Abstract: If $G$ is a reductive algebraic group acting rationally on a smooth affine variety $X$ then it is generally believed that $D(X)^G$ has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this paper we show that this is indeed the case when $G$ is a torus and $X=k^r\times (k^*)^s$. We give a precise description of the primitive ideals in $D(X)^G$ and we study in detail the ring theoretical and homological properties of the minimal primitive quotients of $D(X)^G$. The latter are of the form $D(X)^G/(\mathfrak{g}-\chi(\mathfrak{g}))$ where $\mathfrak{g}=\operatorname{Lie}(G)$, $\chi\in \mathfrak{g}^\ast$ and $\mathfrak{g}-\chi(\mathfrak{g})$ is the set of all $v-\chi(v)$ with $v\in \mathfrak{g}$. They occur as rings of twisted differential operators on toric varieties. As a side result we prove that if $G$ is a torus acting rationally on a smooth affine variety then $D(X/\!\!/ G)$ is a simple ring.
Abstract: In this paper we will think of certain abelian categories with favorable properties as non-commutative surfaces. We show that under certain conditions a point on a non-commutative surface can be blown up. This yields a new non-commutative surface which is in a certain sense birational to the original one. This construction is analogous to blowing up a Poisson surface in a point of the zero-divisor of the Poisson bracket. By blowing up $\le 8$ points in the elliptic quantum plane one obtains global non-commutative deformations of Del-Pezzo surfaces. For example blowing up six points yields a non-commutative cubic surface. Under a number of extra hypotheses we obtain a formula for the number of non-trivial simple objects on such non-commutative surfaces.
Abstract: In this paper we prove that trace rings of generic matrices are Cohen-Macaulay. This is done by relating this problem to a conjecture of Stanley about modules of invariants under a reductive group. We prove a slightly weakened version of this conjecture in special cases. In particular we obtain that the conjecture is true for $SL_2$.
Abstract: In this paper we study rings of differential operators for modules of covariants for one-dimensional tori. In particular we analyze when they are Morita equivalent, when they are simple, and when they have nite global dimension. As a side result we obtain an extension of the Bernstein-Beilinson equivalence to weighted projective spaces.
Abstract: We prove a very slightly weakened version of a conjecture by Richard Stanley
Abstract: Let $U$, $W$ be finite dimensional representations of $G = \mathrm{SL}_2$. We give conditions under which $(U\otimes k[W])^G$ is a Cohen- Macaulay $k[W]^G$-module. In particular we obtain an invariant theoretic proof of the fact that the trace ring of generic $2\times 2$ matrices is Cohen-Macaulay
Abstract: In this paper we derive explicit rational forms for the Poincare series of the commutative and the non-commutative trace ring ($\S$5.3 and $\S$5.5). To this end we use the Molien Weyl formula to reduce the question to a problem about ows in a particular graph.
Abstract: Let D and E be central division algebras over k; let K be the generic splitting field of E; we show that the index of $D \otimes_k K$ is the minimum of the indices of $D \otimes_k E^{\otimes i}$. We use this to calculate the index of D under related central extensions and to construct division algebras with special properties.
Abstract: In this paper we give a new method, in terms of one-parameter sub- groups, to study semi-invariants for algebraic tori. In some cases we obtain extensions to results in [7]. In other cases we obtain different proofs.
Abstract: We compute the Hochschild and cyclic homology of certain three dimensional quantum spaces (type A algebras), introduced by Artin and Schelter. We show that the Hochschild homology is determined by the quasi-classical limit
Abstract: In this note we prove, in the case of $\operatorname{Sl}_2$, a converse to Stanley's conjecture about Cohen-Macaulayness of invariant modules for reductive algebraic groups.
Abstract: Given a family of separable finite dimensional extensions $(L_i)_i$ of a field $k$, we construct a division algebra of dimension $n^2$ over its center which is freely generated over $k$ by the fields $(L_i)_i$.
Abstract: Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let $W$, $U$ two finite dimensional representations of $G$. In this paper we give a survey on the computation of the local cohomology of $(U \otimes SW)^G$.
Abstract: To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (``Sklyanin algebras''), having the Hilbert series of a polynomial algebra. In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other ``Sklyanin-type'' algebras.
