Serre’s conditions and the finite type of classes of modules of bounded projective dimension

The class of modules of projective dimension at most $n$, denoted $\mathcal{P}_n$, is said to be of finite type when its right $\mathsf{Ext}$-orthogonal is exactly the right $\mathsf{Ext}$-orthogonal of the subclass of strongly finitely presented modules in $\mathcal{P}_n$ (recall that the strongly finitely presented modules are the modules with a projective resolution consisting of finitely generated modules). In particular, the finite type of $\mathcal{P}_n$ is equivalent to the right $\mathsf{Ext}$-orthogonal of $\mathcal{P}_n$ being an $n$-tilting class.

The classes $\mathcal{P}_n$ which are of finite type enjoy many additional properties with respect to those which are not, so a next aim is to characterise the rings over which $\mathcal{P}_n$ is of finite type for some $n$. In this talk, we plan to address this question for commutative noetherian rings, and relate this question to a classical criterion of Serre. Explicitly, over a commutative noetherian ring, the class $\mathcal{P}_n$ is of finite type if and only if Serre’s condition $(S_n)$ holds. Additionally, we will also consider the slightly weaker condition of when the class of modules of flat dimension at most $n$ coincides with the direct limit closure of the strongly finitely presented modules in $\mathcal{P}_n$ over commutative noetherian rings, or, in other words, when a ``higher’’ Govorov-Lazard Theorem holds over these rings.

This talk is based on joint work with Michal Hrbek.