Troesch复形与严格多项式函子的扩张
Troesch complexes and extensions of strict polynomial functors
摘要：
本文基于Troesch复形，在严格多项式函子范畴内提出了一种新的扩张演算方法。我们获得了许多经典的Ext-computations以及新的结果新的短的初等证明。特别地，我们得到了GL_n的经典不变不变量理论的“基本定理”的上同调版本，其中n足够大（并且我们给出了n的较小值的一个猜想）。我们还研究了收敛于扭曲函子F^{（r）}和G^{（r）}之间的扩张群Ext^*（F^{（r）}，G^{（r）}）的扭曲谱序列E^{s，t}（F，G，r）.许多经典的Ext-计算仅仅相当于在第二页（由于缺乏原因）折叠这个谱序列，它也是研究Frobenius扭曲对Ext-群的影响的方便工具。我们证明了许多塌缩的情形，并推测塌缩是一个普遍的事实。
英文摘要：
We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical Ext-computations as well as new results. In particular, we get a cohomological version of the `fundamental theorems' from classical invariant invariant theory for GL_n for n big enough (and we give a conjecture for smaller values of n). We also study the `twisting spectral sequence' E^{s,t}(F,G,r) converging to the extension groups Ext^*(F^{(r)}, G^{(r)}) between the twisted functors F^{(r)} and G^{(r)}. Many classical Ext-computations simply amount to the collapsing of this spectral sequence at the second page (for lacunary reasons), and it is also a convenient tool to study the effect of the Frobenius twist on Ext-groups. We prove many cases of collapsing, and we conjecture collapsing is a general fact.
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