Strongly $\mathcal{VW}$-Gorenstein $N$-complexes

Year 2024, Early Access, 1 - 20

Wenjun Guo Honghui Jia Renyu Zhao

https://doi.org/10.24330/ieja.1559027

Abstract

Let $\mathcal{V},\mathcal{W}$ be two classes of $R$-modules. The notion of strongly $\mathcal{VW}$-Gorenstein $N$-complexes is introduced, and under certain mild hypotheses on $\mathcal{V}$ and $\mathcal{W}$, it is shown that an $N$-complex $X$ is strongly $\mathcal{VW}$-Gorenstein if and only if each term of $X$ is a $\mathcal{VW}$-Gorenstein module and $N$-complexes ${\rm Hom}_{R}(V,X)$ and $ Hom_{R}(X,W)$ are $N$-exact for any $V\in\mathcal{V}$ and $W\in\mathcal{W}$. Furthermore, under the same conditions on $\mathcal{V}$ and $\mathcal{W}$, it is proved that an $N$-exact $N$-complex $X$ is $\mathcal{VW}$-Gorenstein if and only if $\mathrm{Z}_{n}^{t}(X)$ is a $\mathcal{VW}$-Gorenstein module for each $n\in\mathbb{Z}$ and each $t=1,2,\ldots,N-1$. Consequently, we show that an $N$-complex $X$ is strongly Gorenstein projective (resp., injective) if and only if $X$ is $N$-exact and ${\rm Z}^t_{n}(X)$ is a Gorenstein projective (resp., injective) module for each $n\in\mathbb{Z}$ and $t=1,2,\ldots,N-1$.

Keywords

$\mathcal{V}\mathcal{W}$-Gorenstein module, strongly $\mathcal{VW}$-Gorenstein $N$-complex, strongly Gorenstein projective $N$-complex, strongly Gorenstein injective $N$-complex

References

• P. Bahiraei, Cotorsion pairs and adjoint functors in the homotopy category of $N$-complexes, J. Algebra Appl., 19(12) (2020), 2050236 (19 pp).
• M. Dubois-Violette, $d^N=0$: generalized homology, $K$-Theory, 14(4) (1998), 371-404.
• E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220(4) (1995), 611-633.
• S. Estrada, Monomial algebras over infinite quivers. Applications to $N$-complexes of modules, Comm. Algebra, 35(10) (2007), 3214-3225.
• Y. Geng and N. Ding, $\mathcal{W}$-Gorenstein modules, J. Algebra, 325 (2011), 132-146.
• J. Gillespie, The homotopy category of $N$-complexes is a homotopy category, J. Homotopy Relat. Struct., 10(1) (2015), 93-106.
• H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
• H. Holm and P. Jorgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205(2) (2006), 423-445.
• H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47(4) (2007), 781-808.
• O. Iyama, K. Kato and J.-i. Miyachi, Derived categories of $N$-complexes, J. Lond. Math. Soc. (2), 96(3) (2017), 687-716.
• M. M. Kapranov, On the $q$-analog of homological algebra, available at arXiv:q-alg/9611005, (1996).
• L. Liang, N. Q. Ding and G. Yang, Some remarks on projective generators and injective cogenerators, Acta Math. Sin. (Engl. Ser.), 30(12) (2014), 2063-2078.
• B. Lu, Strongly $\mathcal W$-Gorenstein complexes, J. Math. Res. Appl., 38(5) (2018), 449-457.
• B. Lu, Cartan-Eilenberg Gorenstein projective $N$-complexes, Comm. Algebra, 49(9) (2021), 3810-3824.
• B. Lu. Gorenstein objects in the category of $N$-complexes, J. Algebra Appl., 20(10) (2021), 2150174 (26 pp).
• B. Lu and A. Daiqing, Cartan-Eilenberg Gorenstein-injective $m$-complexes, AIMS Math., 6(5) (2021), 4306-4318.
• B. Lu and Z. Di, Gorenstein cohomology of $N$-complexes, J. Algebra Appl., 19(9) (2020), 2050174 (14 pp).
• B. Lu, Z. Di and Y. Liu, Cartan-Eilenberg $N$-complexes with respect to self-orthogonal subcategories, Front. Math. China, 15(2) (2020), 351-365.
• B. Lu and Z. Liu, A note on Gorenstein projective complexes, Turkish J. Math., 40(2) (2016), 235-243.
• B. Lu, J. Wei and Z. Di, $\mathcal{W}$-Gorenstein $N$-complexes, Rocky Mountain J. Math., 49(6) (2019), 1973-1992.
• P. Ma and X. Yang, Cohomology of torsion and completion of $N$-complexes, J. Korean Math. Soc., 59(2) (2022), 379-405.
• W. Mayer, A new homology theory. {I}, {II}, Ann. of Math. (2), 43 (1942), 370-380, 594-605.
• S. Sather-Wagstaff, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2), 77(2) (2008), 481-502.
• D. White, Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra, 2(1) (2010), 111-137.
• D. Xin, J. Chen and X. Zhang, Completely $\mathscr W$-resolved complexes, Comm. Algebra, 41(3) (2013), 1094-1106.
• G. Yang, Gorenstein projective, injective and flat complexes, Acta Math. Sinica (Chinese Ser.), 54(3) (2011), 451-460.
• C. Yang and L. Liang, Gorenstein injective and projective complexes with respect to a semidualizing module, Comm. Algebra, 40(9) (2012), 3352-3364.
• X. Yang and N. Ding, The homotopy category and derived category of $N$-complexes, J. Algebra, 426 (2015), 430-476.
• X. Yang and T. Cao, Cotorsion pairs in $\mathcal C_N(\mathcal A)$, Algebra Colloq., 24(4) (2017), 577-602.
• X. Yang and Z. Liu, Gorenstein projective, injective, and flat complexes, Comm. Algebra, 39(5) (2011), 1705-1721.
• G. Zhao and J. Sun, $\mathcal{VW}$-Gorenstein categories, Turkish J. Math., 40(2) (2016), 365-375.
• R. Zhao and W. Ren, $\mathcal{VW}$-Gorenstein complexes, Turkish J. Math., 41(3) (2017), 537-547.