We prove in this paper a noncommutative version of Leray Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with E terms as Ep,q = HPp(A; HPq(B,A)) =⇒ HPp+q(B) and Ep,q = HPp(A; Kq(B,A)) =⇒ Kp+q(B).

We introduce in this paper the notion of noncommutative Serre fibration (shortly, NCSF) and show that up to homotopy, every NCCW complex morphism is some noncommutative Serre fibration. We then deduce a six-term exact sequence for the periodic cyclic homology and for K-theory of an arbitrary noncommutative Serre fibration. We also show how to use this technique to compute K-theory and cyclic th...

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and k Q its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(k Q) := Gr(k Q)/Fdim(k Q) of graded k Q-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(k Q) ≡ ModS(Q) ≡ GrL(Q) ≡ ModL(Q◦)0 ≡ QGr(k Q (n)). Here S(Q) =...

The $\imath$Serre relations and the corresponding Serre-Lusztig are formulated established for arbitrary $\imath$quantum groups arising from quantum symmetric pairs of Kac-Moody type.

A completion of a Cauchy space is obtained without the T2 restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms called s-maps form a subcategory CHY′ of CHY. A completion functor is defined for this subcategory. The completion subcategory of CHY′ turns out to be a bireflective subcategory of CHY′. This t...

“Divide and conquer” has been a common practice to address complex learning tasks such as multi-view object detection. The positive examples are divided into multiple subcategories for training subcategory classifiers individually. However, the subcategory labeling process, either through manual labeling or through clustering, is suboptimal for the overall classification task. In this paper, we...

Anxious-retarded depression is a two-dimensionally defined subcategory of depression based on high scores for both anxiety and retardation. This anxious-retarded subcategory is related to melancholia as defined by DSM-IV. Patients with this diagnosis exhibit elevated plasma arginine vasopressin (AVP) and a high correlation between plasma vasopressin and cortisol, which suggests vasopressinergic...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...