Abstract: For a 3-dimensional Artin-Schelter-regular algebra A with Hilbert series $1/(1-t)^3$ we study central extensions; that is, graded algebras $D$ with a regular central element $z$ in degree $1$, such that $D/(z)=A$. We classify such $D$ and we also classify certain $D$-modules (point modules and line modules) which proved to be important in the study of $3$-dimensional Artin-Schelter-regular algebras.
Abstract: In this paper we prove a translation principle for the central quotients of four-dimensional Sklyanin algebras, which is analogous to the translation principle for semi-simple Lie algebras. In the course of the proof we construct an ``elliptic'' analog of the sheaf of differential operators on $\mathbb{P}^1$. The translation principle may be used to construct the fat points of a four-dimensional, non-PI, Sklyanin algebra.
Abstract: It has been conjectured that the ring of differential operators of the algebraic quotient of a connected smooth affine variety under a reductive group action is simple. This is known in the case that the group in question is the extension of a flinite group with a torus and in the case of classical representation of classical groups. In these notes we present some tools relevant to this conjecture. In particular we show that it is true for some representations of $\operatorname{SL}_2$.
Abstract: Let $G$ be a connected non-special semisimple algebraic group and let $W$ be a finite dimensional $G$-representation such that $W$ has trivial generic stabilizer. Let $\mathfrak{g}=\operatorname{Lie}(G)$. Then the semi-direct product $\mathfrak{g}+W$ is a counter example to the Gel'fand-Kirillov conjecture.
Abstract: The simplest non-trivial division algebras that can be constructed over a rational function field in two variables are those that ramify along a smooth cubic curve. In this note we show that these division algebras are cyclic if they have odd index.
Abstract: Let $A$ be a three dimensional Artin-Schelter regular algebra. We give a description of the category of finitely generated $A$-modules of Gelfand-Kirillov dimension one (modulo those of finite dimension over the ground field). The proof is based upon a result by Gabriel which says that locally finite categories can be described by module categories over topological rings.
Abstract: In this note we prove existence theorems for dualizing complexes over graded and filtered rings, thereby generalizing some results by Zhang, Yekutieli and J{\o}rgensen.
Abstract: Let $W$ be a finite dimensional representation of a linearly reductive group $G$ over a field $k$. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under $G$ of the symmetric algebra of $W$ has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if $R$ is a graded subring of a polynomial ring over a perfect field of characteristic $p>0$ and if the inclusion $R\rightarrow S$ splits, then $D_k(R)$ is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero.
Abstract: Let ``$\operatorname{HH}$'' stand for Hochschild (co)homology. In this note we show that for many rings $A$ there exists $d\in\mathbb{N}$ such that for an arbitrary $A$-bimodule $N$ we have $\operatorname{HH}^i(N)=\operatorname{HH}_{d-i}(N) $. Such a result may be viewed as an analog of Poincare duality. Combining this equality with a computation of Soergel allows one to compute the Hochschildt homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo. (Also see erratum below.)
Abstract: Assume that $S$ is a semigroup generated by ${x_1,...,x_n}$, and let $U$ be the multiplicative free commutative semigroup generated by ${u_1,...,u_n}$. We say that $S$ is of $I$-type if there is a bijection $v:U\rightarrow S$ such that for all a in $U$, ${v(u_1a),....,v(u_na)}={x_1v(a),....,x_nv(a)}$. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of $I$-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of $I$-type appear in the study of the settheoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author.
Abstract: We classify regular algebras of global dimension four that map surjectively onto the two-Veronese of a regular algebra of global dimension three on two generators. We also study the point modules.
Abstract: A conjecture of Kac states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we give a combinatorial reformulation of Kac's conjecture in terms of a property of $q$-multinomial coefficients. As a side result we give a formula for certain inverse Kostka-Foulkes polynomials.
Abstract: We show that it is possible to define reflection isomorphisms on the double of the (twisted) Hall algebra of a quiver. Combining these reflections with Fourier transform yields an alternative construction of Lusztig's braid group action on a quantum enveloping algebra.
Abstract: Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let $W, U$ two finite dimensional representations of $G$. In this paper we compute the local cohomology of $(U \otimes SW)^G$ provided a certain relatively weak technical condition is true.
