Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMorita theory for non-commutative noetherian schemes
73
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study equivalences between the categories of quasi-coherent sheaves on non-commutative noetherian schemes. In particular, give a new proof of Caldararus conjecture about Morita equivalences of Azumaya algebras on noetherian schemes. Moreover, we derive necessary and sufficient condition for two reduced non-commutative curves to be Morita equivalent.
rate research
Read More
In this article we develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra.
In this paper, we develop a geometric approach to study derived tame finite dimensional associative algebras, based on the theory of non-commutative nodal curves.
In arXiv:0907.3784, we introduced a variant of non-commutative Donaldson-Thomas theory in a combinatorial way, which is related with topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide an alternative definition in a geometric way, (2) show that the two definitions agree with each other and (3) compute the invariants using the vertex operator method, following Okounkov-Reshetikhin-Vafa and Young. The stability parameter in the geometric definition determines the order of the vertex operators and hence we can understand the wall-crossing formula in non-commutative Donaldson-Thomas theory as the commutator relation of the vertex operators.
Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that the non-commutative moduli space X(m,S) of (twisted) framed flat R-vector bundles of rank m on a decorated surface S is birationally identified with the moduli spaces of (twisted) flat line bundles on a spectral surface $Sigma_Gamma$ assigned to certain bipartite graphs $Gamma$ on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry a canonical non-commutative Poisson structure. The result above just means that the space X(m, S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group of S. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We develop a parallel dual story of non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry a canonical non-commutative 2-form. The dual non-commutative moduli space A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group of S. The non-commutative cluster A-coordinates on the space A(m,S) are expressed as ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves on surfaces, and use them to give an alternative microlocal proof of the above results.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
