What I have been doing since two months ago is mathematics and sports.  I am now training in marathon and a little bit triathlon. I will attend a 5K-race next weekend!

I feel much better now,much more energy which is good to mathematics. I will write something on quivers, perverse sheaves and their interaction with noncommutative geometry soon.

1. Holonomic D-modules rephrased in noncommutative algebraic geometry and properties.

2. Spectrum of category of coherent sheaves on and reconstruction theorem.

3. Locality theorem, in particular for commutative schemes and holonomic D-modules.

4. Abelian induction problem, representations of finite dimensional Lie algebra.

5. Derived category of coherent sheaves on , t-structures and gluing machinery

6. Constructible sheaves on and perverse sheaves.

An important realization is the spectral theory of “spaces” represented by svetle abelian categories [R,Ch3],which was started in the middle of the eighties. Below follows a brief outline of some of its basic notions and facts.

4.1. Topologizing subcategories and the spectrum .

A full subcategory of an abelian category is called topologizing if it is closed under finite coproducts and subquotients. For an object , we denote by the smallest topologizing subcategory of containing . One can show that objects of are subquotient of finite coproducts of copies of . The spectrum of the “space” consists of all non zero such that for any nonzero suboject of . We endow with the preorder which is called the specialization preorder.

If is a simple object, then the objects of are isomorphic to finite direct sums of copies of and is a minimal element of , if is the category of modules over a commutative unital ring , then the map: is an isomorphism between the prime spectrum of with specialization preorder and .

4.2. Local “space”

An abelian category is called local if it has the smallest nonzero topologizing suncategory. It follows that this subcategory coincides with

Geometry of noncommutative “spaces” and schemes

3.7. Theorem

(a): A pretopology on is subcanonical iff for any presheaf of sets on (in other words,”descent” pretopologies on are precisely subcanonical pretopologies). In this case, ,where is the sheaf on associated with the presheaf and is natural full embedding.

(b). If a pretoplogy is of effective descent, then the above embedding becomes a categorical equivalence.

This theorem says that, roughly speaking, the category of quasi coherent presheaves knows itself which pretopologies to choose. It also indicates where one should look for a correct noncommutative version of the category  (of sheaves of sets on the fpqc site of commutative affine schemes): this should be the category of sheaves of sets on the presite ,where is a pretopology of effective descent. From the minimalistic point of view, the best choice would be the finest pretopology of effective descent. But there is a more important consideration. The main role of a pretopology is that it is used for gluing new “spaces”. The pretopology that seems to be the most relevant for Grassmannians(in particular, for noncommutative projective space) and a number of other smooth noncommutative spaces constructed in [KR5] is the smooth topology introduced in [KR2].

The theorem is quite useful on a pragmatical level. Namely, if   is a sheaf of sets on for an appropriate pretopology of effective descent and is a presheaf of sets on such that its associated sheaf is isomorphic to , and is an exact sequence of presheaves with and representable, then the category (hence the category ) is constructively described (unique up to equivalence) via pair of k-algebra representing . This consideration is used to describe the categories of quasi coherent sheaves on noncommutative “spaces”

3.8. Noncommutative stacks.

There is one more important observation in connection with this theorem: categories which appear in noncommutative algebraic geometry are categories of quasi coherent sheaves on noncommutative stacks.

Geometry of noncommutative “spaces” and schemes

3.4 Theorem

The category is equivalent to the category whose objects are continuous monads on the category and morphisms are conjugacy classes of monads morphisms.

If , then the category is equivalent to the category whose objects are associative unital rings and morphisms are conjugacy classes of ring morphisms. If , then is equivalent to the category whose objects are monoids and morphisms are conjugacy classes of monoids morphism. This shows that the choice of base “space” influences drastically the rest of the story.

3.5. Locally affine relative “space”. Locally affine -”space” are defined in an obvious way, once a notion of a cover(a quasi-pretopology)is fixed. We introduce several canonical quasi-pretopologies on the category . Their common feature is the following: if a set of morphisms to is a cover, then the set of their inverse image functors is conservative and all inverse image functors are exact in a certain mild way. If, in addition, morphisms of covers are continuous, has a finite affine cover, and the category has finite limits, then this requirement suffices to recover the object from the covering date uniquely up to isomorphism(i.e the category is reovered uniquely up to equivalence)via “flat descent”

3.6 “Spaces” determined by presheaves of sets on .