Abstract: We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity $d$ over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by $d\times d$-matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of ``curves'' of degree $d$ in a projective quantum plane.
Abstract: We show that the Gelfand-Kirillov holds for Lie algebras of dimension at most eigth. Recall that in dimension nine the authors have constructed a counter example.
Abstract: Assume that $X$ is a surface over an algebraically closed field $k$. Let $\tilde{X}$ be obtained from $X$ by blowing up a smooth point and let $L$ be the exceptional curve. Let $\operatorname{coh}(X)$ be the category of coherent sheaves on $X$. In this note we show how to recover $\operatorname{coh}({X})$ from $\operatorname{coh}(\tilde{X})$, if we know the object $\mathcal{O}_L(L)$.
Abstract: The representations of dimension vector $\alpha$ of the quiver $Q$ can be parametrised by a vector space $R(Q,\alpha)$ on which an algebraic group $\operatorname{GL}(\alpha)$ acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi--invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi--invariant polynomial function that describes when this map is inverted on the representation and we show that these semi--invariant polynomial functions form a spanning set of all semi--invariant polynomial functions in characteristic $0$. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector $\alpha$. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.
Abstract: In this paper we show that the Hall algebra of a quiver, as defined by Ringel, is the positive part of the quantived enveloping algebra of a generalized Kac-Moody Lie algebra. We give a potential application of this result to a conjecture of Kac which states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra.
Abstract: Let $\mathcal{C}$ be a connected noetherian hereditary abelian category with Serre functor over an algebraically closed field $k$, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from [RV2], we prove that if $\mathcal{C}$ has some object of infinite length, then the Grothendieck group of $\mathcal{C}$ is finitely generated if and only if $\mathcal{C}$ has a tilting object.
Abstract: In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.
Abstract: In this paper we classify $\operatorname{Ext}$-finite noetherian hereditary abelian categories over an algebraically closed field $k$ satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories. As a side result we show that when our hereditary abelian categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
Abstract: The paper "A relation between Hochschild homology and cohomology for Gorenstein rings" contains an error in the sense that Theorem 1 (the "duality theorem") is false in the generality stated. As a result the same is true for its corollaries: Proposition 3 and Corollary 6. The main conclusion, which is an affirmative answer to a question by Patrick Polo, remains valid however
Abstract: In this paper we conjecture that the center of a non-commutative complete regular local ring of global dimension two is a formal power series ring in two variables. We prove this conjecture in the special case of Ore extensions.
Abstract: We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.
Abstract: We give a sufficient condition for an $\operatorname{Ext}$-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
Abstract: A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.
Abstract: {For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b(\operatorname{coh}(Y))$ and $D^b(\operatorname{coh}(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using non-commutative rings. The new proof also covers some mild singular and higher dimensional situations (including the one in the recent paper by Chen: ``Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities''). }
Abstract: In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain three dimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.
Abstract: We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.
Abstract: We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative P^2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford. For the Weyl algebra it was proved by Wilson
Abstract: In this paper we give a direct proof of the properties of the $\mathbb{Z}D_\infty$ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author.
Abstract: This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.
Abstract: In this paper we develop the basic infinitesimal deformation theory of \emph{abelian categories}. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems.
Abstract: In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.
Abstract: The Hilbert scheme of $n$ points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand's result. This allows us to show that the technical condition is not necessary.
Abstract: We give a proof of Yekutieli's global quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
Abstract: In this paper we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure. This follows the fact that appropriately localized path algebras of double quivers carry a certain kind of non-commutative quasi-Hamiltonian structure.
Abstract: Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.
Abstract: In this paper we introduce non-commutative analogues for the quasi-Hamiltonian $G$-spaces introduced by Alekseev, Malkin and Meinrenken. We outline the connection with the non-commutative analogues of quasi-Poisson algebras which the author had introduced earlier.
Abstract: This is a companion note to ``Hochschild cohomology and Atiyah classes'' by Damien Calaque and the author. We compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernouilli numbers. Our computation uses Stokes theorem together with some basic properties proved by Kontsevich.
