The cohomology algebras of finite-dimensional Hopf algebras
The cohomology algebras of finite-dimensional Hopf algebras
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DOI：
10.1090/s0002-9947-1981-0597872-x
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发表时间：
1981
影响因子：
1.3
通讯作者：
C. Wilkerson
中科院分区：
文献类型：
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作者：
C. Wilkerson
证明了域K上有限维分次连通余交换双结合Hopf代数的上同调代数是一个K-G生成的K-代数。反例的Quillen的结果（即非幂零上同调类应具有非零限制的一些交换子Hopf代数）的模拟构造，但这种“检测原理”的有效性的特殊情况下，有限的子Hopf代数的mod 2 Steenrod代数给出了一个初等的证明。作为应用，给出了mod 2 Steenrod代数的有限骨架的上同调代数的Krull维数的一个显式公式.若A是域K上的增广代数，则上同调代数H*（A）定义为ExtA（K，K）。如果A作为K-向量空间是有限维的，则H*（A）仍然可能不是K-生成的K-代数，例如，Lofwall [12].然而，如果A = K[G]，有限群的群代数，则H*（A）是双生成的，Evens [6]和Venkov [21]。余交换Hopf代数是群代数的一种推广，连通分次余交换Hopf代数与有限p-群非常相似。这是这项工作的目的，以推动这种类比尽可能。第一个积极的结果是，有限生成在这种情况下也成立（所有的霍普夫代数在这项工作中提到的是双结合的，要么交换或余交换）。定理A.若A * 是有限维分次连通余交换K-Hopf代数，则H**（A*）= ExtA*（K，K）是一个n-生成K-代数.证明的策略基本上是亚当斯[1]和Liulevicius [10]为计算Steenrod代数的小子Hopf代数的上同调而提出的。一个解决方案A * 作为一个序列的迭代中心扩张的霍普夫代数。每一个这样的扩展有一个相关的频谱序列，和一些持有的差异是由海侵定理有关Steenrod操作的“纤维”和“基地”。本方案与[1]、[10]的方案之间唯一的哲学差异是为了一般性而牺牲了精确的计算结果。1979年12月10日由编辑接收。AMS（MOS）主题分类（1970年）。小学55 G10、55 F35、55 H99、55 J 99、57 F05、18 A24、18 H15。
The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field K is shown to be a finitely generated K-algebra. Counterexamples to the analogue of a result of Quillen (that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub-Hopf algebra) are constructed, but an elementary proof of the validity of this "detection principle" for the special case of finite sub-Hopf algebras of the mod 2 Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the mod 2 Steenrod algebra is given. If A is an augmented algebra over the field K, the cohomology algebra H*(A) is defined as ExtA(K, K). If A is finite dimensional as a K-vector space, H*(A) may still fail to be a finitely generated K-algebra, e.g., Lofwall [12]. However, if A = K[G], the group algebra of a finite group, then H*(A) is finitely generated, Evens [6] and Venkov [21]. Cocommutative Hopf algebras are one generalization of group algebras, and connected graded cocommutative Hopf algebras are closely analogous to finite p-groups. It is the intent of this work to push this analogy as far as possible. The first positive result is that finite generation holds in this context also (all Hopf algebras mentioned in this work are biassociative and either commutative or cocommutative). THEOREM A. If A * is a finite dimensional graded connected cocommutative K-Hopf algebra, then H**(A*) = ExtA* (K, K) is a finitely generated K-algebra. The strategy of the proof is essentially that developed by Adams [1] and Liulevicius [10] for the computation of the cohomology of small sub-Hopf algebras of the Steenrod algebras. One resolves A * as a sequence of iterated central extensions of Hopf algebras. Each such extension has an associated spectral sequence, and some hold on the differentials is provided by the transgression theorem relating Steenrod operations on the "fiber" and "base". The only philosophical difference between the present plan and that of [1], [10] is that precise computational results are sacrificed for the sake of generality. Received by the editors December 10, 1979. AMS (MOS) subject classifications (1970). Primary 55G10, 55F35, 55H99, 55J99, 57F05, 18A24, 18H15.