By definition, the category of noncommutative affine k-schemes is the category opposite to the category of associative unital k-algebras; so that presheaves of sets on are functors from to Sets. The presheaves of sets on appeared in our work with Maxim Kontsevich, for the first time in order to introduce noncommutative projective spaces. It was an attempt to imitate the standard commutative approach realizing schemes(and more general spaces) as sheaves of sets on the category of affine schemes endowed with an appropriate Grothendieck  pretopology. It turned out that it is not clear a priori what an appropriate pretopology in the noncommutative case is: Zariski pretopology is irrelevant, because the noncommutative projective space is not a scheme- it does not have an affine Zariski cover. Flat affine covers seemed to be a as a natural choice, but, they do not form a pretoplogy-invariance under base change fails. Similar story with Grassmannians and other analogs of commutative constructions. The elucidation of this problem is as follows. Consider the fibered category   with the base whose fibers are categories of left modules over corresponding algebras. For every presheaf of sets on , we have the fibered category induced by along the forgetful functor . The category of quasi coherent sheaves on the presheaf is defined as the category opposite to the category of cartesian sections of the fibered category . For a pretopology on , we define the subcategory of quasi-coherent sheaves on .

Geometry of noncommutative “spaces” and schemes

3. Pseudo-geometric start.

The pseudo-geometric noncommutative landscape sketched above is a natural point of departure, by simple reason that it includes most examples of interest. Instead of trying to impose, from the very beginning, general notions of spaces and morphism of spaces, which absorb all the known case, we approach these notions by studying algebraic geometry in certain key pseudo-geometric settings, which are simple enough to not to get lost and, at the same time, sufficient to obtain a rich theory and to see what one should expect or look for in more sophisticated pseudo-geometries.

3.1. “Spaces” represented by categories. In the very first, in a sense the simplest, setting of this kind, “spaces”are represented by svetle(equivalent to small)categories and morphisms of “spaces” are isomorphism classes of(inverse image)functors between the corresponding categories. This defines the category of “spaces”. A morphism of “space” is called continuous if its inverse image functor has a right adjoint (called a direct image functor), and it is called flat, if ,in addition, the inverse image functor is left exact(i.e. preserves finite limits). A continuous morphism is called affine if its direct image functor is conservative(i.e. reflects isomorphism) and has a right adjoint. These notions(introduced in [R])unveil unexpectedly rich algebraic geometry,more precisely, geometries, living inside of . They appear as follows.

3.2. Continuous monads. Fix a “space” S such that the category has cokernels of paris of arrows. We consider of continuous endofunctors of . It is a monoidal category with repsect to the composition of functors whose unit object is the identical functor. The monads in this category are called continuous monads on . In other words, continuous monads on are pairs ,where is continuous functor and is a functor morphism such that  for a unique morphism called the unit of the monad . A monad morphism is given in a natural way. This defines the category of continuous monads on .

If , then the category is naturally equivalent to the category of associative unital rings. If is the category of quasi coherent sheaves on a scheme , then is equivalent to the category of quasi coherent sheaves of rings on endowed with a morphism of sheaves of rings. In particular, the sheaf of rings of (twisted)differential operators can be regarded as a monad on . If is the category of sets, then the category is equivalent to the category of monoids in the usual sense.

3.3. Relative affine “space”. Give a space , we define the category of affine -space as the full subcategory of whose objects are pairs ,where is an affine morphism.

Geometry of noncommutative “spaces” and schemes

2.6 “Spaces” defined by presheaves of sets on the category of noncommutative affine schemes.

