On the Heart of a faithful torsion theory
On the Heart of a faithful torsion theory
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论忠实扭转理论的核心
DOI：
10.1016/j.jalgebra.2006.01.020
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发表时间：
2007
影响因子：
0.9
通讯作者：
F. Mantese
中科院分区：
文献类型：
--
作者：
Riccardo Colpi;E. Gregorio;F. Mantese
在[R.科尔皮Fuller，阿贝尔范畴中的倾斜对象和拟倾斜环，美国数学会，在任意阿贝尔范畴H中引入了倾斜对象，并证明了H与其自同态环上模范畴之间的经典倾斜定理的一个版本。进一步证明了给定Mod-R中的任意忠实挠理论（X，Y），对于环R，相应的HeartH（X，Y）是一个容许倾斜对象的阿贝尔范畴，从而得到HeartH与Mod-R之间的倾斜定理.本文首先证明了H（X，Y）是任意阿贝尔范畴H的一个原型，该范畴在Mod-R中允许一个倾斜对象倾斜到（X，Y）。然后我们研究了Heart的AB型性质和具有直接极限的交换。这使我们能够证明，例如，任何阿贝尔范畴H与倾斜的对象是AB 4，并找到必要和充分条件，保证H是一个格罗滕迪克，甚至是一个模范畴。作为特例，我们研究了两种主要的情形：当（X，Y）是遗传余倾斜时，证明了H（X，Y）是Grothendieck;当（X，Y）是倾斜时，证明了H（X，Y）是模范畴。
In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category H are introduced and are shown to yield a version of the classical tilting theorem between H and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory (X,Y) in Mod-R, for a ring R, the corresponding Heart H(X,Y) is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and Mod-R. In this paper we first prove that H(X,Y) is a prototype for any abelian category H admitting a tilting object which tilts to (X,Y) in Mod-R. Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category H with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that H is a Grothendieck or even a module category. As particular situations, we examine two main cases: when (X,Y) is hereditary cotilting, proving that H(X,Y) is Grothendieck and when (X,Y) is tilting, proving that H(X,Y) is a module category.