Bimodules in geometry

Question

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-Neumark theorem says that the category of measurable spaces is contravariantly equivalent to the category of commutative von Neumann algebras). Even though we do not have a complete geometric description for the noncommutative case, we can still use geometric intuition from the commutative case effectively.

A generalization of this idea is given by Grothendieck's relative point of view, which says that a morphism of rings f: A → B should be regarded geometrically as a bundle of spaces with the total space Spec B fibered over the base space Spec A and all notions defined for individual spaces should be generalized to such bundles fiberwise. For example, for von Neumann algebras we have operator valued weights, relative L^p-spaces etc., which generalize the usual notions of weight, noncommutative L^p-space etc.

In noncommutative geometry this point of view is further generalized to bimodules. A morphism f: A → B can be interpreted as an A-B-bimodule B, with the right action of B given by the multiplication and the left action of A given by f. Geometrically, an A-B-bimodule is like a vector bundle over the product of Spec A and Spec B. If a bimodule comes from a morphism f, then it looks like a trivial line bundle with the support being equal to the graph of f. In particular, the identity morphism corresponds to the trivial line bundle over the diagonal.

For the case of commutative von Neumann algebras all of the above can be made fully rigorous using an appropriate monoidal category of von Neumann algebras.

This bimodule point of view is extremely fruitful in noncommutative geometry (think of Jones' index, Connes' correspondences etc.)

However, I have never seen bimodules in other branches of geometry (scheme theory, smooth manifolds, holomorphic manifolds, topology etc.) used to the same extent as they are used in noncommutative geometry.

Can anybody state some interesting theorems (or theories) involving bimodules in such a setting? Or just give some references to interesting papers? Or if the above sentences refer to the empty set, provide an explanation of this fact?

Answer 1

In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.

When you think in this language, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's Functoriality for Lagrangian correspondences in Floer homology.

Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.

The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.

This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).

Answer 2

The Fourier-Mukai transform comes from a bimodule: the Poincaré bundle. Let $A$ be an abelian variety, the Poincaré bundle $\mathcal{P}$ is a vector bundle on $A \times \hat{A}$ coming from the fact that the points in the dual abelian variety $\hat{A}$ parametrize line bundles on $A$ ($\mathcal{P}$ is the universal family). In the Fourier-Mukai construction, $\mathcal{P}$ is used as a $\mathcal{O}_A$-$\mathcal{O}_{\hat{A}}$-bimodule to produce a functor between the derived categories of coherent sheaves on $A$ and $\hat{A}$ via a push-pull construction.

Answer 3

Here's a theorem from derived algebraic geometry: if A and B are A_∞ algebras (think associative algebras) then giving an A-B-bimodule is the same as giving a functor from {right A-modules} to {right B-modules} which preserves colimits (equivalently, has a right adjoint). The correspondence sends _AM_B to the functor – ⊗_{A A}M_B. Under this correspondence, tensor product of bimodules over the middle algebra is realized by composition of functors.

Answer 4

The paper

Adam Nyman. The Eilenberg-Watts theorem over schemes, available at arXiv

studies the connection between cocontinuous functors $Qcoh(Y) \to Qcoh(X)$, which are there called bimodules, and $Qcoh(X \times Y)$ in detail.

Answer 5

Let S be a scheme of positive characteristic p and X an S-scheme. If F denotes the absolute Frobenius, then we can pull back O_X via F. In the affine case, say S=spec k, X=Spec R, this corresponds to tensoring R over F with R, hence we get a bimodule: for r,f\in R we get fr=r^pf. This bimodule is the beginning of the theory of "F-unit crystals" and a positive characteristic version of the Riemann-Hilbert correspondence! See for example this survey by Emerson-Kisin.

Answer 6

Noncommutative algebraic geometry lives in the nature of bimodules.

There are some work of Grothendieck flavor noncommutative algebraic geometry. Bimodules naturally came in this story.

1.One of the most important concept in NCAG is monad and comonad. From the Barr-Beck theorm(categorical version of Grothendieck flat descent)for noncommutative scheme. We have the following theorem:

Let X be a quasi compact and quasi separated (noncommutative)scheme, u[i]:U[i]--->X is an affine cover

U=coproduct of U[i]--->X A[U]=product of Ou(U[j]),and then Qcoh(U)=coproduct Qcoh(U[i])=A[U]-mod

Then, according to the Beck's theorem. We have Qcoh(X)=G[f]-Comod where G[f]=(M[f] tensoring over A[U], delta) is a comonad on Qcoh(U).

Comonad structure as follows: M[f] tensor over A[U] M[f]<------M[f]---->A[U]

In particular, if the scheme X is semiseparated(say algebraic varieties), M[f] is a A[U] tensor A[U]^op module(it is A[U]-bimodule]. In other words, G[f] is a coalgebra in the monoidal category of A[U] tensor A[U]^op -modules(A[U]-bimodule)

Reference of Beck's theorem for noncommutative scheme(I mentioned above) is

Maxim Kontsevich and Alexander.L.Rosenberg Noncommutative spaces and flat descent.(This paper is in Max Plank preprint series, it is online)

1. Another reference is Maxim Kontsevich and A.L.Rosenberg Noncommutative smooth space http://arxiv.org/PS_cache/math/pdf/9812/9812158v1.pdf

Bimodule is used to define the Covers for noncommutative space

3.If one need to define differential operator in general noncommutative space(such as abelian category), in particular, affine scheme. Noncommutative D-module(in particular, quantum D-module), he needs the differential bimodule. The reference is: V.A.Lunts and A.L.Rosenberg Differential Calculus in Noncommutative Algebraic geometry I and II(These are also in Max-Plank preprint series)

1. One needs Bimodule to develop the machinary to treat noncommutative grassmannian type space and the tannaka formalism in noncommutative nonsymmetric monoidal category. Reference: M.Kontsevich and A.L.Rosenberg Noncommutative Grasmannian and related construction.(MPIM preprint series)

Answer 7

In a paper from 1985, Raeburn and J. Taylor describe how to view all elements of H^2(X,Gm) (etale cohomology) as coming from non-unital Azumaya algebras. The construction relies on bimodule theory for these algebras.

Answer 8

Well, in commutative algebra, you have the fact that any left module is also a right module, so the notion of bimodule can be considered a bit redundant. Any time that an algebraic geometer (or other) uses a module at all, it's generally an A-A bimodule, but we don't think of that, because it's not different form a left A-module or a right A-module, like it is in the noncommutative case.