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noncommutative space as a cover

Noncommutative algebraic geometry studies objects much more general than a single algebra: one wants to glue algebras to more general “systems”. This idea has been around from 1980-s. Of course, one needs a good equivalence among systems. One idea is to replace the algebra by a module and glue categories of modules and then look if the categories are equivalent. But the categories of modules loose the track of the Morita equivalence class, which one would like to keep. For example, one wants to define morphisms between such systems as some class of geometric morphisms, but when one restricts to algebras one would like to have only morphisms of algebras, not the bimodules corresponding to nonaffine morphisms.

This is nicely achieved in a formalism which significantly refines the standard techniques of gluing via corings sketched in chapter 2 of

STEP 1: Category of Covers

Given a kk-algebra RR, a RR-coring M=(M,δ M,ϵ M)M = (M,\delta^M,\epsilon^M) is a comonoid in the category of RR-bimodules (i.e. RR opR\otimes R^{op}-modules). One defines a category of covers as the category of pairs (R,M)(R,M) where RR is a kk-algebra and MM is an RR-coring; one requires that MM is faithfully flat as the right RR-module i.e. tensoring M RM\otimes_R is exact and faithful functor. The morphisms (R 1,M 1)(R 2,M 2)(R_1,M_1)\to (R_2,M_2) are given by pairs of algebraic maps in opposite direction, i.e. an algebra map f R:R 2R 1f_R : R_2\to R_1 and a bimodule map f M:M 2M 1f_M:M_2\to M_1 satisfying the obvious compatibility conditions with the rest of the structure. This defines the category of covers, Cover kCover_k.

One also defines the category of quasicoherent modules QCoh(R,M)QCoh(R,M) as the category of left MM-comodules, i.e. left RR-modules XX together with the coaction XM RXX\to M\otimes_R X.

STEP 2: Introducing structure morphism s Cs_C

The objects of the category of “space covers” Cover k spCover_k^{sp} is the category of pairs (C,s C)(C,s_C) where C=(R,M)=(R,M,δ M,ϵ M)C = (R,M) = (R,M,\delta^M,\epsilon^M) is a cover and s C:R kRMs^C: R\otimes_k R\to M is a surjective homomorphism of coalgebras in the category of RR-bimodules. The morphisms in Cover k spCover_k^{sp} are the morphisms in Cover kCover_k compatible with “structure morphisms” s Cs^C.

STEP 3: Equivalence relation for the morphisms

Still some morphisms induce the same inverse image functors for QCohQCoh categories. To eliminate this, one calls two morphisms f,g:(C,s C)(C,s C)f,g:(C,s_C)\to (C',s_{C'}) equivalent if for any element u=x αy αu = \sum x_\alpha\otimes y_\alpha in Ker(s C)Ker(s_C) the tensor product (fg)(u)=0(f\otimes g)(u) = 0 and (gf)(u)=0(g\otimes f)(u)=0. After moding out this equivalence we obtain Cover˜ k sp\widetilde{Cover}^{sp}_k

STEP 4: Localizing on refinements

Any cover (R,M)(R,M) in Cover kCover_k together with a monomorphism of kk-algebras RSR\hookrightarrow S induces a cover (S,S RM RS)(S,S\otimes_R M\otimes_R S) together with a canonical morphism (S,S RMS)(R,M)(S,S\otimes_R M\otimes S)\to (R,M) in Cover kCover_k. Morphisms of this type are called refinement morphisms; they are closed under composition and pullback. Thus one can define the localized category Cover k[Ref 1]Cover_k[Ref^{-1}]; moreover, (S,S RMS)(S,S\otimes_R M\otimes S) can be canonically equipped with a structure morphism and we can in this vain obtain the category of noncommutative spaces NSpace k:=Cover˜ k sp[Ref 1]NSpace_k := \tilde{Cover}^{sp}_k[Ref^{-1}].

Properties

Theorem. (AR/MK, sec.2.2) The category of separated quasi-compact schemes over kk is equivalent to a full subcategory of NSpace kNSpace_k. The category Alg k opAlg^{op}_k is also equivalent to a full subcategory of NSpace kNSpace_k. Category NSpace kNSpace_k has finite limits.

Theorem. (AR/MK, sec.2.2) Construction (B,M)QCoh(B,M)(B,M)\to QCoh(B,M) extends to a functor from the category NSpace kNSpace_k to the category of abelian kk-linear categories. For separated quasi-compact schemes, it gives usual quasi-coherent sheaves of commutative algebraic geometry. For associative algebras, it gives categories of left modules.

Revised on November 19, 2011 17:41:15 by Zoran Škoda (31.45.198.4)