Infinite Power of Ideals in Abelian Categories

Sinem Odabaşı, Institute of Physics and Mathematics, Science Faculty, The Universidad Austral de Chile (UACh).
Date: 23rd of July, 2019, Tuesday. Time: 11:15 – 12:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: The ‘Phantom phenomenon’ has been sucessufelly carried into abelian setting firstly in [Her07], later in [FGHT13]. In this talk, we claim to introduce ‘ghost phenomenon’ in abelian setting which is also compatible with the existent ones in certain tringulated categories as mentioned above. Besides, we observe that the problem of being zero a finite power of ghost ideal in these triangulated categories is strongly related to being a certain type of object ideals and cotorsion pairs. Using this observation and certain techniques/results on cotorsion pairs, now we are able to ensure that under mild conditions the ideal ‘Ghost’ in an abelian category is always turned out to be ‘zero’ in some infinite power. We then apply this formalism to the ideal Ghost of chain morphisms which induce zero in homology in the category of chain complexes of left R-modules.
This is a joint work-in-progress with Sergio Estrada, X.H. Fu and Ivo Herzog, which has been supported by the grant CONICYT/FONDECYT/Iniciaci\’on/11170394.
[FGHT13] Fu, X. H., Guil Asensio, P. A, Herzog, I. & Torecillas, B. (2013). Ideal approximation theory. Adv. Math. 244, 750-790.
[Her07] Herzog, I. (2007). The phantom cover of a module. Adv. Math. 215, 220–249.