AI 中文总结
本文证明分次余模到对偶代数分次模的协变嵌入在多种情形下保持Ext-群，由此证明对偶Steenrod代数分次余模无非零投射对象，并推广Margolis-Lin公式，去掉有限型条件。
AI 中文摘要
设$\Gamma$是域上的余代数。众所周知，分次$\Gamma$-余模范畴到其对偶代数$\Gamma^*$上的分次模范畴之间存在一个协变嵌入。我们证明，在一系列具有启发性的情形中，该协变嵌入保持Ext-群，包括某些情形（如Steenrod代数）中该协变嵌入在其他方面具有较差的同调性质，并且无法保持底层非分次范畴上的Ext-群。作为一个应用，我们证明了稳定同伦论中的一个旧猜想：在对偶Steenrod代数的分次余模范畴中不存在非零投射对象。作为另一个应用，我们对扩展余模函子的右伴随进行了研究，包括该右伴随的拓扑解释。这推广了Margolis和Lin刻画从Eilenberg-Mac Lane谱到有限型下有界谱的映射的公式；我们的推广去掉了“有限型”假设。
英文摘要
Let $Γ$ be a coalgebra over a field. There is a well-known covariant embedding of the category of graded $Γ$-comodules into the category of graded modules over the dual algebra $Γ^*$. We show that this covariant embedding preserves Ext-groups in a range of motivating cases, including in some cases like Steenrod algebras where the covariant embedding has otherwise bad homological properties and fails to preserve Ext-groups on the underlying ungraded categories. As an application, we prove the old conjecture in stable homotopy theory that there are no nonzero projectives in the category of graded comodules over the dual Steenrod algebra. As another application, we carry out a study of the right adjoint to the extended comodule functor, including a topological interpretation of this right adjoint. This yields generalizations of formulas of Margolis and Lin characterizing maps from Eilenberg-Mac Lane spectra into finite-type bounded-below spectra; our generalizations drop the "finite-type" hypothesis.
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