Hereditary deltas (Q1330016)

A small category is called a delta if it is skeletal and the only endomorphisms are the identities. A small category \({\mathcal C}\) is \(R\)- hereditary if the functor category \((\text{Mod} R)^{\mathcal C}\) has global dimension at most one, where \(\text{Mod} R\) denotes the category of left \(R\)-modules, \(R\) is a ring with an identity. In this paper the authors characterize all deltas that are \(R\)-hereditary. It is not hard to see that if \({\mathcal C}\) is a discrete category then \(\text{gl} \dim (\text{Mod} R)^{\mathcal C} = \text{gl} \dim R\). In this case \({\mathcal C}\) is \(R\)-hereditary if and only if \(R\) is a hereditary ring, i.e., \(\text{gl} \dim R \leq 1\). Hence the main result is interesting.