# Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

Received: 17 June 2019 / Revised: 3 July 2019 / Accepted: 4 July 2019 / Published: 6 July 2019

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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 $\Delta $ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of $\Delta $ . 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 $\pm 1$ coefficients.

*Keywords:*monomial ideal; Stanley-Reisner ring; linear part