Gorenstein and duality pair over triangular matrix rings
H Liu, R Zhu - arXiv preprint arXiv:2202.13148, 2022 - arxiv.org
H Liu, R Zhu
arXiv preprint arXiv:2202.13148, 2022•arxiv.orgLet $ A $, $ B $ be two rings and $ T=\left (\begin {smallmatrix} A & M\\0 & B\\\end
{smallmatrix}\right) $ with $ M $ an $ A $-$ B $-bimodule. We first construct a semi-complete
duality pair $\mathcal {D} _ {T} $ of $ T $-modules using duality pairs in $ A $-Mod and $ B $-
Mod respectively. Then we characterize when a left $ T $-module is Gorenstein $ D_ {T} $-
projective, Gorenstein $ D_ {T} $-injective or Gorenstein $ D_ {T} $-flat. These three class of
$ T $-modules will induce model structures on $ T $-Mod. Finally we show that the homotopy …
{smallmatrix}\right) $ with $ M $ an $ A $-$ B $-bimodule. We first construct a semi-complete
duality pair $\mathcal {D} _ {T} $ of $ T $-modules using duality pairs in $ A $-Mod and $ B $-
Mod respectively. Then we characterize when a left $ T $-module is Gorenstein $ D_ {T} $-
projective, Gorenstein $ D_ {T} $-injective or Gorenstein $ D_ {T} $-flat. These three class of
$ T $-modules will induce model structures on $ T $-Mod. Finally we show that the homotopy …
Let , be two rings and $T=\left(\begin{smallmatrix} A & M \\ 0 & B \\\end{smallmatrix}\right)$ with an --bimodule. We first construct a semi-complete duality pair of -modules using duality pairs in -Mod and -Mod respectively. Then we characterize when a left -module is Gorenstein -projective, Gorenstein -injective or Gorenstein -flat. These three class of -modules will induce model structures on -Mod. Finally we show that the homotopy category of each of model structures above admits a recollement relative to corresponding stable categories. Our results give new characterizations to earlier results in this direction.
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