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Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow.

I'm trying to get some intuition about how much the global dimension can grow when we quotient by some homogeneous ideal of relations I. In general, if Q is acyclic (is this necessary?), then the global dimension is bounded by the number of vertices, but I want something that uses information from I. For simplicity, let's assume that there is at most 1 edge between any two vertices. If I add the relation that a single path of length 2 is 0, then the global dimension goes up to 2, and the same is true if 2 is replaced by any r>2 (right?). I can get higher global dimensions by the following: take some consecutive arrows $a_1, a_2, \dots, a_n$, and require that each path of length 2 $a_{i+1} a_i$ is 0, then the global dimension goes up to n-1.

The way I am trying to picture this is by thinking of projective modules as flowing water which gets stopped by some rock placed where the relations are, and seeing how many times the flow needs to restart before it can reach the end (I don't know if this is a useful comment.)

Anyway, here is my question: is there some simple way to bound the global dimension of Q/I assuming that Q is acyclic and no multiple edges between any two vertices? In this case, we're only allowed to say that certain paths are 0, so I am suspecting this has something to do with "number of overlaps." My guess would be something like, define an overlap to be when an initial segment of one path coincides with an ending segment of another path, and then the global dimension should be less than or equal to number of overlaps + 2.

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up vote 13 down vote accepted

If you are only considering monomial algebras (that is, if you are generating the ideal I by paths) then your intuition about overlaps is correct, once you see which overlaps you need to consider. There is a paper by Bardzell (The alternating behaviour of monomial algebras) where he constructs explicitely a projective resolution of the quotient algebra as a bimodule over itself (whose length bounds the gldim of the algebra) which is constructed precisely by considering overlaps.

By the way, if the graph is not acyclic, then the global dimension can very well be infinite. The simplest example is a quiver with one vertex and a loop, and the ideal geberated by the square of the loop.

Later: Let me be more explicit about what I meant by "once you see which overlaps you need to consider"... Consider the quiver $Q$ which is an oriented path with 15 arrows, and let $I$ be ideal generated by all paths in $Q$ of length 8. There are then 8 minimal relations, they all overlap, but if you work through the construction of minimal projective resolutions of the simple modules of $kQ/I$ you'll see that most of those overlaps do not matter, and that the global dimension is $3$ in this case. You can play this game with longer paths, as long as you divide by not too short relations.

It is not too hard to single out precisely which are the overlaps that do matter when the quiver is a path. The general case is not horribly more complicated, yet it always manages to confuse me.

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