Associated graded rings of one-dimensional analytically irreducible rings II

Lance Bryant noticed in his thesis that there was a flaw in our paper "Associated graded rings of one-dimensional analytically irreducible rings", J. Algebra 304 (2006), 349-358. It can be fixed by adding a condition, called the BF condition. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection.Comment: revised argument for Lemma 1.1, results unchange

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.Comment: Revised version, with new title, new author and new result

Gorenstein rings generated by strongly stable sets of quadratic monomials

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel

The graded Betti numbers of truncation of ideals in polynomial rings

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $\Delta_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $\Delta_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$

Associated graded rings of one-dimensional analytically irreducible rings,

Abstract Lance Bryant noticed in his thesis Mathematics Subject Classification: 13A30 1 The BF condition Let (R, m) be an equicharacteristic analytically irreducible and residually rational local 1-dimensional domain of embedding dimension ν, multiplicity e and residue field k. For the problems we study we may, and will, without loss of generality suppose that R is complete. So our hypotheses are equivalent to supposing R is a subring of , the integral closure of R, is a DVR, every nonzero element of R has a value, and we let S = v(R) = {v(r); r ∈ R, r = 0}. We denote by w 0 , . . . , w e−1 the Apery set of v(R) with respect to e, i.e., the set of smallest values in v(R) in each congruence class (mod e), and we assume w j ≡ j (mod e)

Associated graded rings of one-dimensional analytically irreducible rings,

Abstract Lance Bryant noticed in his thesis Mathematics Subject Classification: 13A30 1 The BF condition Let (R, m) be an equicharacteristic analytically irreducible and residually rational local 1-dimensional domain of embedding dimension ν, multiplicity e and residue field k. For the problems we study we may, and will, without loss of generality suppose that R is complete. So our hypotheses are equivalent to supposing R is a subring of , the integral closure of R, is a DVR, every nonzero element of R has a value, and we let S = v(R) = {v(r); r ∈ R, r = 0}. We denote by w 0 , . . . , w e−1 the Apery set of v(R) with respect to e, i.e., the set of smallest values in v(R) in each congruence class (mod e), and we assume w j ≡ j (mod e)