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 last part of the paper, we describe noncommutative deformation functors for sheaves and quasi-coherent sheaves of modules on a ringed space $(X, \mathcal{A})$. We show that for any good $\mathcal{A}$-affine open cover $\mathsf{U}$ of $X$, the forgetful functor $\mathsf{QCoh}(\mathcal{A}) \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces an isomorphism of noncommutative deformation functors.
\emph{Applications.} We consider noncommutative deformations of quasi-coherent $\mathcal{A}$-modules on $X$ when $(X, \mathcal{A}) = (X, \mathcal{O}_X)$ is a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of $X$ closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left $\mathcal{D}_X$-module $\mathcal{O}_X$ when $X$ is an elliptic curve as an example.
The tensor product of $\mathbb{A}^1$-invariant sheaves with transfers introduced by Voevodsky is generalized to reciprocity sheaves via the theory of modulus presheaves with transfers. We prove several general properties of this construction and compute it in some cases. In particular we obtain new (motivic) presentations of the absolute Kähler differentials and the first infinitesimal neighborhood of the diagonal.
There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works.
The strictly homotopy invariance of the associated Nisnevish sheave $\widetilde{\mathcal F}_{Nis}$ of a homotopy invariant presheave $\mathcal F$ with GW-transfers (or Witt-transfers) on the category of smooth varieties over a prefect field $k$, $char\,k \neq 2$, is proved, i.e. the isomorphism $$H^i_{Nis}(\mathbb A^1\times X,\widetilde{\mathcal F}_{Nis})\simeq H^i_{Nis}(X,\widetilde{\mathcal F}_{Nis})$$ for any $X\in Sm_k$ is obtained. This theorem is necessary for the construction of the triangulated category of GW-motives $\mathbf{DM}^{GW}(k)$ and Witt-motives $\mathbf{DM}^W(k)$ by the Voev…
Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme $X$ in terms of sheaf cohomology on $X$ and $X\times X$. These results are applied to the study of deformations of the sheaf $D_X$ of differential operators on $X$. In particular, in case $X$ is a flag variety we show that any deformation of $D_X$, which is induced by a deformation of $\cO_X$, mu…
We give a description of the tensor product of SC-reciprocity presheaves with transfers in terms of $K$-group of geometric type, and we study a structure of the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_a$. We apply our description to give a description of Chow group of 0-cycles with modulus of products of curves. We also show that the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_m$ is isomorphic to the sheaf $Ω^1$ of Kähler differential forms as reciprocity sheaves over characteristic zero.
A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformatins of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest.
In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
Let (X,D) be a D-scheme in the sense of Beilinson and Bernstein, given by an algebraic variety X and a morphism O_X -> D of sheaves of rings on X. We consider noncommutative deformations of quasi-coherent sheaves of left D-modules on X, and show how to compute their pro-representing hulls. As an application, we compute the noncommutative deformations of the left D_X-module O_X when X is any elliptic curve.
Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.
We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.
We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus $T$ as an ind-object in the category of holomorphic vector bundles on $T$. Extending the results of math.QA/0211262 and math.QA/0308136 we prove that the derived category of quasicoherent sheaves on $T$ is equivalent to the derived category of usual quasicoherent sheaves on the corresponding elliptic curve. We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category $\Qcoh(η_T)$ obtained by taking the quotient of the category of quasicoherent sheaves by …
We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant sheafification functor. Using equivariant sheafification, we develop the general theory of equivariant sheaves of modules over a ring, give explicit constructions of infinite products and introduce an equivariant analogue of skyscraper sheaves. These results underlie recent work by the authors which proves that there is an algebraic model for rational G-spectra …
We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts from commutative geometry (blowups commute, Quot schemes are projective, etc.) carry over. The key new tool in studying these deformations is a relatively simple description of their derived categories and the relevant t-structures; this also allows us to establish nontrivial derived equivalences for deformations of elliptic surfaces. We also establish that…
We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parametrizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build insta…
We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjectur…
A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different …
We prove that the category of representations of the N-Kronecker quiver and that of coherent sheaves on the noncommutative projective scheme of $R=k< X_1,...,X_N >/(\sum^N_{i=1}X_i^2)$ are derived equivalent. This equivalence is easily proved by applying Orlov's Theorem, on the other hand,in our proof,the quadratic relation $\sum_{i=1}^NX_i^2$ naturally arises from Auslander-Reiten Theory.
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.
We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a sy…
We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees …
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…
A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheave…
This is the third paper in a series. In part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces.
The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of c…
The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$Ω$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properti…
Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf O_X to a sheaf of noncommutative algebras and of the sheaves O_Y and O_Z to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf Tor^{O_X}_*(O_Y, O_Z). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as…
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…
