Tate cohomology and Gorensteinness for triangulated categories
Tate cohomology and Gorensteinness for triangulated categories
复制标题
DOI：
10.1016/j.jalgebra.2005.10.024
复制
发表时间：
2006-05
影响因子：
0.9
通讯作者：
J. Asadollahi;Shokrollah Salarian
中科院分区：
文献类型：
--
作者：
J. Asadollahi;Shokrollah Salarian
受Tate上同调的经典结构的启发，我们发展并研究了三角范畴C中的Tate上同调理论。设E是一类真的三角形。通过使用E-投射和E-内射对象，我们给出了这一理论的两种不同的方法，它们通常是不等价的。因此，在本文的第二部分，我们研究了三角范畴，在三角范畴中，这两个理论是等价的。这导致我们研究了所有对象都具有有限E-G投射和有限E-G内射维的范畴。这些范畴将被称为E-Gorenstein三角范畴。我们用两个不变量的有限性刻画了这些范畴：E-silpC，C的E-投射对象的E-内射维度的上确界和E-SpliC，C的E-内射对象的E-投射维度的上确界，其中一个范畴的这两个不变量的有限性意味着另一个不变量的有限。最后，我们证明了在E-Gorenstein三角范畴上，有限E-投射维度的对象类和E-G内射对象类构成了E-完备余挠理论。
Motivated by the classical structure of Tate cohomology, we develop and study a Tate cohomology theory in a triangulated category C. Let E be a proper class of triangles. By using E-projective, as well as E-injective objects, we give two alternative approaches to this theory that, in general, are not equivalent. So, in the second part of the paper, we study triangulated categories in which these two theories are equivalent. This leads us to study the categories in which all objects have finite E-Gprojective as well as finite E-Ginjective dimension. These categories will be called E-Gorenstein triangulated categories. We give a characterization of these categories in terms of the finiteness of two invariants: E-silpC, the supremum of the E-injective dimension of E-projective objects of C and E-spliC, the supremum of the E-projective dimension of E-injective objects of C, where finiteness of each of these invariants for a category implies the finiteness of the other. Finally, we show that over E-Gorenstein triangulated categories, the class of objects of finite E-projective dimension and the class of E-Ginjective objects form an E-complete cotorsion theory.