# 同调代数及广义逆学术报告

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M \\ 0 & B \\\end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Suppose that we are given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(\Phi(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}),\Psi(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary.If we are given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod respectively, then using the result above, we investigate when there exists a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\mathrm{Ho}(\mathcal{M}_{T})$ relative to $\mathrm{Ho}(\mathcal{M}_{A})$ and $\mathrm{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra. This talk is a report on joint work with R.M. Zhu and Y.Y. Peng.

For a ring $R$ and an additive subcategory $\mathcal{C}$ of the category ${\rm Mod}\;R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of ${\rm Mod}\;R$ and the left Gorenstein subcategory of ${\rm Mod}\;R^{op}$ relative to $\mathcal{C}$ form a coproduct-closed duality pair.Let $R,S$ be rings and $C$ a semidualizing $(R,S)$-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing,then $(\mathcal{GF}_C(R),\mathcal{GI}_C(R^{op}))$ is a coproduct-closed duality pair and $\mathcal{GF}_C(R)$ is covering in ${\rm Mod}\;R$, where $\mathcal{G}\mathcal{F}_C(R)$ is the subcategory of ${\rm Mod}\;R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}\mathcal{I}_C(R^{op})$ is the subcategory of ${\rm Mod}\;R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $(\mathcal{A}_C(R^{op}),l\mathcal{G}(\mathcal{F}_C(R)))$ is a coproduct-closed and product-closed duality pair and $\mathcal{A}_C(R^{op})$ is covering and preenveloping in ${\rm Mod}\;R^{op}$, where $\mathcal{A}_C(R^{op})$ is the Auslander class in ${\rm Mod}\;R^{op}$ and $l\mathcal{G}(\mathcal{F}_C(R))$ is the left Gorenstein subcategory of ${\rm Mod}\;R$ relative to $C$-flat modules.

In this talk，we talk about pseudo core inverses of morphisms in an additive category.  One is additive properties of pseudo core inverses. The other is perturbations for the pseudo core inverses.  Some results about core inverses of morphisms are generalized.