Local Cohomology and Degree Complexes of Monomial Ideals

2019/10/30 by Jonathan L. O’Rourke, O'Rourke, Jonathan L.
Computer Science · Mathematics · #05E40 #13A30 #13D45 #14B15 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1910.14140

openalex publication_date 2019/10/30 · openalex created_date 2019/11/08 · openalex updated_date 2026/07/28

Abstract

This paper examines the dimension of the graded local cohomology H_\mathfrakm^p(S/K^s)_γ and H_\mathfrakm^p(S/K^(s)) for a monomial ideal K. This information is encoded in the reduced homology of a simplicial complex called the degree complex. We explicitly compute the degree complexes of ordinary and symbolic powers of sums and fiber products of ideals, as well as the degree complex of the mixed product, in terms of the degree complexes of their components. We then use homological techniques to discuss the cohomology of their quotient rings. In particular, this technique allows for the explicit computation of reg ((I + J + \mathfrakm\mathfrakn)^(s)) in terms of the regularities of I⁽ⁱ⁾ and J^(j).

Citations

• Upper bounds for the regularity of powers of edge ideals of graphs
• Length of local cohomology of powers of ideals

Related

• Degree bounds for local cohomology
• Local Cohomology at Monomial Ideals
• Graded components of local cohomology modules of \mathfrakC-monomial ideals in characteristic zero
• Local Cohomology and complete intersections of rank 1
• A generalized Hochster's formula for local cohomologies of monomial ideals