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SUMMARY:On a conjecture of Auslander and Reiten\, I [MPIM]
DTSTART:20250520T120000Z
DTEND:20250520T133000Z
DTSTAMP:20260415T105400Z
UID:indico-event-416@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
  of finite-dimensional algebras that have been transplanted into commutati
 ve algebra\, and this process has significantly enriched both fields. A no
 table example is the Auslander–Reiten Conjecture\, which asserts that a 
 finitely generated module M over a finite-dimensional algebra A is pro
 jective if the Ext groups $Ext^n(M\,M)$ and $Ext^n(M\,A)$ vanish for all p
 ositive integers n. This long-standing conjecture is closely connected to
  other important conjectures in representation theory\, including the Fini
 tistic Dimension Conjecture and the Tachikawa Conjecture.\nAlthough the Au
 slander–Reiten Conjecture originates in the representation theory of alg
 ebras\, it has attracted considerable interest within commutative algebra.
  In 1994\, Huneke and Wiegand formulated a conjecture about tensor product
 s of torsion-free modules over one-dimensional commutative Noetherian inte
 gral domains. This conjecture\, which remains unresolved\, implies the Aus
 lander–Reiten Conjecture for a broad class of commutative rings.\nAuslan
 der defined a condition\, denoted by (AC)\, to analyze the Finitistic Dime
 nsion Conjecture\, and conjectured that every finite-dimensional algebra s
 atisfies (AC). This conjecture turned out to be false\; interestingly\, th
 e first counterexamples were obtained only after the (AC) conjecture had b
 een transplanted into commutative algebra. Nonetheless\, several classes o
 f rings do satisfy Auslander's condition (AC)\, and remarkable homological
  properties of such classes have been uncovered. For example\, Christensen
  and Holm proved that the Auslander–Reiten Conjecture holds over Noether
 ian rings satisfying (AC).\nIn this talk\, I will survey some of the liter
 ature on these problems and explore the relationship between the Huneke–
 Wiegand Conjecture\, the Auslander–Reiten Conjecture\, and related probl
 ems on the vanishing of Tor.\n \n\nhttps://math-events.uni-bonn.de/event/
 416/
LOCATION:MPIM\, Vivatsgasse\,  7 - Seminar Room (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/416/
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