Tame homomorphisms of polytopal rings.
Let be a pure simplicial complex on the vertex set and its Stanley-Reisner ideal in the polynomial ring . We show that is a matroid (complete intersection) if and only if () is clean for all and this is equivalent to saying that (, respectively) is Cohen-Macaulay for all . By this result, we show that there exists a monomial ideal with (pretty) cleanness property while or is not (pretty) clean for all integer . If , we also prove that () is clean if and only if (,...
For each squarefree monomial ideal , we associate a simple finite graph by using the first linear syzygies of . The nodes of are the generators of , and two vertices and are adjacent if there exist variables such that . In the cases, where is a cycle or a tree, we show that has a linear resolution if and only if has linear quotients and if and only if is variable-decomposable. In addition, with the same assumption on , we characterize all squarefree monomial ideals with a...