t-Structures and cotilting modules over commutative noetherian rings

Abstract

For a commutative noetherian ring $R$, we establish a bijection between the resolving subcategories consisting of finitely generated $R$-modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category $\mathcal {D}(R)$ that contain $R[1]$ in their heart. Under this bijection, the t-structures $(\mathcal U,\mathcal V)$ such that the aisle $\mathcal U$ consists of objects with homology concentrated in degrees $<n$ correspond to the $n$-cotilting classes in ${{\mathrm{Mod}\text {-}R}}$. As a consequence of these results, we prove that the little finitistic dimension findim$R$ of $R$ equals an integer $n$ if and only if the direct sum $\bigoplus _{k=0}^n E_k(R)$ of the first $n+1$ terms in a minimal injective coresolution $0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots$ of $R$ is an injective cogenerator of ${{\mathrm{Mod}\text {-}R}}$.