函子上的Reynolds算子
Reynolds Operator on functors
摘要：
设$G= {\rm Spec} A$是仿射$R$ -幺半群方案.证明了交换$R$ -模的对偶函子范畴与交换${\mathcal A}^*$ -模的对偶函子范畴等价.证明了$G$是不变正合的当且仅当$A^*= R \times B^*$是$R$ -代数，且第一投影$A^* \to R$是$A$的单位.若$\mathbb M$是$G$ -模和$w_G := (1,0) \in R \times B^* = A^*$的对偶函子，则证明了$\mathbb M^G = w_G \cdot \mathbb M$和$\mathbb F = w_G \cdot \mathbb M \oplus (1-w_G) \cdot \mathbb M$，从而Reynolds算子可以定义在$\mathcal M$上.
英文摘要：
Let $G= {\rm Spec} A$ be an affine $R$-monoid scheme. We prove that the category of dual functors (over the category of commutative $R$-algebras) of $G$-modules is equivalent to the category of dual functors of ${\mathcal A}^*$-modules. We prove that $G$ is invariant exact if and only if $A^*= R \times B^*$ as $R$-algebras and the first projection $A^* \to R$ is the unit of $A$. If $\mathbb M$ is a dual functor of $G$-modules and $w_G := (1,0) \in R \times B^* = A^*$, we prove that $\mathbb M^G = w_G \cdot \mathbb M$ and $\mathbb F = w_G \cdot \mathbb M \oplus (1-w_G) \cdot \mathbb M$; hence, the Reynolds operator can defined on $\mathcal M$.
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