加法函子的内射稳定化。II。（余）挠与Auslander-Gruson-Jensen函子
Injective stabilization of additive functors. II. (Co)torsion and the Auslander-Gruson-Jensen functor
摘要：
利用加法函子的内射稳定性定义了模的挠子模。它适用于任意环上的任意模。对于交换整环上的任意模，它与经典挠一致，对于任意环上的R-R-表示模，它与Bass挠一致。一个正式的对偶方法-基于投影稳定-产生了一个新的概念：一个模块的余挠商模块。这是在完成的一般性-新的概念是定义为任何模在任何环。与挠不同，余挠没有经典原型。建立了这些结构的一般性质。证明了Auslander-Gruson-Jensen函子应用于余挠函子返回挠函子。因此，一个环是单侧绝对纯的当且仅当另一侧的每个纯内射是上挠自由的。如果环的内射包络是双表示的，则Auslander-Gruson-Jensen函子的右伴随应用于挠函子返回余挠函子。这种对应建立了此类环上挠和余挠之间的对偶。特别地，这种对偶性适用于阿廷代数。证明了在任意环上，模的挠的特征模同构于模的特征模的余挠。在环的内射包络上的各种有限性条件下，计算了挠和余挠的导出函子。
英文摘要：
The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach -- based on projective stabilization -- gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality -- the new concept is defined for any module over any ring. Unlike torsion, cotorsion does not have classical prototypes. General properties of these constructs are established. It is shown that the Auslander-Gruson-Jensen functor applied to the cotorsion functor returns the torsion functor. As a consequence, a ring is one-sided absolutely pure if and only if each pure injective on the other side is cotorsion-free. If the injective envelope of the ring is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor applied to the torsion functor returns the cotorsion functor. This correspondence establishes a duality between torsion and cotorsion over such rings. In particular, this duality applies to artin algebras. It is also shown that, over any ring, the character module of the torsion of a module is isomorphic to the cotorsion of the character module of the module. Under various finiteness conditions on the injective envelope of the ring, the derived functors of torsion and cotorsion are computed.
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