In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is Hecke algebra a reductive $p$-adic group, to case it finite central extension group. The most important being that concentrated in single degree for irreducible representations and gives rise Schneider--Stuhler Ext groups (a Serre like property). A...

Aghapournahr and Melkersson introduced the notion of condition on a Serre subcategory category modules over commutative Noetherian ring. This paper investigates structure set prime ideals satisfying subcategory. We try to calculate members this for consisting extension in two given subcategories. Moreover, we classify sets rings with small dimension.

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...

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the étale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois represen...

We study relations between the Serre dimension defined as growth of entropy functor and global Bridgeland stability conditions due to Ikeda–Qiu. A fundamental inequality infimum dimensions is proved. Moreover, we characterize Gepner type on fractional Calabi–Yau categories via dimension, classify triangulated lower than one with a condition.

We define A∞-bimodules similarly to Tradler and show that this notion is equivalent to an A∞-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A∞-functors are defined via A∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A∞-category A over a field k adm...

We define A∞-bimodules similarly to Tradler and show that this notion is equivalent to an A∞-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A∞-functors are defined via A∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A∞-category A over a field k adm...

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 2 terms as E 2 p,q = HP p (A; HP q (B, A)) =⇒ HP p+q (B) and E 2 p,q = HP p (A; Kq(B, A)) =⇒ Kp+q(B).

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity ε, ut +F (u )x = ε(B(u)ux)x. When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that uε converges to a solution of the inviscid system before the formation of shocks if th...

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).