Componentwise linear powers and the x-condition - Herzog, Jürgen; Hibi, Takayuki; Moradi, Somayeh (MATHEMATICA SCANDINAVICA, 2022)
Let S=K[x_1,…,x_n] be the polynomial ring over a field and A a standard graded S-algebra. In terms of the Gröbner basis of the defining ideal J of A we give a condition, called the x-condition, which implies that all graded components A_k of A have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring R(I) of a graded ideal or the symmetric algebra Sym(M) of a module M. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.