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Green walks in a hypergraph. (English) Zbl 0938.16008

A finite cycle hypergraph \(H\) consists of finite sets \(C\) and \(E\), together with a bijection \(\pi\colon C\to C\) and a surjection \(\varepsilon\colon C\to E\). The vertices of \(H\) are the cycles (\(\pi\)-orbits) of \(C\), the edges of \(H\) are the elements of \(E\), and \(\varepsilon\) defines the incidence between vertices and edges. Given \(H\) and a complete discrete valuation ring \(R\), the author constructs a hereditary \(R\)-order \(\Gamma_H\) and a Bäckström order \(\Lambda_H\) in \(\Gamma_H\). Then he gives an abstract characterization of the \(R\)-orders \(\Lambda_H\) obtained in this way. More generally, the author defines a class of semiperfect rings (called prohereditary) and certain semiperfect subrings of prohereditary rings (called rings with Green walk). Such a pair \(\Lambda\subseteq\Gamma\) is shown to give rise to a finite cycle hypergraph. Moreover, projective resolutions can be constructed by a generalization of Green’s “walk around the Brauer tree”.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
20C11 \(p\)-adic representations of finite groups
05C65 Hypergraphs
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16L30 Noncommutative local and semilocal rings, perfect rings
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