Abstract: In this paper we prove that the sheaf of $\mathcal{L}$-differential operators for a locally free Lie algebroid $\mathcal{L}$ is formal when viewed as a sheaf of $G_\infty$-algebras via Tamarkin's morphism of DG-operads $B_\infty\rightarrow G_\infty$.
Abstract: In this paper we provide a proof of a result announced by Kontsevich that the HKR-morphism, twisted by the square root of the Todd genus, gives an isomorphism between the Hochschild cohomology of a smooth algebraic variety and the cohomology of poly-vector fields, both considered as Gerstenhaber algebras. Our proof is set in the framework of Lie algebroids and so applies in more general settings as well.
Abstract: We show that determinantal varieties defined by maximal minors of a generic matrix have a kind of non-commutative desingularization. More precisely we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.
Abstract: We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.
Abstract: For Lie algebras whose Poisson semi-center is a polynomial ring we give a bound for the sum of the degrees of the generating semi-invariants. This bound was previously known in many special cases.
Abstract: We generalize and clarify Gerstenhaber and Schack's "Special Cohomology Comparison Theorem". More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category $U$ and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do not have to assume that $U$ is a poset.
Abstract: In this paper we describe non-commutative versions of $\mathbb{P}^1\times \mathbb{P}^1$. These contain the class of non-commutative deformations of $\mathbb{P}^1\times\mathbb{P}^1$.
Abstract: In the recent paper "Mutation in triangulated categories and rigid Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.
Abstract: In this paper we give a new definition for non-commutative $\mathbb{P}^1$-bundles over commutative schemes. The main motivation for this is that every non-commutative deformation of a Hirzebruch surface is given by a non-commutative $\mathbb{P}^1$-bundle over $\mathbb{P}^1$.
Abstract: In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - that Shoikhet's quasi-isomorphism is compatible with cap product after twisting with a Maurer-Cartan element.
Abstract: In these notes we provide the foundation for the deformation theoretic parts of ``Non-commutative quadrics'' and ``Non-commutative $\mathbb{P}^1$-bundles over commutative schemes''.
Abstract: The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of non-commutative quadric surfaces, We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by non-commutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of $\operatorname{Ext}$-groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a "curve" having self-intersection number -2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen-Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of non-commutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional non-commutative quadric hypersurfaces will apply to higher dimensional non-commutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.
Abstract: We prove a technical result which allows us to establish the non-degeneracy of potentials on quivers in some previously unknown cases. Our result applies to McKay quivers and also to potentials derived from geometric helices on Del Pezzo surfaces. On the other hand we also give an example of a skew group ring with a degenerate potential. This shows that for 3-CY orders Iyama-Reiten mutations cannot always be iterated indefinitely.
Abstract: We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes.
Abstract: We discuss the representation theory of the bialgebra $\operatorname{end}(A)$ introduced by Manin. As a side result we give a new proof that Koszul algebras are distributive and furthermore we show that some well-known $N$-Koszul algebras are also distributive.
Abstract: We prove that complete $d$-Calabi-Yau algebras in the sense of Ginzburg are derived from superpotentials.
Abstract: In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov's well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.
Abstract: The $p$-support of a holonomic D-module was introduced by Kontsevich. Thomas Bitoun in his PhD thesis proved several properties of $p$-support conjectured by Kontsevich. In this note we give an alternative proof for involutivity by reducing it to a slight extension of Gabber's theorem on the integrability of the characteristic variety. For the benefit of the reader we review how this extension follows from Kaledin's proof of Gabber's theorem.
Abstract: Let $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, $A$ a sheaf of Azumaya algebras over $X$ of rank $r$, and $\operatorname{Hmo}(R)$ the category of noncommutative motives with $R$-coefficients. Assume that $1/r$ belongs to $R$. Under this assumption, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we show that all the $R$-linear additive invariants of $X$ and $A$ are exactly the same. Examples include (nonconnective) algebraic $K$-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then computer the $R$-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional $k$-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element of the Grothendieck group of $X$ which has rank $r$ becomes invertible in the $R$-linearized Grothendieck group, and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.
Abstract: The famous representability theorem by Orlov asserts that any fully faithful functor between the derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. This result has been extendedby Lunts and Orlov to include functors from perfect complexes to quasicoherent complexes. In this paper we show that the latter extension is false without the full faithfulness hypothesis. Our results are based on the properties of scalar extensions of derived categories whose investigation was started by Pawel Sosna and the first author.
