Nonnoetherian geometry and toric superpotential algebras

Details Nonnoetherian geometry and toric superpotential algebras Nonnoetherian geometry and toric superpotential algebras

2011-09-21

arxiv.org/abs/1109.4601v2

Mathematics

Algebraic Geometry

21 pages. A section on toric superpotential algebras has been added. Comments welcome!

Scientific paper

We show that nonnoetherian subalgebras of affine coordinate rings can be realized geometrically as affine varieties that contain positive dimensional subvarieties which are identified as closed points. We introduce the notion of the 'geometric dimension' of a point, and characterize the unique largest subset for which the closed points are zero dimensional. The following application is then considered: Let A be a non-cancellative superpotential algebra of a brane tiling quiver Q, and suppose a cancellative algebra A' (a 'superconformal quiver theory') is obtained by contracting ('Higgsing') an adequate set of arrows in Q to vertices. We show that under a certain new isomorphism, the nonnoetherian center Z of A will be generated by the intersection of the cycles in Q, and birational to the noetherian ring generated by the union of these cycles (the 'mesonic chiral ring'). Further, we show that the latter ring will be isomorphic to the center of A', and therefore Z will be an affine toric Gorenstein singularity with a positive dimensional closed point.

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