Quasi-projective dimension

Gheibi, Mohsen; Jorgensen, David A.; Takahashi, Ryo

doi:10.2140/pjm.2021.312.113

Cited by 6 publications

(11 citation statements)

References 25 publications

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“…For all equipresented rings, and more generally all rings of large enough cohomological support, we also answer a question of Gheibi, Jorgensen, and Takahashi [24, Question 3.9]. This question proposes yet another characterization of complete intersections: that R is a local complete intersection if and only if every finitely generated R -module has finite quasi-projective dimension (see the end of Section 5 and [24] for a definition and other details).…”

Section: Introductionmentioning

confidence: 77%

“…In [24, Corollary 3.8], it is shown that if R is complete intersection, then every finitely generated R -module has finite quasi-projective dimension. Furthermore, every module of finite quasi-projective dimension is proxy small (see [24, Proposition 3.11]); however, finite quasi-projective dimension is not equivalent to a module being proxy small, as shown in [24, Example 4.9]. Regardless, Theorem 4.8 answers Question 4.15 in the affirmative in the following setting.…”

Section: Resultsmentioning

confidence: 99%

“…We now discuss a connection with a definition introduced and investigated by Gheibi, Jorgensen, and Takahashi in [24]. Definition 4.14.…”

Section: By [36 Theorem 522] If R Is Not a Complete Intersection And Dimmentioning

confidence: 99%

“…This should be understood as a weakening of the small property. Since its introduction, the proxy small property has been studied by various authors [14], [18], [21], [22], [24], [26], [32], [35], [38].…”

Section: §1 Introductionmentioning

confidence: 99%

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Constructing Nonproxy Small Test Modules for the Complete Intersection Property

Briggs¹,

Grifo²,

Pollitz³

2021

Nagoya Math. J.

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A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.

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Section: Introductionmentioning

confidence: 77%

Section: Resultsmentioning

confidence: 99%

“…We now discuss a connection with a definition introduced and investigated by Gheibi, Jorgensen, and Takahashi in [24]. Definition 4.14.…”

Section: By [36 Theorem 522] If R Is Not a Complete Intersection And Dimmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Constructing Nonproxy Small Test Modules for the Complete Intersection Property

Briggs¹,

Grifo²,

Pollitz³

2021

Nagoya Math. J.

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confidence: 99%

Quasi-injective dimension

Gheibi¹

2022

Preprint

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Following our previous work about quasi-projective dimension [9], in this paper we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective dimension and Gorenstein injective dimension in the context of quasi-injective dimension such as the followings. (a) If quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension with maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a non zero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.

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