Abstract: We give concrete DG-descriptions of certain stable categories of maximal Cohen-Macaulay modules. This makes in possible to describe the latter as generalized cluster categories in certain cases.
Abstract: In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the \emph{existence} of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a well-known conjecture of Bondal and Orlov in this special case.
Abstract: In this article we study in detail the category of noncommutative motives of separable algebras $\operatorname{Sep}(k)$ over a base field $k$. We start by constructing four different models of the full subcategory of commutative separable algebras $\operatorname{CSep}(k)$. Making use of these models, we then explain how the category $\operatorname{Sep}(k)$ can be described as a "fibered $Z$-order" over $\operatorname{CSep}(k)$. This viewpoint leads to several computations and structural properties of the category $\operatorname{Sep}(k)$. For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (=CSA) and sequences of elements in the Brauer group of $k$. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic $K$-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras.
Abstract: In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.
Abstract: In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.
Abstract: For any Koszul Artin-Schelter regular algebra $A$, we consider a version of the universal Hopf algebra $\operatorname{aut}(A)$ coacting on $A$, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor $M$ to finite dimensional vector spaces such that $\operatorname{aut}(A)= \operatorname{coend}_U(M)$. Using this pair $(U,M)$ we show that $\operatorname{aut}(A)$ is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of $U$.
Abstract: In our companion paper "The Manin Hopf algebra of a Koszul Artin-Schelter regular algebra is quasi-hereditary" we used the Tannaka-Krein formalism to study the universal coacting Hopf algebra $\operatorname{aut}(A)$ for a Koszul Artin-Schelter regular algebra $A$. In this paper we study in detail the case of a polynomial ring in two variables. In particular we give a more precise description of the standard and costandard representations of $\operatorname{aut}(A)$ as a coalgebra and we show that the latter can be obtained by induction from a Borel quotient algebra. Finally we give a combinatorial characterization of the simple $\operatorname{aut}(A)$-representations as tensor products of $\operatorname{end}(A)$-representations and their duals.
Abstract: In this paper we relate the deformation theory of Ginzburg Calabi-Yau algebras to negative cyclic homology. We do this by exhibiting a DG-Lie algebra that controls this deformation theory and whose homology is negative cyclic homology. We show that the bracket induced on negative cyclic homology coincides with Menichi's string topology bracket. We show in addition that the obstructions against deforming Calabi-Yau algebras are annihilated by the map to periodic cyclic homology. In the commutative we show that our DG-Lie algebra is homotopy equivalent to $(T^{\operatorname{poly}}[[u]],-u \operatorname{div})$.
Abstract: Let $X$ be a smooth scheme, $Z$ a smooth closed subscheme, and U the open complement. Given any localizing and $A^1$-homotopy invariant of dg categories $E$, we construct an associated Gysin triangle relating the value of $E$ at the dg categories of perfect complexes of $X$, $Z$, and $U$. In the particular case where $E$ is homotopy $K$-theory, this Gysin triangle yields a new proof of Quillen's localization theorem, which avoids the use of devissage. As a first application, we prove that the value of $E$ at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of E at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an etale descent result concerning noncommutative mixed motives with rational coefficients.
Abstract: In this article, using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely ``fixed-point data''. As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic $K$-theory, of Baranovsky concerning cyclic homology, of the second author and Polishchuk concerning Hochschild homology, and of Baranovsky-Petrov and Caldararu-Arinkin (unpublished) concerning twisted Hochschild homology. As an application, we verify Grothendieck's standard conjectures of type $C^+$ and $D$, as well as Voevodsky's smash-nilpotence conjecture, in the case of ``low-dimensional'' orbifolds. Finally, we establish a result of independent interest concerning nilpotency in the Grothendieck ring of an orbifold.
Abstract: Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.
Abstract: We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group $G$. In some cases, including Weyl groups of type $A$, $B$, $G_2$, $F_4$, as well as the groups $G(m,1,n)$, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces $V^g/C(g)$, where $C(g)$ is the centralizer subgroup of $g$ in $G$. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on $C^n$, where $C$ is a smooth curve.
