On the complete relative cohomology of Frobenius extensions
On the complete relative cohomology of Frobenius extensions
复制标题
论Frobenius扩张的完全相对上同调
DOI：
10.21099/tkbjm/1496162133
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
T. Nozawa
中科院分区：
文献类型：
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作者：
T. Nozawa
设A是交换环K上的代数，F是子代数.假设扩张A/F是Frobenius扩张。然后在[3，第3节]中，对任意的左J-模M和reZ，引入了完备的相对上同调群Hl（niM，-）.我们分别用A0和F°表示A和F的相对环。设P=A KA°，用δ表示F（Z）KF 0在P中的自然像，则扩张P/S也是Frobenius扩张。由于A是一个左P-模，具有自然的方式，我们有HlP，s->（A，-）。对于[6，第3节]，我们将用Hr（A，F，-）表示此Hu>.s>（A，-）。本文将研究这类完全相对上同调H{A，F，-）.第一节研究A的相对完全分解，第二节给出[4，命题1和定理1]中关于完全相对上同调群的基本正合列的对偶。在第3节中，我们将研究类似于[9，第2节]的内积，我们称之为杯积。若Frobenius扩张的基环是交换的，则本文中的杯积与[2，第十一章练习2]中的积V重合
Let A be an algebra over a commutative ring K and F a subalgebra. Suppose that the extension A/F is a Frobenius extension. Then in [3, section 3], the complete relative cohomology group Hl^niM, ―) is introduced for an arbitrary left J-module M and reZ. We denote the opposite rings of A and F by A0 and F° respectively. Put P=A KA° and let 5 denote the natural image of F(Z)KF0 in P. Then the extension P/S is also a Frobenius extension. Since A is a left P-module with the natural way, we have HlP,s->(A,―). We will denote this Hu>.s>(A, ―) by Hr(A, F, -) for [6, section 3]. In this paper, we will study this complete relative cohomology H{A, F, ―). In section 1, we will study relative complete resolutions of A and in section 2, we willintroduce the dual of the fundamental exact sequence of [4, Proposition 1 and Theorem 1] for complete relative cohomology groups. In section 3, we will study an internal product like as in [9, section 2] which we will call the cup product. If the basic ring of the Frobenius extension is commutative, the cup product in this paper coincides with the product V in [2, Exercise 2 of Chapter XI]