##
**Geometric domains with specified pseudo-dimensions.**
*(English)*
Zbl 1062.13001

This paper is concerned with the pseudo-supports of a finitely generated module over a finite-dimensional commutative Noetherian ring. The authors are concerned with the properties of pseudo-supports of geometric local domains (local domains of finite type over an algebraically closed field). Because they work with localization of affine domains they extend the definition of pseudo-support given by M. P. Brodmann and R. Y. Sharp [Nagoya Math. J. 167, 217–233 (2002; Zbl 1044.13007)] and apply it to any finite-dimensional commutative Noetherian ring. The pseudo-supports of such geometric local domains have properties that can be summarized either by an occurrence diagram of the ring or by a sequence of pseudo-dimensions.

First the authors obtain (in some circumstances) a geometric local domain with a specified occurrence diagram from an affine algebra (over an algebraically closed field) with the same specified occurrence diagram. Standard properties of local cohomology modules enable the authors to conclude that there are obvious restrictions on the type of a subset of \(\mathbb{N}_{0}\times \mathbb{N}_{0}\) that can be an occurrence diagram in this way. Much of the paper is devoted to the consideration of whether a subset of \(\mathbb{N}_{0}\times \mathbb{N}_{0}\) that meets those obvious restrictions is an occurrence diagram of some geometric normal local domain.

The authors show how to produce examples of normal geometric local domains with occurrence diagrams that are more complicated than the 0-level and \(h\)-level \(n\)-diagrams and include ones in which the heights of the non-empty columns fail to form a non-decreasing sequence. For this they use Segre products of standard normal graded Noetherian domains defined over an algebraically closed field.

In the final section the authors present some connections between occurrence diagrams and certain sets that are related to Grothendieck’s finiteness theorem for local cohomology. They provide many geometric normal local domains and show that a question raised by C. Huneke [Res. Notes Math. 2, 93–108 (1992; Zbl 0782.13015)] about certain sets related to the Grothendieck theorem has a negative answer. It is possible to deduce from the appearance of the occurrence diagram for \(M\), a finitely generated module over a finite dimensional local ring \((R,m)\) that the inclusion \(\Lambda _{m}(M)\subseteq F_{m}(M)\) is strict. Here \[ F_m(M)=:\{i\in\mathbb{N}\mid H_m^i(M)\text{ is not finitely generated}\} \] and \[ \Lambda_m(M)=:\text{\{depth }M_m+\text{ht}(m+p)/p\mid p\in\text{Supp}(M)\setminus \text{Var}(m)\} \] where \(\text{Var}(m)\) means the variety of \(m\).

First the authors obtain (in some circumstances) a geometric local domain with a specified occurrence diagram from an affine algebra (over an algebraically closed field) with the same specified occurrence diagram. Standard properties of local cohomology modules enable the authors to conclude that there are obvious restrictions on the type of a subset of \(\mathbb{N}_{0}\times \mathbb{N}_{0}\) that can be an occurrence diagram in this way. Much of the paper is devoted to the consideration of whether a subset of \(\mathbb{N}_{0}\times \mathbb{N}_{0}\) that meets those obvious restrictions is an occurrence diagram of some geometric normal local domain.

The authors show how to produce examples of normal geometric local domains with occurrence diagrams that are more complicated than the 0-level and \(h\)-level \(n\)-diagrams and include ones in which the heights of the non-empty columns fail to form a non-decreasing sequence. For this they use Segre products of standard normal graded Noetherian domains defined over an algebraically closed field.

In the final section the authors present some connections between occurrence diagrams and certain sets that are related to Grothendieck’s finiteness theorem for local cohomology. They provide many geometric normal local domains and show that a question raised by C. Huneke [Res. Notes Math. 2, 93–108 (1992; Zbl 0782.13015)] about certain sets related to the Grothendieck theorem has a negative answer. It is possible to deduce from the appearance of the occurrence diagram for \(M\), a finitely generated module over a finite dimensional local ring \((R,m)\) that the inclusion \(\Lambda _{m}(M)\subseteq F_{m}(M)\) is strict. Here \[ F_m(M)=:\{i\in\mathbb{N}\mid H_m^i(M)\text{ is not finitely generated}\} \] and \[ \Lambda_m(M)=:\text{\{depth }M_m+\text{ht}(m+p)/p\mid p\in\text{Supp}(M)\setminus \text{Var}(m)\} \] where \(\text{Var}(m)\) means the variety of \(m\).

Reviewer: Corina Mohorianu (Iaşi)

### MSC:

13A02 | Graded rings |

13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |

13D45 | Local cohomology and commutative rings |

13E05 | Commutative Noetherian rings and modules |

13E15 | Commutative rings and modules of finite generation or presentation; number of generators |

### Keywords:

catenary local domain; finitely generated module; finite dimensional commutative Noetherian ring; occurrence diagram
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\textit{M. P. Brodmann} and \textit{R. Y. Sharp}, J. Pure Appl. Algebra 182, No. 2--3, 151--164 (2003; Zbl 1062.13001)

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### References:

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[2] | Brodmann, M. P.; Sharp, R. Y., On the dimension and multiplicity of local cohomology modules, Nagoya Math. J., 167, 217-233 (2002) · Zbl 1044.13007 |

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[10] | O. Zariski, P. Samuel, Commutative Algebra, Vol. I, Graduate Texts in Mathematics, Vol. 28, Springer, Berlin, 1975.; O. Zariski, P. Samuel, Commutative Algebra, Vol. I, Graduate Texts in Mathematics, Vol. 28, Springer, Berlin, 1975. · Zbl 0313.13001 |

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