Abstract: We give an easy example of a triangulated category, linear over a field $k$, with two different enhancements, linear over $k$, answering a question of Canonaco and Stellari.
Abstract: Let $R$ be the homogeneous coordinate ring of the Grassmannian $G=\operatorname{Gr}(2,n)$ defined over an algebraically closed field of characteristic $p>0$. In this paper we give a completely characteristic free description of the decomposition of $R$, considered as a graded $R^p$-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of $G$ and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that $R$ provides a "noncommutative resolution" for $R^p$ when $p \ge n−2$, generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirković on Frobenius decompositions for partial flag varieities. We are hopeful that it is an instance of a more general "$p$-uniformity" principle.
Abstract: We prove a version of the classical 'generic smoothness' theorem with smooth varieties replaced by non-commutative resolutions of singular varieties. This in particular implies a non-commutative version of the Bertini theorem.
Abstract: We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR.
Abstract: Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety $\operatorname{Spec} R$ admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.
Abstract: In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.
Abstract: In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact quasi-separated scheme $X$, we prove that (after inverting some elements of representation ring of $G$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is "concentrated" in the subscheme of $G$-fixed points $X^G$. Moreover, in the particular case where $G$ is connected and the action has finite stabilizers, we compute the additive invariants of $[X/G]$ using solely the subgroups of roots of unity of $G$. As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards.
Abstract: Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open GIT substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.
Abstract: If $G$ is a reductive group which acts on a linearized smooth scheme $X$ then we show that under suitable standard conditions the derived category of coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on the categorical quotient $X^{ss}/\!\!/G$ which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of $X^{ss}/\!\!/G$ constructed earlier by the authors. The results in this paper also complement a result by Halpern-Leistner (and similar results by Ballard-Favero-Katzarkov and Donovan-Segal) that asserts the existence of a semi-orthogonal decomposition of the derived category of $X/G$ in which one of the components is the derived category of $X^{ss}/G$.
Abstract: In the paper "Deformation theory of abelian categories", the last two authors proved that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects. We show a generalization of this to pretriangulated dg-categories with a left bounded non-degenerate t-structure with enough derived injectives, the latter being derived enhancements of the injective objects in the heart of the t-structure. Such dg-categories (with an additional hypothesis of closure under suitable products) can be completely described in terms of left bounded twisted complexes of their derived injectives.
Abstract: We show that it is possible to freely adjoin the right dual to an object in a monoidal category and moreover that the resulting tautological functor is fully faithful.
Abstract: We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.
Abstract: Let R be the homogeneous coordinate ring of the Grassmannian $G=\operatorname{Gr}(2,n)$ defined over an algebraically closed field $k$ of characteristic $p≥\operatorname{max}\{n−2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r\ge 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $G$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$.
Abstract: Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such "quasi-symmetric" GKZ hypergeometric systems.
Abstract: In this paper we prove that any smooth projective variety of dimension $\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier-Mukai functor between smooth projective schemes.
Abstract: Consider a monoidal category which is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_\infty$-algebra which is $A_\infty$-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This $B_\infty$-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted $A_\infty$-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined $B_\infty$-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of $B_\infty$-algebras.
Abstract: We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or "non-commutative algebraic varieties which are finite over their centres". The case of toric Brauer pairs is discussed in further detail.
Abstract: Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation $X$ of a reductive group $G$, extending the local system of triangulated categories established by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack $X/G$.
Abstract: We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of $m$-parking functions of length $n$. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m,n)$-Dyck paths, the number of which is given by the Fuss-Catalan number $A_n(m,1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes
Abstract: We interpret all Maurer-Cartan elements in the formal Hochschild complex of a small dg category which is cohomologically bounded above in terms of torsion Morita deformations. This solves the "curvature problem", i.e. the phenomenon that such Maurer-Cartan elements naturally parameterize curved A_infinity deformations. In the infinitesimal setup, we show how ($n+1$)-th order curved deformations give rise to $n$-th order uncurved Morita deformations.
