Olgur Celikbas

Auslander–Reiten conjecture and Auslander–Reiten duality

Abstract Motivated by a result of Araya, we extend the Auslander–Reiten duality theorem to Cohen–Macaulay local rings. We also study the Auslander–Reiten conjecture, which is rooted in Nakayamaʼs work on finite dimensional algebras. One of our results detects a certain condition that forces the conjecture to hold over local rings of positive depth.

Vanishing of Tor, and why we care about it

Abstract Given finitely generated modules M and N over a local ring R , the tensor product M ⊗ R N typically has nonzero torsion. Indeed, the assumption that the tensor product is torsion-free influences the structure and vanishing of the modules Tor i R ( M , N ) for all i ≥ 1 . In turn, the vanishing of Tor i R ( M , N ) imposes restrictions on the depth properties of the modules M and N . These connections made their first appearance in Auslanders 1961 paper “Modules over unramified regular local rings”. We will survey the literature on these topics, with emphasis on progress during the past twenty years.

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let

Homological dimensions of rigid modules

R

Complete homology over associative rings

R be a local complete intersection ring of codimension

Bounds on depth of tensor products of modules

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