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A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxbys result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $mathcal{F}_C$-projective dimension, and finite $mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
We introduce and study a class of objects that encompasses Christensen and Foxbys semidualizing modules and complexes and Kubiks quasi-dualizing modules: the class of $mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equiva
It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain
A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $Rto Tgets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the emph{connected sum} $R#_TS$ is defined to be the l
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t
In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and if so, wha