Associated graded rings of one-dimensional analytically irreducible rings
by
Valentina Barucci
Dipartimento di Matematica, Universita di Roma La Sapienza
Coauthors: Ralf Fröberg

Let (R, m) be a one-dimensional analytically irreducible and residually rational ring, e.g. the local ring of an irreducible curve singularity. Since the integral closure of R is a DVR V, if v is the valuation on V, we can consider S=v(R)={v(r);  0 ¹ r Î R }, which is a numerical semigroup. If e is the multiplicity of R, we call a subset {f₀, ..., f_e-1} of R an Apery basis of R if, for each i, 0 £ i £ e-1, {v(f₀), ..., v(f_e-1)} is the Apery set of S with respect to e, i.e. the set of the smallest elements in S in the e congruence classes mod e.

If {f¢₀, ..., f¢_e-1} is an Apery basis of R¢, the first neighborhood ring of R, i.e. the overring È_{n ³ 0}(mⁿ:mⁿ), for each i, 0 £ i £ e-1, we define the integers a_i and b_i in the following way:

a_i by v(f¢_i) = v(f_i)-a_ie

b_i as the largest integer j such that f_i Î m^j.

We prove the following criterion:

The associated graded ring gr(R)=Å_{n ³ 0}mⁿ/mⁿ⁺¹ is Cohen Macaulay if and only if a_i=b_i, for each i.

Some applications of the criterion will also be given.

Date received: March 23, 2005