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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.

alt text http://img696.imageshack.us/img696/5738/16817893.png

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.

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.

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.

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.

alt text http://img696.imageshack.us/img696/5738/16817893.png

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.

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 to 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.

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.