Relative Hopf modules—Equivalences and freeness criteria
Relative Hopf modules—Equivalences and freeness criteria
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DOI：
10.1016/0021-8693(79)90093-0
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发表时间：
1979-10
影响因子：
0.9
通讯作者：
M. Takeuchi
中科院分区：
文献类型：
--
作者：
M. Takeuchi
设A是域k上的Hopf代数，BCA是Hopf子代数。右(A，B)-Hopf模的概念是由我[L]提出的，并研究了右(A，B)-Hopf模的范畴，证明了如果A是交换的或余交换的，则A是忠实平坦的B-模。最近，Kadford[9]利用这个概念来知道A何时是自由(或投射)B-模。本文在两个方向上推广了相对Hopf模的概念，并利用它得到了A在B上的许多自由性或投射性判据。首先，我们注意到定义了右(A，B)-Hopf模，如果只有B是A的右余理想子代数，这意味着B是d(B)Cb@A的子代数。在第一节中，我们证明了右(A，B)-Hopf模的范畴等价于某个余模范畴，左(a(A)，A)-Hopf模的范畴等价于某个模范畴，但对平坦性作了一点假设。Sweedler的定理，即右(A，A)-Hopf模范畴等价于k-向量空间范畴，由此引出。将这些等价应用于交换情形，证明了A为忠实平坦模的右上理想子代数与A为忠实余平坦左(或等价右)余模的商Hopf代数之间存在L1对应Bt，2.这意味着如果G是仿射k-群方案，HCG是闭子群方案，则左陪集H？，G[7，Chap.III，第三节，7.21是仿射的当且仅当Fline环O(G)是忠实余平坦的左或/和右O(H)余模。在余交换的情况下，证明了当A是忠实余平坦余模时，Hopf子代数与商左A-模余代数之间存在L1对应Rct2。
Let A be a Hopf algebra over a field k and let BCA be a Hopf subalgebra. The notion of right (A, B)-Hopf modules was introduced by me [l] and the category of those modules was studied to prove that A is a faithfully flat B-module, if A is either commutative or cocommutative. Recently Kadford [9] used this notion to know when A is a free (or projective) B-module. In this paper we generalize the notion of relative Hopf modules in two directions, and apply it to obtain many freeness or projectivity criteria for A over B. First, we note that right (A, B)-Hopf modules are defined, if only B is a right coideal subalgebra of A, which means that B is such a subalgebra that d (B) CB@ A. Dually, we can define left (T (A), A)-Hopf modules, where w A-+ r (A) is a surjection of coalgebras and left A-modules. In Section 1, we show that the category of right (A, B)-Hopf modules is equivalent to some comodule category, and the category-of left (a (A), A)-Hopf modules is equivalent to some module category, with a little assumption on flatness. The theorem of Sweedler, which states that the category of right (A, A)-Hopf modules is equivalent to the category of k-vector spaces, follows from this. These equivalences are applied in the commutative case to prove that there is a ll correspondence B t, 2 between right coideal subalgebras over which A is a faithfully flat module and quotient Hopf algebras over which A is a faithfully coflat left (or equivalently right) comodule. This means that if G is an affine k-group scheme and H CG a closed subgroup scheme, then the dur k-sheaf of left cosets H?, G [7, Chap. III, Sect. 3, 7.21 is affine if and only if the afline ring O (G) is a faithfully coflat left or/and right O (H)-comodule. In the cocommutative case, it follows that there is a ll correspondence R ct 2 between Hopf subalgebras and quotient left A-module coalgebras over which A is a faithfully coflat comodule.