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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Abstract

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

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

Dates
First available in Project Euclid: 2 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1533230834

Digital Object Identifier
doi:10.1216/RMJ-2018-48-3-947

Mathematical Reviews number (MathSciNet)
MR3835581

Zentralblatt MATH identifier
06917356

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.rmjm/1533230834


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References

  • B. Benedetti, B. Bolognese and M. Varbaro, Regulating Hartshorn's connectedness theorem, J. Alg. Comb. 46 (2017), 33–50.
  • B. Benedetti and M. Varbaro, On the dual graph of Cohen-Macaulay algebras, Int. Math. Res. Not. 2015 (2015), 8085–8115.
  • M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Comm. Algebra 40 (2012), 4295–4305.
  • G. Faltings, Some theorems about formal functions, Publ. Res. Inst. Math. Sci. 16 (1980), 721–737.
  • F. Harary, Graph theory, Addison-Wesley Publishing Co., Reading, MA, 1969.
  • R. Hartshorne, Complete intersections and connectedness, Amer. J. Math. 84 (1962), 497–508.
  • M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, Contemp. Math. 159 (1992), 197–208.
  • B. Holmes, On simplicial complexes with Serre property $S_2$, in preparation.
  • G. Lyubeznik, On some local cohomology invariants of local rings, Math. Z. 254 (2006), 627–640.
  • R. Naimi and J. Shaw, Induced subgraphs of Johnson graphs, Involve 5 (2012), 25–37.
  • S. Sather-Wagstaff and S. Spiroff, Torsion in kernels of induced maps on divisor class groups, J. Alg. Appl. 14 (2015), 1500123, 23 pages.
  • P. Schenzel, On connectedness and indecomposibility of local cohomology modules, Manuscr. Math. 128 (2009), 315–327.
  • U. Walther, On the Lyubeznik numbers of a local ring, Proc. Amer. Math. Soc. 129 (2001), 1631–1634.
  • W. Zhang, On the highest Lyubeznik number of a local ring, Compos. Math. 143 (2007), 82–88.