On maximality of Gorenstein sequences

Abstract:

It is well known that if $A$ is a Gorenstein ring, then every ideal generated by a regular sequence ${\text {x}} \subset A$ has irreducible (minimal) primary components. This feature led us to define a Gorenstein sequence of a ring $A$ to be any ordered regular sequence ${\text {x}} = \{ {x_1}, \ldots ,{x_r}\} \subset A$ such that for every $i \in \{ 1, \ldots ,r\}$ the ideal $({x_1}, \ldots ,{x_i})$ has irreducible minimal primary components. We showed for Gorenstein sequences (${\mathbf {G}}$-sequences for short) some parallels of well-known properties of regular sequences and moreover by means of ${\mathbf {G}}$-sequences we gave the following natural characterization of local Gorenstein rings: “A local ring $(A,\;\mathfrak {m})$ is Gorenstein iff $\mathfrak {m}$ contains a ${\mathbf {G}}{\text { - sequence of length = }}K{\text { - }}\dim A$". In this note we are going to give some information about “maximality” of ${\mathbf {G}}$-sequences in a local ring $A$, producing sufficient conditions on $A$ in order that the maximal ${\mathbf {G}}$-sequences of $A$ all have the same length, i.e. in order to give a “good” definition of ${\mathbf {G}}$-depth $A$. Furthermore, we will state some results about the ${\mathbf {G}}$-depth behavior with respect to local flat ring homomorphisms.