Triangular Matrix Categories over path Categories and Quasi-hereditary Categories, as well as one point extensions by Projectives

Abstract

In this paper, we prove that the lower triangular matrix category $\Lambda =\left[\begin{array}{cc}\mathcal{T}& 0\\ M& \mathcal{U}\end{array}\right]$, where $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary $\mathrm{Hom}$-finite Krull-Schmidt $K$-categories and $M$ is a $\mathcal{U}{\otimes }_K{\mathcal{T}}^{op}$-module that satisfies suitable conditions, is quasi-hereditary in the sense of \cite{LGOS1} and \cite{Martin}. Moreover, we solve the problem of finding quotients of path categories isomorphic to the lower triangular matrix category $\Lambda$, where $\mathcal{T}=K\mathcal{R}/\mathcal{J}$ and $\mathcal{U}=K\mathcal{Q}/\mathcal{I}$ are path categories of infinity quivers modulo admissible ideals. Finally, we study the case where $\Lambda$ is a path category of a quiver $Q$ with relations and $\mathcal{U}$ is the full additive subcategory of $\Lambda$ obtained by deleting a source vertex $\ast$ in $Q$ and $\mathcal{T}=\mathrm{add}\{\ast \}$. We then show that there exists an adjoint pair of functors $(\mathcal{R},\mathcal{E})$ between the functor categories $\mathrm{mod}\Lambda$ and $\mathrm{mod}\mathcal{U}$ that preserve orthogonality and exceptionality; see \cite{Assem1}. We then give some examples of how to extend classical tilting subcategories of $\mathcal{U}$-modules to classical tilting subcategories of $\Lambda$-modules.

References in corpus (2)

• Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extension
• Standardly stratified lower triangular $\mathbb{K}$-algebras with enough idempotents