A Minimal Morse Resolution of Path Ideals of Lines of Projective Dimension 2

Abstract

We study the connection between a special class of monomial ideals and CW complexes. Let I denote the ideal $I_t(L_{2t+1})$, i.e., a path ideal of line graph of projective dimension 2. We study cellular resolutions and discrete Morse theory as a tool to find a CW complex that supports the minimal free resolution of $I$. As a result, we have constructed an explicit Morse matching that induces a CW complex supporting the minimal free resolution of $I$. We also used the results from Bayer and Sturmfels to prove that the minimal free resolution of $I$ is supported on a solid $(t+2)$-gon.