Relative tilting modules with respect to a semidualizing module. (English) Zbl 1513.13041

Salimi, Maryam

Czech. Math. J. 69, No. 3, 781-800 (2019).

Let \(R\) be a commutative Noetherian ring with identity. Recall that a finitely generated \(R\)-module \(C\) is called semidualizing if the homothety map \(\chi_C^R:R\rightarrow\text{Hom}_R\left(C,C\right)\) is an isomorphism, and \(\text{Ext}^i_R\left(C,C\right)=0\) for all \(i>0\). Also, recall that an \(R\)-module \(T\) is said to be generalized tilting if:

(i)

\(\text{pd}_RT<\infty\).

(ii)

\(\text{Ext}_R^i(T,T^{(k)})=0\) for every cardinal \(k\) and all \(i\geq 1\).

(iii)

there exists an exact sequence \(0\rightarrow R\rightarrow T_0\rightarrow T_1\rightarrow\cdots\rightarrow T_r\rightarrow 0,\) where each \(T_i\) is a direct summand of a direct sum of copies of \(T\).

Let \(C\) be a semidualizing module of \(R\). In this paper, the notion of \(C\)-tilting \(R\)-modules is introduced and studied. The author reveals some connections between \(C\)-tilting \(R\)-modules and (generalized) tilting \(R\)-modules. Also, she extends some properties of (generalized) tilting \(R\)-modules to \(C\)-tilting \(R\)-modules. Her definition of \(C\)-tilting \(R\)-modules and her main results require the following definitions:
It is known that every \(R\)-module \(M\) admits a complex \[L^+=\cdots\rightarrow C\otimes_R P_n\rightarrow\cdots\rightarrow C\otimes_R P_0\rightarrow M\rightarrow 0\] such that every \(P_i\) is projective, and the complex \[\text{Hom}_R(C,L^+)=\cdots\rightarrow P_n\rightarrow\cdots\rightarrow P_0\rightarrow\text{Hom}_R(C,M)\rightarrow 0\] is exact. The complex \[L=\cdots\rightarrow C\otimes_R P_n\rightarrow\cdots\rightarrow C\otimes_R P_0\rightarrow 0\] is called a proper \(\mathscr{P}_C\)-projective resolution of \(M\). Let \(M\) and \(N\) be two \(R\)-modules. For each non-negative integer \(n\), the \(n\)th relative Ext module of \(M\) and \(N\) is defined by \(\text{Ext}^n_{\mathscr{P}_C}(M,N):=\text{H}_{-n}(\text{Hom}_R(L,N))\), where \(L\) is a proper \(\mathscr{P}_C\)-projective resolution of \(M\).
According to the author, an \(R\)-module \(T\) is called \(C\)-tilting if:

(i)

\(\text{Ext}^n_{\mathscr{P}_C}(T,-)=0\) for \(n\gg 0\).

(ii)

\(\text{Ext}_{\mathscr{P}_C}^i(T,T^{(k)})=0\) for every cardinal \(k\) and all \(i\geq 1\).

(iii)

there exists an exact sequence \(0\rightarrow C\rightarrow T_0\rightarrow T_1\rightarrow\cdots\rightarrow T_r\rightarrow 0\), where each \(T_i\) is a direct summand of a direct sum of copies of \(T\).

Let \(\mathcal{T}_C\left(R\right)\) stands for the full subcategory of \(C\)-tilting \(R\)-modules. It is evident that \(R\) is a semidualizing module of \(R\) and \(\mathcal{T}_R\left(R\right)\) is the class of all generalized tilting \(R\)-modules.
The Auslander class \(\mathscr{A}_C\left(R\right)\) is the class of all \(R\)-modules \(M\) for which the natural map \[\gamma_M^C:M\rightarrow\text{Hom}_R\left(C,C\otimes_RM\right)\] is an isomorphism, and \[\text{Tor}^R_i\left(C,M\right)=0=\text{Ext}_R^i\left(C,C\otimes_RM\right)\] for all \(i\geq 1\). Also, the Bass class \(\mathscr{B}_C\left(R\right)\) is the class of all \(R\)-modules \(M\) for which the evaluation map \[\xi_M^C:C\otimes_R\text{Hom}_R\left(C,M\right)\rightarrow M\] is an isomorphism, and \[\text{Ext}^i_R\left(C,M\right)=0=\text{Tor}^R_i\left(C,\text{Hom}_R\left(C,M\right)\right)\] for all \(i\geq 1\).
The author shows that \(\mathcal{T}_R\left(R\right)\subseteq\mathscr{A}_C\left(R\right)\), and if \(T\) is a generalized tilting \(R\)-module, then \(C\otimes_R T\) is \(C\)-tilting. Also, she proves that \(\mathcal{T}_C\left(R\right)\subseteq\mathscr{B}_C\left(R\right)\) and if \(T\) is a \(C\)-tilting \(R\)-module, then \(\text{Hom}_R(C,T)\) is a generalized tilting \(R\)-module.

