Dg Algebra Research Articles

A Note on Formality and Singularities of Moduli Spaces

This paper studies formality of the differential graded algebra $RHom(E,E)$, where $E$ is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves $E$, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by $E$. This paper also explains why Kaledin's theorem fails to apply in the remaining case.

On the homological mirror symmetry conjecture for pairs of pants

The $n$-dimensional pair of pants is defined to be the complement of $n + 2$ generic hyperplanes in $\mathbb{CP}n$. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism $A_\infty$ algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with $n+2$ generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model $(\mathbb{C}^{n+2},W)$, where $W = z_1...z_{n+2}$. We show that the endomorphism $A_\infty$ algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.

Compact generators in categories of matrix factorizations

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.

DG algebra structures on AS-regular algebras of dimension 2

Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H(A). Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We (J Algebra, 2008, 320: 2934–2962). Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DG A-module with a free underlying graded A#-module may not be semi-projective.

Cone Length for DG Modules and Global Dimension of DG Algebras

As the definition of free class of differential modules over a commutative ring in [1], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A e -module A.

Deformed Calabi–Yau completions

We define and investigate deformed n-Calabi-Yau completions of homolog- ically smooth dierential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi-Yau com- pletions do have the Calabi-Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi-Yau property. We show that deformed 3-Calabi-Yau completions of algebras of global dimension at most 2 are quasi-isomorphic to Ginzburg dg algebras and apply this to the study of cluster-tilted algebras and to the construction of derived equivalences as- sociated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses non commutative dierential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi-Yau property.

Cluster tilting objects in generalized higher cluster categories

We prove the existence of an m -cluster tilting object in a generalized m -cluster category which is ( m + 1 ) -Calabi–Yau and Hom-finite, arising from an ( m + 2 ) -Calabi–Yau dg algebra. This is a generalization of the result for the m = 1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension.

Derived equivalences from mutations of quivers with potential

We show that Derksen–Weyman–Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi–Yau triangulated categories. Our approach is related to that of Iyama–Reiten and ‘Koszul dual’ to that of Kontsevich–Soibelman. It improves on previous work by Vitória. In Appendix A, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.

Perfect Derived Categories of Positively Graded DG Algebras

We investigate the perfect derived category \({{\rm dgPer}}(\mathcal{A})\) of a positively graded differential graded (dg) algebra \(\mathcal{A}\) whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of \({{{\rm dgPer}}}(\mathcal{A})\) whose objects are easy to describe, define a t-structure on \({{{\rm dgPer}}}(\mathcal{A})\) and study its heart. We show that \({{{\rm dgPer}}}(\mathcal{A})\) is a Krull–Remak–Schmidt category. Then we consider the heart in the case that \(\mathcal{A}\) is a Koszul ring with differential zero satisfying some finiteness conditions.

Koszul differential graded algebras and BGG correspondence II

The concept of Koszul differential graded (DG for short) algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module A k is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] (see also Proposition 2.2), A k is not compact if H(A) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

Formality of DG algebras (after Kaledin)

We provide proper foundations and proofs for the main results of Kaledin (2007) [Ka]. The results include a flat base change for formality and behavior of formality in flat families of A ( ∞ ) and DG algebras.

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor

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.

On the (non)vanishing of some “derived” categories of curved dg algebras

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.

Formality of the Constructible Derived Category for Spheres: A Combinatorial and a Geometric Approach

We describe the constructible derived category of sheaves on the n-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods. As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg algebra in terms of differential forms, which allows us to prove formality for n-spheres, for real or complex coefficients.

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra \({(\mathcal{R}, \mathcal{E})}\) we associate a differential graded algebra \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) in a functorial way by means of explicit formulas. We describe two canonical filtrations on \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) , and derive an analogue of the Cartan relations for derivations of \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) ; we classify central extensions of \({\mathcal{E}}\) in terms of \({H^2(\mathcal{E}, \mathcal{R})}\) and study the canonical cocycle \({\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}\) whose class \({[\Theta]}\) obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra \({\mathcal{C}(\mathcal{E}, \mathcal{R})}\) is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.

Koszul differential graded modules

The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module kA lies in Dc(A), then the heart of the standard t-structure on Dc(A) is anti-equivalent to the category of finitely generated modules over some finite dimensional algebra. As a corollary, Dc(A) is equivalent to the bounded derived category of its heart as triangulated categories.

Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ((10), 1972) and Gromov ((4), 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In (6), we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

Deformation theory of objects in homotopy and derived categories I: General theory

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we define four deformation functors Def h ( E ) , coDef h ( E ) , Def ( E ) , coDef ( E ) . The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two – in the derived category. We study their properties and relations. These functors are defined on the category of artinian (not necessarily commutative) DG algebras.

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.

Homological Invariants for Connected DG Algebras

In this article, we study various homological invariants of differential graded (DG for short) modules over a connected DG algebra following Frankild–Jørgensen. Two different versions of homological dimensions (resolutional and functorial) are defined. In some cases, they are proved to be simply the bound of the cohomology of the DG module. Some homological identities, such as Auslander–Buchsbaum formula and Bass formula, are proved for compact DG modules over a connected DG algebra.