复形范畴中的覆盖包络理论与Avramov-Foxby猜测

The classical theory of covers and envelopes mainly concerns the category of modules. We notice that the category of complexes is an abelian category that is larger than the category of modules (since the category of modules can be viewed as a full subcategory of the category of complexes). Thus studying covers and envelopes in the category of complexes has become a hot research field recently. However, further complications arise when we extend our attention from the category of modules to the category of complexes. For example, with respect to injective modules, there are #-injective complexes, DG-injective complexes and injective complexes, etc. Avramov-Foxby conjecture is actually a question about the relation between #-injective complexes and DG-injective complexes. Our recent work shows that the conjecture is also a question about the relation between exact #-injective complexes and injective complexes. Recently, the work of Iacob shows that the conjecture is very closely related to covers and envelopes in the category of complexes. Based on it, we continue studying Avramov-Foxby conjecture by studying covers and envelopes in the category of complexes.

经典的覆盖包络理论从本质上讲是在模范畴中展开的，而复形范畴是比模范畴“更大”的Abel 范畴（因为模范畴可以看成是复形范畴的全子范畴），因此在复形范畴中开展覆盖包络理论的研究已成为了当前的热点课题。然而，当我们把研究对象由模范畴扩展到复形范畴时，许多问题将变得非常复杂。例如，和模范畴中的内射对象相对应，在复形范畴中有#-内射复形、DG-内射复形、内射复形等。Avramov-Foxby 猜测说的就是#-内射复形和DG-内射复形之间的关系问题。我们最近的工作表明，这一猜测的本质也是考虑正合的#-内射复形和内射复形之间的关系问题。近来，Iacob 的研究表明这一猜测与复形范畴中的覆盖包络理论有着密切的联系，我们将以此为基础，通过研究复形范畴中的覆盖包络理论继续研究Avramov-Foxby猜测。

本项目主要借助复形范畴中的覆盖包络理论研究 Avramov-Foxby 猜测、Tate 上同调理论、复形的各种同调不变量等。主要研究成果如下：（1）证明了在诺特环上任意复形都有#-内射包络。（2）研究了 Cartan-Eilenberg Gorenstein 投射复形的性质，给出了它的一些同调刻画。（3）在一般的 Abel 范畴中证明了 Tate 上同调的平衡性, 改进了 Sather-Wagstaff 等人的结果。（4）研究了复形的限制投射维数，给出了它的一些等价刻画。（5）进一步研究了 Ding 投射模和 Ding 内射模的性质，证明了 Ding 投射模类是投射可解的，而Ding内射模类是内射可解的。