Hilbert coefficients of quadratic algebras

Abstract

If $R=k[x_1,\dots ,x_n]/I$ is a graded artinian algebra, then the length of $k[x_1,\dots ,x_n]/I^s$ becomes a polynomial in $s$ of degree $n$ for large $s$. If we write this polynomial as ${\sum }_{i=0}^n(-1{)}^i{e}_i(\genfrac{}{}{0ex}{}{s+n-i-1}{n-i})$, then the $e_i$'s are called Hilbert coefficients of $I$. We will study length and Hilbert coefficients of some classes of quadratic algebras.