Standardly stratified lower triangular K-algebras with enough idempotents

In this paper we study the lower triangular matrix $\mathbb{K}$-algebra $\mathrm{\Lambda }:=\begin{array}{cc}\hfill T\hfill & \hfill 0\hfill \\ \hfill M\hfill & \hfill U\hfill \end{array}]$, where U and T are basic $\mathbb{K}$-algebras with enough idempotents and M is an U-T-bimodule where $\mathbb{K}$ acts centrally. Moreover, we characterise in terms of U, T and M when, on one hand, the lower triangular matrix $\mathbb{K}$-algebra Λ is standardly stratified in the sense of 15]; and on the other hand, when Λ is locally bounded in the sense of Gabriel 10]. Finally, we also study several properties relating the projective dimensions in the categories of finitely generated modules $\mathrm{mod}(U)$, $\mathrm{mod}(T)$ and $\mathrm{mod}(\mathrm{\Lambda })$.