## Rocky Mountain Journal of Mathematics

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

#### 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

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

#### 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

#### 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.
• B. Holmes, On simplicial complexes with Serre property $S_2$, in preparation.