establish its significance in determining the minimal resolution of $M$.
Topik & Kata Kunci
Penulis (2)
H. Charalambous
A. Tchernev
Akses Cepat
- Tahun Terbit
- 2002
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