论文标题
分级$ p $ - 极环和$ω^nς^nx $的同源
Graded $p$-polar rings and the homology of $Ω^nΣ^nX$
论文作者
论文摘要
作为先前未分级工作的扩展，我们将一个分级的$ p $ polar环定义为对级别的通勤环的类似物，在该级别的元素中，仅允许乘以$ p $ - tuplace（而不是对）等级的元素。我们表明，在$ k $ -s的代数上定义了免费的Aggine $ p $ adiC Group foundor以及免费的正式组函数，用于$ k $ - 代数$ k $的特征性$ p $，通过$ p $ -polar $ -polar $ k $ -k $ -algebras。因此，对于任何Aggine $ p $ - adiC或正式组函数，尤其是$ p $ typical typical witt vectors的函数，也是如此。作为一个应用程序，我们表明免费的$ e_n $ -algebra $ h^*的同源性（ω^nς^nd x; \ m athbf f_p）$作为hopf代数，仅取决于$ p $ - 波尔的结构$ h^*（x; \ \ sathbf f_p）$以功能性的方式。
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^*(Ω^nΣ^n X;\mathbf F_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf F_p)$ in a functorial way.
