arXiv subjects
Publications and source records attributed to John William MacQuarrie.
We give a practical, algorithmic method to calculate minimal projective resolutions of simple modules for a finite dimensional incidence $k$-algebra $Λ$, where $k$ is a field. We apply the method to the calculation of Ext groups between simple $Λ$-modules, Hochschild cohomology groups $\HH^i(Λ, Λ)$, and singular cohomology groups of finite $T_0$ topological spaces with coefficients in $k$.
We prove that if $B\subseteq A$ is an extension of finite dimensional algebras such that the projective dimension of $A/B$ as a $B$-bimodule is finite, if $A$ has finite finitistic dimension, then so does $B$. We exhibit examples demonstrating that the algebra $B$ appearing in such an extension can be more complicated than $A$.
We define the path coalgebra and Gabriel quiver constructions as functors between the category of $k$-quivers and the category of pointed $k$-coalgebras, for $k$ a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel $k$-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and algebras in terms of path objects are unique, giving an application to homogeneous algebras.