Reviewer: Kamran Divaani-Aazar (Tehran)

MSC:

13D05	Homological dimension and commutative rings
13D45	Local cohomology and commutative rings

Keywords:

\(C\)-projective module; semidualizing module; tilting module

References:

[1]	Avramov, L. L.; Foxby, H.-B., Ring homomorphisms and finite Gorenstein dimension, Proc. Lond. Math. Soc. III. 75 (1997), 241-270 · Zbl 0901.13011
[2]	Bazzoni, S.; Mantese, F.; Tonolo, A., Derived equivalence induced by infinitely generated {\(n\)}-tilting modules, Proc. Am. Math. Soc. 139 (2011), 4225-4234 · Zbl 1232.16004
[3]	Bongratz, K., Tilted algebras, Representations of Algebras. Proc. 3rd Int. Conf., Puebla, 1980 Lect. Notes Math. 903, Springer, Berlin (1981), 26-38 · Zbl 0478.16025
[4]	Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc. 353 (2001), 1839-1883 · Zbl 0969.13006
[5]	Enochs, E. E.; Jenda, O. M. G., Relative Homological Algebra, Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2011) · Zbl 1238.13001
[6]	Foxby, H.-B., Gorenstein modules and related modules, Math. Scand. 31 (1973), 267-284 · Zbl 0272.13009
[7]	Golod, E. S., \(G\)-dimension and generalized perfect ideals, Tr. Mat. Inst. Steklova 165 Russian (1984), 62-66 · Zbl 0577.13008
[8]	Happel, D.; Ringel, C. M., Tilted algebras, Trans. Am. Math. Soc. 274 (1982), 399-443 · Zbl 0503.16024
[9]	Holm, H.; Jø{r}gensen, P., Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), 423-445 · Zbl 1094.13021
[10]	Holm, H.; White, D., Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47 (2007), 781-808 · Zbl 1154.16007
[11]	Miyashita, Y., Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113-146 · Zbl 0578.16015
[12]	Salimi, M., On relative Gorenstein homological dimensions with respect to a dualizing module, Mat. Vesnik 69 (2017), 118-125 · Zbl 1474.13031
[13]	Salimi, M.; Sather-Wagstaff, S.; Tavasoli, E.; Yassemi, S., Relative Tor functors with respect to a semidualizing module, Algebr. Represent. Theory 17 (2014), 103-120 · Zbl 1295.13023
[14]	Salimi, M.; Tavasoli, E.; Yassemi, S., Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module, Arch. Math. 98 (2012), 299-305 · Zbl 1246.13021
[15]	Sather-Wagstaff, S., Semidualizing Modules, Available at https://ssather.people.clemson.edu/DOCS/sdm.pdf · Zbl 1282.13021
[16]	Sather-Wagstaff, S.; Sharif, T.; White, D., Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010), 571-600 · Zbl 1190.13007
[17]	Takahashi, R.; White, D., Homological aspects of semidualizing modules, Math. Scand. 106 (2010), 5-22 · Zbl 1193.13012
[18]	Tang, X., New characterizations of dualizing modules, Commun. Algebra 40 (2012), 845-861 · Zbl 1246.13022
[19]	Vasconcelos, W. V., Divisor Theory in Module Categories, North-Holland Mathematics Studies 14, Elsevier, Amsterdam (1974) · Zbl 0296.13005
[20]	White, D., Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra 2 (2010), 111-137 · Zbl 1237.13029