On the H-finite cohomology
On the H-finite cohomology
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关于 H-有限上同调
DOI：
10.1016/j.jalgebra.2003.09.040
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
T. Guédénon
中科院分区：
文献类型：
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作者：
T. Guédénon
设k是一个场，H是k上的Hopf代数，有一个对映对。假设H作用于一个结合（左诺etherian） k代数R，使得R是一个H模代数。我们以相容的方式考虑了所有h模的范畴，h局部有限模的子范畴，以及每个r模的子范畴。这些范畴都是有足够单射的阿贝尔范畴，我们在其中推导出与Ext *(−,−)有关的谱序列。现在设（−）h表示取h不变量，设S=RH。我们从ModSto中定义了一个函子LS(R,−)，它对于内射对象具有良好的行为。我们还证明了函子（−）h携带一些单射到单射。当R是可交换的，H是协交换的，k是射影的，在有限维H模的范畴中，我们得到了更精确的结果，例如比较Picard群PicR（R，H）和Pic(S)。
Let k be a field and H a Hopf algebra over k with a bijective antipode. Suppose that H acts on an associative (left noetherian) k-algebra R such that R is an H-module algebra. We consider the categories of all H-modules, the subcategory of those which are H-locally finite, and the subcategories of each which are also R-modules in a compatible way. These categories are all abelian with enough injectives and we derive spectral sequences relating Ext∗(−,−) in them. Now let (−)Hdenote taking H-invariants and set S=RH. We define a functor LS(R,−) from ModSto ModR(#H)that has good behavior with respect to injective objects. We also show that the functor (−)Hcarries some injectives to injectives. When R is commutative, H is cocommutative, and k is projective in the category of finite-dimensional H-modules, we obtain more precise results, comparing, for example, the Picard groups PicR(R,H) and Pic(S).