### Abstract

We study h-vectors of simplicial complexes which satisfy Serre's condition (S_{r}). Let r be a positive integer. We say that a simplicial complex △ satisfies Serre's condition (S_{r}) if H̃ _{i}(lk_{△} (F);K) = 0 for all F ∈ △ and for all i < min{r-1, dim lk_{△} (F)}, where lk_{△} (F) is the link of △ with respect to F and where H̃_{i}(△;K) is the reduced homology groups of △ over a field K. The main result of this paper is that if △ satisfies Serre's condition (S_{r}) then (i) h_{k}(△) is non-negative for k = 0, 1, . . ., r and (ii) ∑_{k}≥r h_{k}(△) is non-negative.

Original language | English |
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Pages (from-to) | 1015-1028 |

Number of pages | 14 |

Journal | Mathematical Research Letters |

Volume | 16 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Graded Betti numbers
- H-vectors
- Serre's conditions
- Stanley-Reisner rings

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*16*(6), 1015-1028. https://doi.org/10.4310/MRL.2009.v16.n6.a10

**H-vectors of simplicial complexes with Serre's conditions.** / Murai, Satoshi; Terai, Naoki.

Research output: Contribution to journal › Article

*Mathematical Research Letters*, vol. 16, no. 6, pp. 1015-1028. https://doi.org/10.4310/MRL.2009.v16.n6.a10

}

TY - JOUR

T1 - H-vectors of simplicial complexes with Serre's conditions

AU - Murai, Satoshi

AU - Terai, Naoki

PY - 2009/1/1

Y1 - 2009/1/1

N2 - We study h-vectors of simplicial complexes which satisfy Serre's condition (Sr). Let r be a positive integer. We say that a simplicial complex △ satisfies Serre's condition (Sr) if H̃ i(lk△ (F);K) = 0 for all F ∈ △ and for all i < min{r-1, dim lk△ (F)}, where lk△ (F) is the link of △ with respect to F and where H̃i(△;K) is the reduced homology groups of △ over a field K. The main result of this paper is that if △ satisfies Serre's condition (Sr) then (i) hk(△) is non-negative for k = 0, 1, . . ., r and (ii) ∑k≥r hk(△) is non-negative.

AB - We study h-vectors of simplicial complexes which satisfy Serre's condition (Sr). Let r be a positive integer. We say that a simplicial complex △ satisfies Serre's condition (Sr) if H̃ i(lk△ (F);K) = 0 for all F ∈ △ and for all i < min{r-1, dim lk△ (F)}, where lk△ (F) is the link of △ with respect to F and where H̃i(△;K) is the reduced homology groups of △ over a field K. The main result of this paper is that if △ satisfies Serre's condition (Sr) then (i) hk(△) is non-negative for k = 0, 1, . . ., r and (ii) ∑k≥r hk(△) is non-negative.

KW - Graded Betti numbers

KW - H-vectors

KW - Serre's conditions

KW - Stanley-Reisner rings

UR - http://www.scopus.com/inward/record.url?scp=73849089456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73849089456&partnerID=8YFLogxK

U2 - 10.4310/MRL.2009.v16.n6.a10

DO - 10.4310/MRL.2009.v16.n6.a10

M3 - Article

AN - SCOPUS:73849089456

VL - 16

SP - 1015

EP - 1028

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 6

ER -