AI 中文总结
本文研究特征p>0域上无限变量除幂代数的GL-等变模，证明其为GL-凝聚代数，建立有限表现模的转移定理，进而得到其有界导出范畴的半正交分解，关键利用其为GL-诺特子代数的平坦余极限的性质。
AI 中文摘要
本文研究特征p>0的代数闭域k上无限变量除幂代数$D = \text{Div}(k^{\infty})$上的GL-等变模。与此前分析的GL-代数不同，除幂代数不是诺特环，甚至不是有限生成的。我们证明D是GL-凝聚的，并对有限表现D-模证明了一个“转移定理”。利用该定理，我们得到其有界导出范畴的（半无限）半正交分解，其中每个分量对应D的弗罗贝尼乌斯扭转$D^{(r)}$。我们方法的关键在于D是GL-诺特子代数的平坦余极限这一事实。
英文摘要
In this paper, we study GL-equivariant modules over the infinite-variable divided power algebra $D = \text{Div}(k^{\infty})$ with $k$ an algebraically closed field of characteristic $p > 0$. Unlike previously analyzed GL-algebras, the divided power algebra is not noetherian or even finitely generated. We show that $D$ is GL-coherent and prove a ``shift theorem'' for finitely presented $D$-modules. Using this, we obtain a (semi-infinite) semi-orthogonal decomposition of its bounded derived category with one piece corresponding to each Frobenius twist $D^{(r)}$ of $D$. Crucial to our approach is the fact that $D$ is a flat colimit of subalgebras which are GL-noetherian.
Comments34 pages, no figures. Comments welcome!
