Abstract

Let $\mathcal{F}$ (correspondingly $\mathcal{P}$) denote the abelian category of functors (strict polynomial functors in the sense of Friedlander and Suslin) from finite dimensional vector spaces over F_p to vector spaces over F_p. These two categories are related via the exact forgetful functor

$\iota :\mathcal{P}\to \mathcal{F}.$

The category $\mathcal{F}$ is strongly related to topology and representation theory of symmetric and general linear groups but the homological algebra in $\mathcal{F}$ is rather mysterious. The category $\mathcal{P}$ is easier for cohomological calculations. The known $Ex{t}_{\mathcal{F}}(.,.)$ calculations are obtained only for functors which belong to the image of ι and are performed using comparison of $Ex{t}_{\mathcal{P}}$- and $Ex{t}_{\mathcal{F}}$-groups induced by ι. The aim of the following note is to find cohomological conditions which guarantee that a given functor $F\in \mathcal{F}$ comes from $\mathcal{P}$ via ι.

摘要

让 $\mathcal{F}$ （相应地 $\mathcal{磷}$）表示的函子从有限维向量空间在阿贝尔范畴（在弗里德兰德和Suslin感严格多项式函子）˚F _p到向量空间中的过˚F _p。这两个类别通过确切的健忘函子相关

$一：\mathcal{磷}\to \mathcal{F}.$

类别 $\mathcal{F}$ 与对称和一般线性群的拓扑和表示理论密切相关，但同调代数 $\mathcal{F}$比较神秘。类别$\mathcal{磷}$更容易进行上同调计算。已知的$乙X{吨}_{\mathcal{F}}(.,.)$仅对属于ι图像的函子进行计算，并使用$乙X{吨}_{\mathcal{磷}}$- 和 $乙X{吨}_{\mathcal{F}}$- 由ι诱导的组。以下注释的目的是找到上同调条件，以保证给定的函子$F\in \mathcal{F}$ 来自 $\mathcal{磷}$通过ι。