2016/05/31 by Alexandre Tchernev, Tchernev, Alexandre, Marco Varisco +1
Computer Science · Mathematics · #05E40 #13D02 #14M25 #18G10 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:05E40 #msc:13D02 #msc:14M25 #msc:18G10
paper · pdf · doi:10.48550/arxiv.1605.09748
24 pages
arxiv created 2016/05/31 · openalex publication_date 2016/05/31 · arxiv updated 2016/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theory applies to the toric case, and provide an analog for toric rings of the lcm-lattice for monomial ideals.