## Strongly tilting truncated path algebras

For any truncated path algebra Λ, we give a structural description of the modules in the categories $${\mathcal{P}^{

For any truncated path algebra Λ, we give a structural description of the modules in the categories $${\mathcal{P}^{

For any truncated path algebra Λ, we give a structural description of the modules in the categories $${\mathcal{P}^{

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer-Sturmfels in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of toric algebras in dimension three by associating a superpotential and a notion of consistency...

Published in Advances in Mathematics

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight θ of Q, the moduli space M ( Q , d ) θ s s of θ-semi-stable d-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight...

Published in Advances in Mathematics

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0 → U → M → V → 0 with indecomposable ends that a...

Published in Advances in Mathematics

We solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated Tits integral quadratic forms, and derive some consequences.