Flat covers and flat cotorsion modules
Flat covers and flat cotorsion modules
复制标题
DOI：
10.1090/s0002-9939-1984-0754698-x
复制
发表时间：
1984-02
期刊：
影响因子：
--
通讯作者：
E. Enochs
中科院分区：
文献类型：
--
作者：
E. Enochs
不知道任意环上的模是否有平坦覆盖，但是对于交换诺特环上的某些模，它们可以被证明存在。这些覆盖，反过来，有一个有趣的联系，平坦的余挠模。一个完整的描述平坦的余挠模类似于哈里森的扭转自由，余挠阿贝尔群将被给予。在本文中，R表示交换诺特环。记法。如果R是局部环，m（R）表示它的极大理想。对于p ∈ Spec（R），Rp表示局部环Rp的完备化，对于任何R-模，M表示具有m（Rp）-adic拓扑的R-模Mp的（分离的）完备化。k（p）表示Rp（_ Rp/m（Rp））的剩余域。E（M）是M的内射包络。如果R是局部的，则MV表示M的Matlis对偶Hom（M，E（R/m（R），并且M被称为自反的，如果MV V M自然。对于集合X，Mx是所有函数X-M的模，M（x）是那些具有有限支集的函数的子模。Soc（M）将表示M的基座。1.平面封面。定义（见[1]）。一个线性映射0：F-M称为M的平坦预覆盖，如果F是平坦的，并且如果有图
It is not known whether modules over an arbitrary ring have flat covers, however for certain modules over commutative noetherian rings they can be shown to exist. These covers, in turn, have an interesting connection with flat cotorsion modules. A complete description of flat cotorsion modules analogous to that given by Harrison for torsion free, cotorsion abelian groups will be given. In this article, R will denote a commutative noetherian ring. NOTATION. If R is a local ring, m(R) will denote its maximal ideal. For p e Spec(R), Rp will denote the completion of the local ring Rp, and for any R-module, M will denote the (separated) completion of the R module Mp with the m(Rp)-adic topology. k(p) denotes the residue field of Rp (_ Rp/m(Rp)). E(M) will be an injective envelope of M. If R is local, MV denotes the Matlis dual, Hom(M, E(R/m(R))), of M, and M is said to be reflexive if MV V M naturally. For a set X, Mx is the module of all functions X -M and M(x) the submodule of those functions with finite support. Soc(M) will denote the socle of M. 1. Flat covers. DEFINITION (SEE [1]). A linear map 0: F -M is said to be a flat precover of M if F is flat and if any diagram