The small finitistic dimensions of commutative rings, III
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914377349726208 |
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| author | Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei |
| contents | The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04060 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The small finitistic dimensions of commutative rings, III Zhang, Xiaolei Commutative Algebra The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type. |
| title | The small finitistic dimensions of commutative rings, III |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2603.04060 |