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Gorenstein homological dimensions for extriangulated categories

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Pith's one-line read For an extriangulated category with a proper class of triangles, the supremum of xi-Gorenstein projective dimensions equals the supremum of xi-Gorenstein injective dimensions.

arxiv 1908.00931 v1 pith:XZXZHDJZ submitted 2019-08-01 math.RT math.CT

Jiangsheng Hu , Dongdong Zhang , Panyue Zhou This is my paper · ORCID
classification math.RTmath.CT
keywords mathcaltextrmrespextriangulatedprojectivecategoriesdimensiondimensions
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The reading

Homological algebra measures how far objects are from being built out of simple building blocks. Gorenstein dimensions measure how far an object is from having a special kind of resolution. This paper works in extriangulated categories, a common setting introduced by Nakaoka and Palu that contains exact categories and triangulated categories as special cases. The authors use a chosen class xi of distinguished triangles and define relative Gorenstein projective and injective dimensions based on it.
Extended reading notes

Core claim

Theorem 4.7 states that for an extriangulated category C with P(xi) generating and I(xi) cogenerating, and under Condition (star), the conditions (1) sup xi-GpdM <= m for all M, (2) sup xi-GidM <= m for all M, and (3) xi-spliC = xi-silpC <= m are equivalent. Corollary 4.8 then yields the equality sup xi-GpdM = sup xi-GidM. If true, this generalizes the Bennis-Mahdou result for rings and the Ren-Liu result for triangulated categories.

Load-bearing premise

Condition (star), stated before Theorem 4.7, requires that whenever higher xixt groups vanish for an object M of finite xi-projective dimension, the natural map C(M,N) to xixt^0(M,N) is an isomorphism, together with a dual statement. This condition is load-bearing because it converts the diagram arguments in Theorem 4.7 into actual xi-Ginjectivity. The paper asserts Condition (star) for exact categories with only 'one can check' and sketches it for triangulated categories, and it offers no non-exact, non-triangulated example where the condition is verified.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no numerical or fitted parameters because this is a pure mathematics paper. The central claim depends on the framework axioms of extriangulated categories, properness of xi, enough projectives and injectives, generating and cogenerating subcategories, Condition (star), and the companion-paper results [5]. No new entities are postulated.

assumptions (6)
  • domain assumption Extriangulated category axioms for (C, E, s), including the compatibility conditions from [6, Definition 2.12].
    The paper operates entirely in Nakaoka-Palu extriangulated categories; without these axioms the whole framework is undefined.
  • domain assumption xi is a proper class of E-triangles: closed under base change and cobase change, saturated, closed under finite coproducts, and containing split triangles.
    This is assumed throughout after Definition 2.3 and is needed for closure in Lemma 3.6 and for the resolution arguments.
  • domain assumption C has enough xi-projectives and enough xi-injectives and satisfies weak idempotent completeness.
    Stated at the start of Section 3; needed for existence of resolutions, comparison theorems, and well-definedness of xixt.
  • domain assumption P(xi) is a generating subcategory and I(xi) is a cogenerating subcategory of C.
    Hypotheses of Theorem 4.7 and Corollary 4.8; used in Lemma 4.4 to turn finiteness of one Gorenstein dimension into finiteness of silp and spli.
  • ad hoc to paper Condition (star): vanishing of higher xixt groups from or to a finite-dimension object implies the natural map to xixt^0 is an isomorphism.
    Introduced in Theorem 4.7, proved only for exact and triangulated categories in Example 4.10. Without it, the converses in Theorem 4.7 are not established.
  • domain assumption Validity of the lemmas and propositions quoted from the authors' companion paper [5].
    Definitions 2.3 through 2.9, the Horseshoe Lemma 3.3, Lemma 3.7, and Proposition 5.6 are imported from [5]; if any of those results have errors, the present proofs would collapse.

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Pith. "Pith review of Gorenstein homological dimensions for extriangulated categories." pith.science (2026). https://pith.science/paper/XZXZHDJZ

@misc{pith2026190800931,
  author       = {Pith},
  title        = {Pith review of: Gorenstein homological dimensions for extriangulated categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZXZHDJZ}},
  note         = {Machine review of arXiv:1908.00931}
}
abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. The authors introduced and studied $\xi$-$\mathcal{G}$projective and $\xi$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $\xi$-$\mathcal{G}$projective dimension by using derived functors on $\mathcal{C}$. Second, let $\mathcal{P}(\xi)$ (resp. $\mathcal{I}(\xi)$) be a generating (resp. cogenerating) subcategory of $\mathcal{C}$. We show that the following equality holds under some assumptions: $$\sup\{\xi\textrm{-}\mathcal{G}{\rm pd}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\}=\sup\{\xi\textrm{-}\mathcal{G}{\rm id}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\},$$ where $\xi\textrm{-}\mathcal{G}{\rm pd}M$ (resp. $\xi\textrm{-}\mathcal{G}{\rm id}M$) denotes $\xi$-$\mathcal{G}$projective (resp. $\xi$-$\mathcal{G}$injective) dimension of $M$. As an application, our main results generalize their work by Bennis-Mahdou and Ren-Liu. Moreover, our proof is not far from the usual module or triangulated case.

