The derived dimensions and syzygy finite type

TL;DR

This paper establishes an upper bound for the derived dimension of an artin algebra based on syzygy type and projective dimension, generalizing a known result relating derived and global dimensions.

Contribution

It introduces a new bound for the derived dimension of artin algebras using syzygy type and simple modules, extending previous results that linked derived and global dimensions.

Findings

Derived dimension is bounded by syzygy type and projective dimension.

If global dimension is finite, the bound simplifies to a known inequality.

Generalizes the relation between derived and global dimensions for artin algebras.

Abstract

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 $\Lambda$.