We give a sufficient condition for an 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.
This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.
Let $k[x]_{(x)}$ be the polynomial ring $k[x]$ localized in the maximal ideal $(x)\subseteq k[x]$. We study the Hilbert functor parameterizing ideals of colength $n$ in this ring {\it having support at the origin}. The main result of this article is that this functor is not representable. We also give a complete description of the functor as a limit of representable functors.
Recently N. Nitsure showed that for a coherent sheaf F on a noetherian scheme the automorphism functor Aut_F is representable if and only if F is locally free. Here we remove the noetherian hypothesis and show that the same result holds for the endomorphism functor End_F even if one asks for representability by an algebraic space.
In this paper we prove that a finite product of Brauer--Severi varieties is categorical representable in dimension zero if and only if it admits a $k$-rational point if and only if it is rational over $k$. The same is true for certain isotropic involution varieties over a field $k$ of characteristic different from two. For finite products of generalized Brauer--Severi varieties, categorical representability in dimension zero is equivalent to the existence of a full exceptional collection. In this case however categorical representability in dimension zero is not equivalent to the existence of …
We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor from the bounded derived category of a smooth projective variety over a field to the category of L-modules, to the case where L is a field extension of the base field k of the variety X, with L of transcendence degree less than or equal to one or L purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor between the bounded derived categories of coherent sheaves of X and Y, with X and Y smooth projective and Y of dimension less than or equal …
Let $S$ be a noetherian scheme, and let $E$ be a coherent sheaf on it. We define a group-valued contravariant functor $GL_E$ on $S$-schemes by associating to any $S$-scheme $T$ the group $GL_E(T)$ of all linear automorphisms of the pullback of $E$ to $T$. This functor is clearly a sheaf in the fpqc topology. We prove that $GL_E$ is representable by a group-scheme over $S$ if and only if the sheaf $E$ is locally free.
Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify for many applications. Provided a derived analogue of Schlessinger's condition holds, the theorem reduces to verifying conditions on the underived part and on cohomology groups. Another simplification is that functors need only be defined on nilpotent extensions of discrete rings. Finally, there is a pre-representability theorem, which can be applied to associ…
We prove a representability theorem for moduli functors of framed torsion-free sheaves on nonsingular complex projective surfaces, using formal geometry along a curve in the surface. This has as a consequence that a certain restriction morphism for moduli stacks of sheaves has fibers that are schemes.
The classification of affine line bundles on a compact complex space $X$ is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. For a fixed Chern class $c$, we introduce the affine Picard functor $Picaff_{X,x_0}^c:An^{op}\to Set$ which assigns to a complex space $T$ the set of families of linearly $x_0$-framed affine line bundles on $X$ with Chern class $c$ parameterized by $T$. Our main result states that this functor is representable if and only if the map $h^0:Pic^c(X)\to\mathbb{N}$ is constant. If this is the case, the …
We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.
We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor.…
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.
Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that th…
Given two algebraic groups $G$, $H$ over a field $k$, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not étale. When $G$ is linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$ is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting…
We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified under…
In this paper we study contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to M_n(C) for n >= 3. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gelfand spectrum functor for C*-algebras is also proved.
We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures, and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category, and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the la…
We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context…
We investigate the possibility of constructing Kochen-Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen-Specker coloring of the $n \times n$ idempotent integer matrices for $n \geq 3$, thereby illustrating that Kochen-Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topologic…
The aim of this paper is to put some recent results of Huang-Pandzic (conjectured by Vogan) and Kostant on Dirac cohomology in a broader perspective.This is achieved by introducing an induction functor in the noncommutative equivariant cohomology. In this context, the results of Huang-Pandzic and Kostant are interpreted as special cases (corresponding to the manifold being a point) of more general results on noncommutative equivariant cohomology introduced by Alekseev-Meinrenken.
A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic AF-algebras.
The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integral coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with i…
In this article we study the Picard functor and the Picard stack of an algebraic stack. We give a new and direct proof of the representability of the Picard stack. We prove that it is quasi-separated, and that the connected component of the identity is proper when the fibers of the stack are geometrically normal. We study some examples of Picard functors of classical stacks. In an appendix, we review the lisse-etale cohomology of abelian sheaves on an algebraic stack.
We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud's relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Neron-Severi groups or of the Picard group itself. We give some examples and applications. In an appendix, we prove the semicontinuity theorem for a (non necessarily tame) algebraic stack.
These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.
We establish the representability of the general linear ${\mathbb Z}_2^n$-group and use the restricted functor of points - whose test category is the category of ${\mathbb Z}_2^n$-manifolds over a single topological point - to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.
This paper gives an elementary introduction to noncommutative deformations of modules. The main results of this deformation theory are due to Laudal. Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M_1, ..., M_p} be a finite family of left A-modules. We study the simultaneous formal deformations of the family M, described by the noncommutative deformation functor Def(M): a(p) -> Sets introduced by Laudal. In particular, we prove that the deformation functor Def(M) has a pro-representing hull H(M), unique up to non-canonical isomorphism, and…
In this paper, we construct a refined, relative version of the étale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $\infty$-category of motivic spaces over a base scheme $S$ to the $\infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $\infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $\infty$-category is generated under cofiltered limits by those spaces whose …
We construct a functor from the category of elliptic curves to a category of noncommutative tori. Our proof is based on an isomorphism between the Sklyanin algebras and dense sub-algebras of the noncommutative tori.
In this article one extends the classical theory of (intermediate) Jacobians to the "noncommutative world". Concretely, one constructs a Q-linear additive Jacobian functor J(-) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf(X)) (associated to the derived dg category perf(X) of a smooth projective scheme X) identifies with the un…
There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go re…
We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known exa…
In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the sym…
A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the…
In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct non…
The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'', locally described by noncommutative rings and their categories of one-sided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. T…
We continue the work initiated in arXiv:1206.3645, where we introduced a new stable symmetric monoidal $(\infty,1)$-category $SH_{nc}$ encoding a motivic stable homotopy theory for the noncommutative spaces of Kontsevich and obtained a canonical monoidal colimit-preserving functor $SH\to SH_{nc}$ relating this new theory to the $(\infty,1)$-category encoding the stable motivic $\mathbb{A}^1$ theory of Morel-Voevodsky. For a scheme $X$ this map recovers the dg-derived category of perfect complexes $L_{pe}(X)$. In this sequel we address the study of the different flavours of algebraic $K$-theory…
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in …
