Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

• Main Author: Lukas Katthän
• Format: Article
• Language: English
• Published: MDPI AG 2019-07-01
• Series: Mathematics
• Subjects: monomial ideal; Stanley-Reisner ring; linear part
• Online Access: https://www.mdpi.com/2227-7390/7/7/605

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex <inline-formula> <math display="inline"> <semantics> <mo>&#916;</mo> </semantics> </math> </inline-formula>. Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of <inline-formula> <math display="inline"> <semantics> <mo>&#916;</mo> </semantics> </math> </inline-formula>. Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only <inline-formula> <math display="inline"> <semantics> <mrow> <mo>&#177;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> coefficients.