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  1. ; ; ; (, International Mathematics Research Notices)
    Zeev Rudnick (Ed.)
    Abstract We introduce the ozone group of a noncommutative algebra $$A$$, defined as the group of automorphisms of $$A$$, which fix every element of its center. In order to initiate the study of ozone groups, we study polynomial identity (PI) skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial ring to be Gorenstein (resp. regular) in low dimension. 
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  2. ; ; (, Transactions of the American Mathematical Society)
    Dan Abramovich (Ed.)
    Let A A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A A -modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. 
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