Auslander范畴和自由正规化扩张

§1. Introduction
Let R be a ring. M od-R denotes the category of all right R-modules, and R-M od denotes the category of all left R-modules.
Over a commutative, noetherian local ring, semidualizing modules provide a common generalization of a dualizing module and a free module of rank one. Foxby [6] first defined them (PG-modules of rank one). In [8], Henric Holm and Diana White extended the definition of semidualizing (S,R)-bimodules, where R and S are arbitrary associative rings, which are defined as follows: An (R,S)-module RCS is semidualizing if
S=
n i=1
ai
R
where
aiR = Rai
for
i = 1,2,...n.
Finite
normalizing
extensions
have
been
studied
in many papers such as [9–15]. S is called a free normalizing extension of R if S =
to non-commutative non-noetherian rings. The Auslander class AC(S) is defined as follows:
the Auslander class with respect to C, denoted by AC or AC(S), consists of all S-modules M satisfying
(a2) CS admits a degreewise finite Sop-projective resolution.
Received date: 2021-02-21 Foundation item: Supported by the Natural Science Foundation of Anhui Province (Grant No. 2008085QA03). Biographies: GU Qin-qin (1979-), female, native of Laiwu, Shandong, associate professor ofห้องสมุดไป่ตู้Anhui University of Technology, engages in algebra; ZHUO Yuan-fan (1997-), male, native of Baoying, Jiangsu, postgraduate student of Anhui University of Technology, engages in algebra. Corresponding author: GU Qin-qin.
and
right
free
normalizing
extensions
of R and S respectively, where each ai centralizes the elements of R and bj centralizes the
elements of S. Suppose rijkc = csijk for all c ∈ C and all i, j, k; where put aiaj =
motivation for our Theorem 2.1. Let S be a ring and let R be a subring of S (with the same 1).
S is called a finite normalizing extension of R if there exist elements a1,a2,...an ∈ S such that
n i=1
aiR
is
a
normalizing extension of R and S is free with basis {a1 = 1,a2,...,an} as both a right R-module
and a left R-module.
Let A and B be the free normalizing extensions of R and S respectively. Let SCR be a semidualizing (R,S)-bimodule. We get the semidualizing (A,B)-bimodule with respect to RCS. Under a suitable condition on RCS, we develop a generalized Morita theory for Auslander categories.
(a1) RC admits a degreewise finite R-projective resolution (i.e., there exists a resolution ··· → P1 → P0 → M → 0 where each Pi is finitely generated R-projective).
204
No. 2
GU Qin-qin, ZHUO Yuan-fan: Auslander Categories and Free Normalizing Extensions 205
(b1) The natural homothety map RRR → HomSop (C,C) is an isomorphism.
§2. Auslander categories and free normalizing extensions
In this section, we will give our main results.
Theorem 2.1. Let A =
n i=1
Rai
and
B
=
n j=1
S
bj
be
left
Chin. Quart. J. of Math. 2021, 36(2): 204–209
Auslander Categories and Free Normalizing Extensions
GU Qin-qin, ZHUO Yuan-fan
(School of Mathematics and Physics, Anhui University of Technology, Maanshan 243032, China)
Abstract: Let RCS be a semidualizing (R,S)-bimodule. Then RCS induces an equivalent between the Auslander class AC (S) and the Bass class BC (R). Let A and B be free normalizing extensions of R and S respectively. In this paper, we prove that HomS(BBS,R CS) is a semidualizing (A,B)-bimodule under some suitable conditions, and so HomS(BBS,R CS) induces an equivalence between the Auslander class
of the category of R-modules, namely the so-called Auslander class AC(R) and Bass class BC(R) defined by Avramov and Foxby [1, 6]. Semidualizing modules and their Auslander
Hom(B,R CS )B ⊗ − : AHom(B,RCS)(B) BHom(B,RCS)(A) : Hom(Hom(B,R CS )B,−)
and bibj =
n h=1
sijhbh.
(1) If RCS is a bimodule, S ∼= EndR(RCS) and R ∼= EndS(RCS), then
n h=1
rijhah
A ∼= EndB(Hom(BBS,R CS)B) and B ∼= EndR(Hom(BBS,R CS))
(resp., Bass) classes have caught the attention of several authors, see for example [1–7]. Henric
Holm and Diana White also extended the definitions of Auslander classes and Bass classes
(a) T or≥R1(C,M ) = 0. (b) Ext≥R1(C,C ⊗R M ) = 0. (c) The natural map µCCM : M → HomR(C,C ⊗R M ) is an isomorphism.
We will write µM = µCCM if there is no confusion. Dually, we can define BC (R).
(b2) The natural homothety map SSS → HomR(C,C) is an isomorphism. (c1) Ext≥R1(C,C) = 0. (c2) Ext≥So1p (C,C) = 0. A semidualizing module over a commutative noetherian ring gives rise two full subcategories
as ring isomorphism.
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CHINESE QUARTERLY JOURNAL OF MATHEMATICS
Vol. 36
(2) If RCS is a semidualizing (R,S)-bimodule, then Hom(B,R CS) is a semidualizing (A,B)bimodule, and
AHomS (B BS ,RCS )(B)
and the Bass class BHomS (B BS ,RCS )(A).
Furthermore, under a suitable condition on RCS, we develop a generalized Morita theory for Auslander categories. Keywords: Semidualizing module; Auslander class; Excellent extension 2000 MR Subject Classification: 13B02, 13B10, 16D20, 16D90 CLC number: O154.2 Document code: A Article ID: 1002-0462 (2021) 02-0204-06 DOI: 10.13371/j.cnki.chin.q.j.m.2021.02.009
Let R be commutative and noetherian. Christensen [2] proved that if ψ : R → S is local and
flat, then C is semidualizing for R if and only if C ⊗R S is semidualizing for S. This result is the