论文标题
针对小学Abelian $ P $ -Groups的同谋的显式公式
Explicit formulas for the cohomology of the elementary abelian $p$-groups
论文作者
论文摘要
令$ g $为小学的Abelian $ P $ -Group，$ g \ cong {\ Mathbb f} _p^r $，让$ s_1，\ ldots，s_r $是$ g $ over $ {\ mathbb f} _p $的基础。 令$ v $为$ g $，$ v = {\ rm hom}（g，{\ mathbb f} _p）_p）= h^1（g，{\ mathbb f} _p）$。令$ x_1，\ ldots，x_r $为$ {\ mathbb f} _p $的$ v $的基础，该_p $是基础$ s_1，\ ldots，s_r $ g $的基础。对于$ 1 \ leq i \ leq r $，我们用$ y_i =β（x_i）\在h^2（g，{\ m athbb f} _p）$中，其中$β：h^1（g，{\ mathbb f} _p} _p} _p} _p）\ to h^2（g to h^2（g^2（g，g，mathbb f iass） 环$（h^*（g，{\ mathbb f} _p），+，\ cup）$满足$$ h^*（g，g，{\ mathbb f} _p）\ cong \ begin {cases} {\ mathbb f} λ（x_1，\ ldots，x_r）\ otimes {\ mathbb f} _p [y_1，\ ldots，y_r]＆p> 2 \ end end {cases} $$ 当$ p = 2 $同构$τ：{\ mathbb f} _p [x_1，\ ldots，x_r] \ to h^*（g，{\ mathbb f} _p）$由$ x_ {i_1}} {i_1} \ cd__n} c cd_ cup x________________________ x_ {i_n} \ in H^n（g，{\ mathbb f} _p）$。当$ p> 3 $同构$τ：λ（x_1，\ ldots，x_r）\ outimes {\ mathbb f} _p [y_1，\ ldots，y_r] \ to h^*（g，g，wed wed wed）给予$ x_______ x_ {i_l} \ otimes y_ {j_1} \ cdots y_ {j_k} \ mapsto x_ {i_1} \ cup \ cdots \ cup x_ {i_l} {i_l} \ cup cup y_ _ {j_1} h^{2k+l}（g，{\ mathbb f} _p）$。 在本文中，我们为逆同构$τ^{ - 1} $提供明确的公式。 $ h^*（g，{\ mathbb f} _p）$的元素是根据归一化的Cochains编写的。在证明期间，我们使用另一种方法来描述标准化的科目。也就是说，对于每一个$ g $ -module $ m $，我们都有$ c^n（g，m）\ cong {\ rm hom}（t^n（t^n（{\ nathcal i}），m）$，其中$ {\ nathcal i} $是$ g $，$ g $，$ g $，$ g $ {\ nathcal i} v v v v v v v v v var c的增强率。 z} [g] \ to {\ mathbb z} $。
Let $G$ be an elementary abelian $p$-group, $G\cong{\mathbb F}_p^r$ and let $s_1,\ldots,s_r$ be a basis of $G$ over ${\mathbb F}_p$. Let $V$ be the dual of $G$, $V={\rm Hom}(G,{\mathbb F}_p)=H^1(G,{\mathbb F}_p)$. Let $x_1,\ldots,x_r$ be the basis of $V$ over ${\mathbb F}_p$ which is dual to the basis $s_1,\ldots,s_r$ of $G$. For $1\leq i\leq r$ we denote by $y_i=β(x_i)\in H^2(G,{\mathbb F}_p)$, where $β:H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$ is the connecting Bockstein map. The ring $(H^*(G,{\mathbb F}_p),+,\cup )$ satisfies $$H^*(G,{\mathbb F}_p)\cong\begin{cases}{\mathbb F}_p[x_1,\ldots,x_r]&p=2\\ Λ(x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]&p>2\end{cases}.$$ When $p=2$ the isomorphism $τ:{\mathbb F}_p[x_1,\ldots,x_r]\to H^*(G,{\mathbb F}_p)$ is given by $x_{i_1}\cdots x_{i_n}\mapsto x_{i_1}\cup\cdots\cup x_{i_n}\in H^n(G,{\mathbb F}_p)$. When $p>3$ the isomorphism $τ:Λ(x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H^*(G,{\mathbb F}_p)$ is given by $x_{i_1}\wedge\cdots\wedge x_{i_l}\otimes y_{j_1}\cdots y_{j_k}\mapsto x_{i_1}\cup\cdots\cup x_{i_l}\cup y_{j_1}\cup\cdots\cup y_{j_k}\in H^{2k+l}(G,{\mathbb F}_p)$. In this paper we give explicit formulas for the inverse isomorphism $τ^{-1}$. The elements of $H^*(G,{\mathbb F}_p)$ are written in terms of normalized cochains. During the proof we use an alternative way to describe the normalized cochains. Namely, for every $G$-module $M$ we have $C^n(G,M)\cong{\rm Hom}(T^n({\mathcal I}),M)$, where ${\mathcal I}$ is the augmented ideal of $G$, ${\mathcal I}=\ker\varepsilon :{\mathbb Z}[G]\to{\mathbb Z}$.
