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Register a new userDing modules and dimensions over formal triangular matrix rings
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Lixin Mao
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Let $T=biggl(begin{matrix} A&0 U&B end{matrix}biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $biggl(begin{matrix} M_1 M_2end{matrix}biggr)_{varphi^M}$ is Ding projective if and only if $M_1$ and $M_2/{rm im}(varphi^M)$ are Ding projective and the morphism $varphi^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $ker(widetilde{{varphi_{W}}})$ are Ding injective and the morphism $widetilde{{varphi_{W}}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.
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Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=left(begin{smallmatrix} A & 0 U & B end{smallmatrix}right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
Let $T=left( begin{array}{cc} R & M 0 & S end{array} right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cort{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $mathbb{D}^b(T{text-} Mod)$ restricts to a recollement of its subcategory $mathbb{D}^b(T{text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $underline{T{text-} GProj}$ and recollements of the Gorenstein defect category $mathbb{D}_{def}(T{text-} Mod)$.
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Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^circ$ in $R^A$.
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