Rocky Mountain Journal of Mathematics

On the structure of $S_2$-ifications of complete local rings

Sean Sather-Wagstaff and Sandra Spiroff

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Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.

Article information

Rocky Mountain J. Math., Volume 48, Number 3 (2018), 947-965.

First available in Project Euclid: 2 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
Secondary: 13D45: Local cohomology [See also 14B15] 13J10: Complete rings, completion [See also 13B35]

$S_2$-ifications graph labeling monomial ideals connected components canonical modules


Sather-Wagstaff, Sean; Spiroff, Sandra. On the structure of $S_2$-ifications of complete local rings. Rocky Mountain J. Math. 48 (2018), no. 3, 947--965. doi:10.1216/RMJ-2018-48-3-947.

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