Tilting and Silting Theory of Noetherian Algebras

Abstract

We develop silting theory of a Noetherian algebra $\Lambda $ over a commutative Noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $\Lambda $, and as a consequence, we see that mutation exists. As in the case of finite-dimensional algebras, functorially finite torsion classes of $\Lambda $ bijectively correspond to silting $\Lambda $-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $\Lambda $, and by using this theorem, we study torsion classes of the module category of $\Lambda $. When $R$ has Krull dimension one, we describe the set of torsion classes of $\Lambda $ explicitly by using the set of torsion classes of finite-dimensional algebras.