Let $${\mathfrak {g}}$$ be a simple complex Lie algebra. In this paper we study the BGG category $${\mathcal {O}}_q$$ for quantum group $$U_q({\mathfrak {g}})$$ with q being root of unity in field K characteristic $$p >0$$ . We first consider modules and prove Steinberg tensor product theorem them. This result reduces problem determining corresponding irreducible characters to same finite subse...

We describe new classes of noetherian local rings R whose finitely generated modules M have the property that ToriR(M,M)=0 for i≫0 implies has finite projective dimension, or ExtRi(M,M)=0 dimension injective dimension.

Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

Let R be a ring and M a right R-module. Ng (1984) defined the finitely presented dimension f p dim M of M as inf n there exists an exact sequence Pn+1 → Pn → · · · → P0 → M → 0 of right R-modules, where each Pi is projective, and Pn+1 Pn are finitely generated . If no such sequence exists for any n, set f p dim M = . The right finitely presented dimension r f p dim R of R is defined as sup f p ...

Recently, Yekutieli introduced projective dimension, injective dimension and flat of DG-modules by generalizing the characterization ordinary modules vanishing Ext or Tor-groups. In this paper, we introduce a DG-version projective, resolution for over connective DG-algebra which are different from known resolutions. An important feature these resolutions is that, roughly speaking, “length” give...