Infinite powers of approximating ideals
报告题目(Title)：Infinite powers of approximating ideals
报告人(Speaker)：扶先辉 (东北师范大学)
地点(Place)：后主楼1220
时间(Time)：2021年9月8日16:00-17:00
邀请人(Inviter)：胡维
报告摘要
Let (A, E) be an exact category such that each continuous system in (A, E) has colimits. In this talk, a theory of infinite powers of approximating ideals in the exact category (Arr(A),ME) is presented. A special case of the morphism version of Eklof’s Lemma states that if J is an ideal, and a is an arrow which is μ-⊥J -filtered in (Arr(A),ME), then a belongs to ⊥(J (μ)) . This is used to show that if J is a special preenveloping ideal with Ω−1(J ) a cosyzygy ideal of J , and if Ω−1μ (J ) is the class consisting of arrows μ-filtered by Ω−1(J ) in (Arr(A),ME), then J (μ) is a special preenveloping ideal with Ω−1μ (J ) a J (μ)-cosyzygy ideal.
This theory is used to study Generalized Generating Hypothesis. In particular, (1) it is used to show a dual of a result of Xu: if the class of pure projective right R-modules is closed under extensions, then every FP-projective right R-module is pure projective; and (2) it is used to study the ghost ideal in the category of complexes. This talk is based on an ongoing project with S. Estrada, I. Herzog, and S. Odabasi.
References
1. S. Estrada, X.H. Fu, I. Herzog, and S. Odaba¸si, Infinite powers of approximating ideals, In preparation.
2. X.H. Fu, P. A. Guil Asensio, I. Herzog and B. Torrecillas, Ideal approximation theory, Adv. Math. 244(2013) 750-790.
3. X.H. Fu and I. Herzog, Powers of the phantom ideal, Proc. London Math. Soc. 112 (2016) 714-752.
4. X.H. Fu, I. Herzog, J.S. Hu, and H.Y. Zhu, Lattice theoretic properties of approximation ideals, J. Pure Appl. Algebra, provisionally accepted.
