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SUMMARY:On a conjecture of Auslander and Reiten\, II [MPIM]
DTSTART:20250521T120000Z
DTEND:20250521T133000Z
DTSTAMP:20260416T081600Z
UID:indico-event-417@math-events.uni-bonn.de
DESCRIPTION:Speakers: Olgur Celikbas (West Virginia University/MPIM)\n\nAl
 gebra Seminar\nThere are many conjectures from the representation theory o
 f finite-dimensional algebras that have been transplanted into commutative
  algebra\, and this process has significantly enriched both fields. A nota
 ble example is the Auslander–Reiten Conjecture\, which asserts that a fi
 nitely generated module M over a finite-dimensional algebra A is proje
 ctive if the Ext groups $Ext^n(M\,M)$ and $Ext^n(M\,A)$ vanish for all pos
 itive integers n. This long-standing conjecture is closely connected to o
 ther important conjectures in representation theory\, including the Finiti
 stic Dimension Conjecture and the Tachikawa Conjecture.\nAlthough the Ausl
 ander–Reiten Conjecture originates in the representation theory of algeb
 ras\, it has attracted considerable interest within commutative algebra. I
 n 1994\, Huneke and Wiegand formulated a conjecture about tensor products 
 of torsion-free modules over one-dimensional commutative Noetherian integr
 al domains. This conjecture\, which remains unresolved\, implies the Ausla
 nder–Reiten Conjecture for a broad class of commutative rings.\nAuslande
 r defined a condition\, denoted by (AC)\, to analyze the Finitistic Dimens
 ion Conjecture\, and conjectured that every finite-dimensional algebra sat
 isfies (AC). This conjecture turned out to be false\; interestingly\, the 
 first counterexamples were obtained only after the (AC) conjecture had bee
 n transplanted into commutative algebra. Nonetheless\, several classes of 
 rings do satisfy Auslander's condition (AC)\, and remarkable homological p
 roperties of such classes have been uncovered. For example\, Christensen a
 nd Holm proved that the Auslander–Reiten Conjecture holds over Noetheria
 n rings satisfying (AC).\nIn this talk\, I will survey some of the literat
 ure on these problems and explore the relationship between the Huneke–Wi
 egand Conjecture\, the Auslander–Reiten Conjecture\, and related problem
 s on the vanishing of Tor.\n \n\nhttps://math-events.uni-bonn.de/event/41
 7/
LOCATION:MPIM\, Vivatsgasse\,  7 - Seminar Room (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/417/
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