arXiv subjects
Publications and source records attributed to Justin Lyle.
We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[Δ]$ as well as the $f$-vector and $h$-vector of $Δ$. We present, as an application, a formula for computing regularity of monomial ideals.
Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $\mbox{Hom}_R(M,N)$ has some nice properties and $\mbox{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$, then $M$ (and sometimes $N$) must be be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.