# What is a “staircase” of an ideal in a polynomial ring?

I've come across the term "staircase" applied to an ideal in a polynomial ring over a field. Can someone explain this?

If a bit more formalism is required: Let $k$ be a field, let $R = k[X_1, \ldots, X_n]$ for an integer $n$ and let $I$ be an ideal in $R$. Then the "staircase of $I$" is mentioned.

-
It is usually the ideal spanned by the initial monomials of the elements of $I$, or equivalently, the ideal generated by the initial monomials of a Groebner basis of $I$. –  Mariano Suárez-Alvarez Nov 10 '11 at 16:16
Thanks! That makes perfect sense in the context of what I'm reading. –  Bristol Nov 10 '11 at 16:26
(If you draw the initial monomials of the elements in the initial ideal, in the case of two variables, as dots in $\mathbb N^2$, then the reason for the name should be obvious. Sometimes it is this picture which is called the staircase.) –  Mariano Suárez-Alvarez Nov 10 '11 at 16:31