Abstract: Recently the Euler forms on numerical Grothendieck groups of rank $4$ whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the existence of noncommutative surfaces which up to that point had not been considered for $m\ge 2$. These have been constructed for $m=2$ using noncommutative $\mathbb{P}^1$-bundles, and for all $m\ge 2$ by a different construction using maximal orders on $\operatorname{Bl}_x\mathbb{P}^2$. In this article we compare the constructions for $m=2$, i.e. we compare the categories arising from half-ruled del Pezzo quaternion orders on $F_1$ with noncommutative $\mathbb{P}^1$-bundles on $\mathbb{P}^1$. This can be seen as a noncommutative instance of the classical isomorphism $F_1\cong \operatorname{Bl}_x\mathbb{P}^2$.
Abstract: Let a reductive group G act on a smooth variety X such that a good quotient X//G exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X//G, obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X//G. In fact the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne-Mumford stacks.
Abstract: Let $T$ be a torus, $X$ a smooth quasi-compact separated scheme equipped with a $T$-action, and $[X/T]$ the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories $E$, we prove that the derived completion of $E([X/T])$ at the augmentation ideal $I$ of the representation ring $R(T)$ of $T$ agrees with the Borel construction associated to the $T$-action on $X$. Moreover, for certain localizing $A^1$-homotopy invariants, we extend this result to the case of a linearly reductive group scheme $G$. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic $K$-theory, of Thomason's completion theorem in étale $K$-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological $K$-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in $l$-adic étale $K$-theory, in (real) semi-topological $K$-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, $l$-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy $K$-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological $K$-theory.
Abstract: Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kaparanov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of the Laurent polynomials associated to the faces of $P$ that are hypersurfaces. In this short note we show that it is also the straightforward union of all the discriminant loci, i.e. we may include those of higher codimension, and there is no need to take closures. This answers a question by Kite and Segal.
Abstract: Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of the usual crepant resolutions that appear in algebraic geometry. In this paper we survey some results around NCCRs.
Abstract: We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\rightarrow \hat{\mathcal{H}}^\sigma$ where $\hat{\mathcal{H}}^\sigma$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^\sigma−\operatorname{Mod}_{lf}≃RQ−\operatorname{Mod}_{lf}$. Examples of locally finite $\hat{\mathcal{H}}^\sigma$-, resp. $\mathcal{R}_{\mathbb{Q}}$-modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.
Abstract: This paper is a sequel to "t-structures and twisted complexes on derived injectives" by the same authors. We develop the foundations of the infinitesimal derived deformation theory of pretriangulated dg-categories endowed with t-structures. This generalizes the deformation theory of abelian categories developed by the last two authors. We show how deformations of dg-categories of derived injectives yield derived deformations of the associated t-structures.
Abstract: These are some notes I wrote many years ago. I occasionally get requests for them so I decided to tidy them up and put them here. The main motivation is that there are two possible definitions for the notion of a $G$-equivariant $\mathcal{D}$-module. The first one is the standard category theoretic definition in terms of the schemes $G\times X$ and $G\times G\times X$. The second, more convenient one, is in terms of the Lie algebra of $G$. In these notes we show that these two definitions are equivalent for a connected group. This is certainly well-known but I haven't been able to locate an elementary proof.
Abstract: These are notes on de Jong's proof of the period$=$index theorem over fields of transcendence degree two. They are actually about the ``simplified'' proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school ``Central Simple Algebras over Function Fields of Surfaces'' at the Universität Konstanz between August, 26 and September, 1 2007 but I did not finish them in time.
Abstract: We classify the solutions to a system of equations, introduced by Bondal, which encode numerical constraints on full exceptional collections of length 4 on surfaces. The corresponding result for length 3 is well-known and states that there is essentially one solution, namely the one corresponding to the standard exceptional collection on the surface $\mathbb{P}^2$. This was essentially proven by Markov in 1879. It turns out that in the length 4 case, there is one special solution which corresponds to $\mathbb{P}^1×\mathbb{P}^1$ whereas the other solutions are obtained from $\mathbb{P}^2$ by a procedure we call numerical blowup. Among these solutions, three are of geometric origin (the union of $\mathbb{P}^2$ and a point, $\mathbb{P}^1×\mathbb{P}^1$ and the ordinary blowup of $\mathbb{P}^2$ at a point). The other solutions are parametrized by the natural numbers and very likely do not correspond to commutative surfaces. However they can be realized as noncommutative surfaces, as was recently shown by Dennis Presotto and the first author.