Symbolic powers of edge ideals.
Given a (finite simple) graph \(G\) with vertex set \(\{X_1,\dots, X_n\}\), a set of indeterminates generating the polynomial ring \(R= k[X_1,\dots, X_r]\), the ideal \(I(G)= (\{X_iX_j|\, X_i\) adjacent to \(X_j\})\) and the ring \(R/I(G)\) becomes of interest in the setting of commutative algebra/algebraic geometry as well as graph theory, since it is one more way to connect an algebraic object in a natural way to a graph. Thus one may relate parameters such as the stability number \(\alpha\) of \(G\) to properties of \(R/I(G)\) for example. A similar idea with \(R= k\langle X_1,\dots, X_n\rangle\) and the variables being non-commutative, including digraphs, has also been considered earlier than items in the reference list. In this case the setting of the paper under review is that of arbitrary ideals \(I\) which are specialized to ideals generated by square-free monomials (polynomials) of degree two (i.e., ideals of type \(I(G)\)). If \(I^{(m)}\) is the intersection of all primary components of \(I^m\) corresponding to minimal primes of \(I\), \(\sum_r(I)=\sum_r \sum^{r-1}_{i=1} I^{(i)} I^{(r- i)},\) and \(F(I)=\bigoplus{\strut}_{r\geq 2}(I^{(r)}/\sum_r (I))\), the essentially symbolic \(R\)-module of \(I\), then an element \(f\in I^{(r)}\setminus\sum_r(I)\), an essential generator of order \(r\) of \(F(I)\), is the prey mostly to be stalked in order to answer questions which in the \(I(G)\) case provide other answers/estimates for suitable types of simple graphs \(G\). A typical result (among several main results) might be that \(n\leq 2h-1\), where \(h\) is the height of \(I(G)\). While recapturing previous results of \textit{A. Simis}, \textit{W. V. Vasconcelos} and \textit{R. H. Villareal} [J. Algebra 167, No. 2, 389--416 (1994; Zbl 0816.13003)] as well, arguments in this well-done paper tend to have the ``free algebra'' flavor with ``side conditions'' leading to specifics as is common to similar contexts including this one.
scientific article; zbMATH DE number 2055681