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Works this paper leans on

25 extracted references · 23 canonical work pages

  1. [1]

    Asadollahi and S

    J. Asadollahi and S. Salarian, Gorenstein objects in triangulated categories, J. Algebra 281 (2004) 264-286

    work page 2004

  2. [2]

    Asadollahi and S

    J. Asadollahi and S. Salarian, Tate cohomology and Gorensteinness for triangulated categories, J. Algebra 299 (2006) 480-502

    work page 2006

  3. [3]

    Auslander and M

    M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc., Vol. 94, 1969

    work page 1969

  4. [4]

    L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85(3) (2002) 393-440

    work page 2002

  5. [5]

    Beligiannis, Relative homological algebra and purity in triangulated categories, J

    A. Beligiannis, Relative homological algebra and purity in triangulated categories, J. Algebra 227(1) (2000) 268-361

    work page 2000

  6. [6]

    Bennis and N

    D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138(2) (2010) 461-465

    work page 2010

  7. [7]

    B\" u hler, Exact categories, Expo

    T. B\" u hler, Exact categories, Expo. Math. 28 (2010) 1-69

    work page 2010

  8. [8]

    Celikbas, L

    O. Celikbas, L. W. Christensen, L. Liang and G. Piepmeyer, Stable homology over associative rings, Trans. Amer. Math. Soc. 369 (2017) 8061-8086

    work page 2017

Show all 25 references
  1. [9]

    E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projectivve modules, Math. Z. 220 (1995) 611-633

    1995

  2. [10]

    E. E. Enochs and O. M. G. Jenda, Relative

  3. [11]

    Gillespie, Model structures on exact categories, J

    J. Gillespie, Model structures on exact categories, J. Pure Appl. Algebra 215 (2011) 2892-2902

    2011

  4. [12]

    Holm, Gorenstein homological dimensions, J

    H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167-193

    2004

  5. [13]

    Hovey, Cotorsion pairs, model category structures, and representation theory ,

    M. Hovey, Cotorsion pairs, model category structures, and representation theory ,

  6. [14]

    Hovey, Cotorsion pairs and model categories ,

    M. Hovey, Cotorsion pairs and model categories ,

  7. [15]

    T. V. Gedrich, K. W. Gruenberg, Complete cohomological functors on groups. Topol. Appl. 25 (1987) 203-223

    1987

  8. [16]

    J. S. Hu, D. D. Zhang, P. Y. Zhou, Proper classes and Gorensteinness in extriangulated categories, arXiv:1906.10989, 2019

    1906 arXiv

  9. [17]

    Liu and H

    Y. Liu and H. Nakaoka. Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra 528 (2019) 96-149

    2019

  10. [18]

    Krause, Smashing subcategories and the telescope conjecture: An algebraic approach,

    H. Krause, Smashing subcategories and the telescope conjecture: An algebraic approach,

  11. [19]

    Nakaoka and Y

    H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cahiers de Topologie et Geometrie Differentielle Categoriques, Volume LX-2 (2019) 117-193

    2019

  12. [20]

    Ren and Z

    W. Ren and Z. K. Liu, Gorenstein homological dimensions for triangulated categories, J. Algebra 410 (2014) 258-276

    2014

  13. [21]

    Ren and Z

    W. Ren and Z. K. Liu, Balance of Tate cohomology in triangulated categories, Appl. Categ. Struct. 23(6) (2015) 819-828

    2015

  14. [22]

    W. Ren, R. Y. Zhao and Z. K. Liu, Cohomology theoreies in triangulated categories, Acta Math. Sin. Engl. Ser. 32(11) (2016) 1377-1390

    2016

  15. [23]

    J. L. Verdier, Cat\' e gories d\' e riv\' e es: \' e tat 0, in: SGA 4 1 2 , in: Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 262-311

    1977

  16. [24]

    X. Y. Yang, Model structures on triangulated categories. Glasg. Math. J. 57(2) (2015) 263-284

    2015

  17. [25]

    P. Y. Zhou and B. Zhu, Triangulated quotient categories revisited, J. Algebra 502 (2018) 196-232

    2018

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