The cyclic homology of the group rings.
The derived functors of and protorsion modules
The dimension of a quasi-hereditary algebra
The dimension of the derived category of elliptic curves and tubular weighted projective lines
We show that the dimension of the derived category of an elliptic curve or a tubular weighted projective line is one. We give explicit generators realizing this number, and show that they are in a certain sense minimal.
The Dirac complex on abstract vector variables: megaforms.
The Divisor Class Group of Ordinary and Symbolic Blow-ups.
The duality of Auslander-Reiten quiver of path algebras
Let be a finite union of Dynkin quivers, a finite abelian group, the generalized McKay quiver of and the Auslander-Reiten quiver of . Then acts functorially on the quiver . We show that the Auslander-Reiten quiver of coincides with the generalized McKay quiver of .
The duality theorem for twisted smash products of Hopf algebras and its applications
Let denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for , and as an application we establish the relationship between the homological dimensions of and A if H and its dual H* are both semisimple.
The elliptic dynamical quantum group as an -Hopf algebroid.
The Endomorphism Ring Theorem for Frobenius Extensions.
The Equivalence of Various Generalizations of Group Rings and Modules.
The existence of envelopes
The existence of Galois sets
The existence of identity of orthodox semigroup rings.
The existence of relative pure injective envelopes
Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.
The Exterior Derivation and the Orders of Differential Forms on Contractible Algebras.
The Extraordinary Derived Category.
The finite dimension property of the irreducible representations of Noetherian -Lie algebras.
The flat dimensions of injective modules.
The free -ring is a graded -ring.