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Register a new userCentral extensions of some linear cycle sets
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For each member $mathcal{A}$ of a family of linear cycle sets whose underlying abelian group is cyclic of order a power of a prime number, we compute all the central extensions of $mathcal{A}$ by an arbitrary abelian group.
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For a central perfect extension of groups $A rightarrowtail G twoheadrightarrow Q$, we study the maps $H_3(A,mathbb{Z}) to H_3(G, mathbb{Z})$ and $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$ provided that $Asubseteq G$. First we show that the image of $H_3(A, mathbb{Z})to H_3(G, mathbb{Z})/rho_ast(Aotimes_mathbb{Z} H_2(G, mathbb{Z}))$ is $2$-torsion where $rho: A times G to G$ is the usual product map. When $BQ^+$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$.
Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to ker(p). This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.
We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in a subsequent paper to study twisted deformation quantization on algebraic varieties.
We consider a class of extensions of both abstract and pseudocompact algebras, which we refer to as strongly proj-bounded extensions. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we show that the finiteness of the big left finitistic dimension is preserved by strongly proj-bounded extensions. In order to construct examples, we describe a new class of extensions of algebras of finite relative global dimension, which may be of independent interest.
This paper studies the K-theory of categories of partially cancellative monoid sets, which is better behaved than that of all finitely generated monoid sets. A number of foundational results are proved, making use of the formalism of CGW-categories due to Campbell and Zakharevich, and numerous example computations are provided.
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