arXiv subjects
Publications and source records attributed to S. Paul Smith.
We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.
We examine maps between noncommutative projective spaces. A surjection of graded rings A-->A/J induces a closed immersion Proj(A/J)-->Proj(A). A homomorphism f:A-->B between graded rings induces an affine map U --> Proj(A) from a non-empty open subspace U of Proj(B). If A^{(n)} is the n-th Veronese subalgebra of a graded ring A there is a map Proj(A)-->Proj(A^{(n)}); we identify open subspaces on which this map is an isomorphism. Applying these results when A is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.
This paper examines various kinds of subspaces of the non-commutative spaces that are modelled on quasi-projective commutative schemes. It is shown how intersections and unions of weakly closed subspaces, closed subspaces, their weakly open and open complements, behave. Intersections with effective divisors are also examined, and so are closed points.
This paper introduces a notion of integrality that is suitable for non-commutative varieties. It is compatible with the usual notion of integrality for schemes. The function field and generic point of a non-commutative integral space are also defined. These agree with the usual notion for noetherian schemes. It is shown that these notions behave as one would wish. For example, various non-commutative analogues of projective space are integral.