The category of affine noncommutative k-schemes is the category opposite to the category of associative unital k-algebras. Some of the important examples of noncommutative “space”, such as noncommutative Grassmannians, flag varieties and many others [KR1],[KR2],[KR3], are defined in two steps. The first step is a construction of a presheaf of sets on (i.e. a functor from category of unital associative k-algebras to the category of sets). In commutative algebraic geometry, the second step is taking the associated sheaf with respect to an appropriate(fpqc or Zariski)topology on . In noncommutative geometry, we assign, instead, to every presheaf of sets on a fibered category whose fibers are categories of modules over k-algebras and define the category of quasi-coherent sheaves on this presheaf as the category opposite to the category of cartesian sections of this fibered category [KR4]. The category of quasi coherent presheaves represents the “space” corresponding to the presheaf of sets.

2.7. Commutative “spaces” ,which “live” in symmetric monoidal categories.

After the formalism of Tannakian categories appeared at the end of the sixties-beginning of the seventies [Sa],[DeM], and super-mathematics approximately at the same time, the idea of mathematics(or at least algebra and geometry),which uses general symmetric monoidal categories, instead of the symmetric monoidal category of vector spaces, became familiar. In [De], Deligne presented a sketch of a fragment of commutative projective geometry  in symmetric monoidal k-linear abelian categories as a part of his proof of the characterization of rigid monoidal abelian categories having a fiber functor.

Manin defined the(category of coherent sheaves on the)Proj of a commutative -graded algebra in a symmetric monoidal abelian category endowed with a fiber functor [M1]using, once again, the Serre’s description of the category of coherent sheaves on a projective variety as its definition.

2.8. Quantized enveloping algebras and algebraic geometry  in braided monoidal categories.

While working(in 1995) on a quantum analog of Beilinson-Bernstein localization construction, it was discovered  that “spaces” of noncommutative algebraic geometry could be something different from just abelian or Grothendieck categories. In this particular situation, the natural action of the quantized enveloping algebra if a semisimple Lie algebra on its quantum base affine space becomes differential only if the whole picture is put into the monoidal category of -graded modules endowed with a braiding determined by the Cartan matrix of the Lie algebra (see [LR2],[LR3],[LR4]) . This list(which is far from being complete)shows that the range of objects-spaces and morphisms of spaces, of noncommutative algebraic geometry is considerably larger than the range of objects of commutative algebraic geometry.

One of the benefits of the pseudo-geometric viewpoint in noncommutative algebraic geometry  is a considerable increase of its range. Roughly, the picture is as follows.

2.1. Spaces and algebras. The duality between compact topological spaces and commutative unital algebra is a fundamental fact of functional analysis discovered by I.M.Gelfand in the late thirties. A.Connes extended formally this duality to the noncommutative setting identifying “noncommutative space” with noncommutative algebras. This eventually led to the creation of noncommutative differential geometry [C1],[C2]. Following Connes’s example. V.Drinfeld [Dr] defined the category of noncommutative affine schemes(he called them “quantum space”) in a similar way, as the category dual to the category of unital associative algebras,forcing to the noncommutative case the duality

[algebras <--->affine schemes]

of commutative algebraic geometry.

2.2 Noncommutative Proj. Noncommutative projective spaces were introduced(by Manin’s suggestion)via a formal extension of the Serre’s description of the category of quasi-coherent sheaves on a projective variety [S]: the category of quasi coherent sheaves on the projective spectrum of an associative graded ring R is the quotient category of the category of graded R-modules by the subcategory of locally finite ones(this approach was further developed in [V1],[V2],[A2],[AZ],[OW], and in a number of other works)

Thus, a noncommutaive projective space is represented by a category , which is regarded as category of quasi coherent sheaves on . This point of view is well adapted to the affine case: for any associative ring R, the category of quasi coherent sheaves on the corresponding affine schemes is identified with the category of left R-modules

2.3. “Spaces” represented by abelian categories. From the perspective of the above mentioned developments, a point of view which looked plausible at the end of the eighties(and was later, after appearance of [R1] and [R], adopted by most mathematicians working in the area)is that “spaces” of noncommutative algebraic geometry are represented by abelian categories(thought as their category of quasi coherent or coherent sheaves). If X and Y are “spaces” represented by abelian categories, respectively and , then morphism from X to Y are isomorphism classes of additive functors called inverse image functors of the morphism they represent.

