The derived dimensions and syzygy finite type

Let $\Lambda$ be an artin algebra, and $\mathcal{V}$ a subset of all simple modules in $\mod\Lambda$. Suppose that $\Lambda/\rad \Lambda$ has finite syzygy type, then the derived dimension of $\Lambda$ is at most $\ell\ell^{t_{\mathcal{V}}}(\Lambda_{\Lambda})+\pd\mathcal{V}.$ In particular, if the global dimension of $\Lambda$ is finite, then the derived dimension of $\Lambda$ is at most $\ell\ell^{t_{\mathcal{V}}}(\Lambda_{\Lambda})+\pd\mathcal{V}.$ This generalized the famous result which state that the derived dimension of $\Lambda$ is less than or equal to the global dimension of $\Lambda$.