Narrow Results

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the $0$th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the $E_2$-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

In this paper we determine all the Hopf $q$-brace structures on rank one pointed Hopf algebras and compute the socle of each one of them. We also identify which among them are Hopf skew-braces. Then we determine when two Hopf $q$-brace structures on rank one pointed Hopf algebras are isomorphic, and, finally, we compute all the weak braiding operators on these Hopf algebras.

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽_q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then a_n(L) is the number of restricted subalgebras H ⊂ L such that dim_𝔽q L/H = n, n ≥ 0. We compute the numbers a_n(F_d) explicitly and find asymptotics, where F_d is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

In this paper we describe the structure of prime and semiprime R-modules M such that R/Ann_R(M) is artinian. The obtained results are then applied to describe the structure of prime and semiprime right comodules over a coring C under some assumption, where a right comodule M is said to be prime (semiprime) if the corresponding left *C-module is prime (semiprime). Finally we apply the results to coalgebras over commutative rings.

We introduce the notion of right strictly quasi-finite coalgebras, as coalgebras with the property that the class of quasi-finite right comodules is closed under factor comodules, and study its properties. A major tool in this study is the local techniques, in the sense of abstract localization.

The superposition of cooperations satisfies the well-known clone axioms (C1), (C2) and (C3). We define terms for indexed coalgebras of type τ, cooperations induced by those terms, and prove that the set of all induced cooperations forms a clone. This clone is equal to the clone of all cooperations generated by the fundamental cooperations of an indexed coalgebra. Finally, we introduce the concept of rational equivalence for coalgebras and determine all two-element coalgebras up to rational equivalence.