2.4 “Spaces” represented by triangulated categories. Another viewpoint motivated in the first place by representation theory of reductive groups, and later (around 1993) by problems of mathematical physics(-homological mirror symmetry) is to consider “spaces” represented by(enhanced)triangulated categories, which sometimes can be thought as derived categories of quasi coherent sheaves on thses “space”

2.5. “Spaces”represented by A-infinity categories. At the end of nineties, working on deformation theory, M.Kontsevich expanded geometric flavor by considering “spaces” represented y A-infinity categories.

It is worth mentioning that Cohn’s philosophy serves as a base for works of Gelfand and Retakh and their collaboratos on birational noncommutative algebras. Recently, Cohn’s universal localizations found applications in topology.

1.6. Imposing naive geometric spaces. The above mentioned approaches to noncommutative algebraic geometry insisted on a naive generalization of the standard pattern of commutative scheme theory: noncommutative version of schemes were sought as geometric spaces, and the latter were understood as topological spaces endowed with a structure sheaf of associative rings. This holds for D-scheme of Beilinson-Bernstein and for much more recent Kapranov’s version of formal Noncommutative geometry[Ka], because, by nature, D-schemes as well as Kapronov’s NC-schemes, are quasi coherent sheaves of associative algebras on commutative schemes. But, an arbitrary left noetherian associative algebra is not isomorphic to the algebra of global sections of the corresponding structure sheaf on Gabriel’s or Cohn’s (or any other)spectrum. It is therefore not surprising that imposing ringed space as the frame work for noncommutative algebraic geometry and trying to literally mimic the pattern of commutative algebra and algebraic geometry, led to considerable difficulties already on a very basic level.

1.7. Pseudo-geometry verus geometry.

The discovery of quantum groups triggered a flow of new examples supplied mostly by mathematical physics and attributed to noncommutative geometry, reviving some stagnating areas(e.g. Hopf algebra) and involving a big number of mathemaicians and theoretical physicists fascinated by the geometric flavor of this suddenly wide open field of research. This rise of the interest in noncommutative algebraic geometry was marked by the transition from attempts to build the foundations relying on naive generalizations of geometric spaces to the opposite extreme-viewing noncommutative algebraic geometry as pseudo-geometry, that is geometry in which spaces are replaced by something else. The transition was greatly influenced by Connes’ s approach to noncommutative differential geometry. On a more advanced stage, its root can be found in the pseudo-geometric development of Grothendieck’s algebraic geometry between the end of fifties and the beginning of the seventies-going from the category of geometric spaces(that is locally ringed)to the category Esp of spaces which are sheaves of sets on the fpqc presite of affine schemes, then expanding to topoes, algebraic spaces and stacks. Note that in commutative algebraic geometry, all these notions and points of view coexisted and complemented each other.

1.8. Points from commutative algebraic geometry. The abandon of the geometric point of view was due not so much to the limitations of Gabriel’s injective spectrum and shortcomings in the attempts of using it, but, mostly due to the fact the Gabriel’s spectrum was known and appreciated by a few algebraist,while the dominating paradigm of a point came from commutative algebraic geometry: points of commutative schemes are equivalence classes of geometric points, i.e. the morphism from spectra of fields. A naive noncommutative generalization of this notion is obtained by replacing fields by skew fields. Thus the naive points of an affine ‘space’ corresponding to an associative untial ring R are morphisms from R to skew fields, and the equivalence classes of morphisms from R to skew fields are in natural bijective correspondence with complete prime two-sided ideals of the ring R. Noncommutative rings usually have very few completely prime two-sided ideals. One consequence of this other transplantation of a commutative paradigm into noncommutative setting, was a widely adopted opinion that noncommutative algebraic geometry is essentially a geometry without points. Such a viewpoint reduces noncommutative algebraic geometry to the condition of a poor relative of its commutative predecessor: one can not count on a noncommutative version of local algebra, in particular, one can not count on a local study of spaces and morphisms of a spaces, which constitute at least a half of the content of commutative algebraic geometry. Fortunately, this opinion is wrong.