Dg Algebra Research Articles

Derived localisation of algebras and modules

For any dg algebra A, not necessarily commutative, and a subset S in H(A), the homology of A, we construct its derived localisation LS(A) together with a map A→LS(A), well-defined in the homotopy category of dg algebras, which possesses a universal property, similar to that of the ordinary localisation, but formulated in homotopy invariant terms. Even if A is an ordinary ring, LS(A) may have non-trivial homology. Unlike the commutative case, the localisation functor does not commute, in general, with homology but instead there is a spectral sequence relating H(LS(A)) and LS(H(A)); this spectral sequence collapses when, e.g. S is an Ore set or when A is a free ring.We prove that LS(A) could also be regarded as a Bousfield localisation of A viewed as a left or right dg module over itself. Combined with the results of Dwyer–Kan on simplicial localisation, this leads to a simple and conceptual proof of the topological group completion theorem. Further applications include algebraic K-theory, cyclic and Hochschild homology, strictification of homotopy unital algebras, idempotent ideals, the stable homology of various mapping class groups and Kontsevich's graph homology.

English

We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a natural `discretization' $C_\infty$ quasi-isomorphism $\varphi$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $\varphi$ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for $\varphi$, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Plebanski.

A higher homotopic extension of persistent (co)homology

Our objective is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of a simplicial set embedded in \({\mathbb {R}}^{n}\) induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique \(A_{\infty }\)-algebra structure on the complete persistent cohomology of the filtered simplicial set. We then construct of a (pseudo)metric on the set of all barcodes of all cohomological degrees enriched with the \(A_{\infty }\)-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular \(A_{\infty }\)-algebra structure chosen. We also compute this distance for some basic examples. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology, that was observed by de Silva, Morozov, and Vejdemo-Johansson under some restricted assumptions which we do not suppose.

Derived categories of graded gentle one-cycle algebras

Let A be a graded algebra. It is shown that the derived category of dg modules over A (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded A-modules. This is applied to study derived categories of graded gentle one-cycle algebras.

Noncommutative formality implies commutative and Lie formality

Over a field of characteristic zero we prove two formality conditions. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.

Scalar extensions of categorical resolutions of singularities

Let X be a quasi-compact, separated scheme over a field k and we can consider the categorical resolution of singularities of X. In this paper let k′/k be a field extension and we study the scalar extension of a categorical resolution of singularities of X and we show how it gives a categorical resolution of the base change scheme Xk′. Our construction involves the scalar extension of derived categories of DG-modules over a DG algebra. As an application we use the technique of scalar extension developed in this paper to prove the non-existence of full exceptional collections of categorical resolutions for a projective curve of genus ≥1 over a non-algebraically closed field.

Path A∞ algebras of positively graded quivers

Let A be a path A∞-algebra over a positively graded quiver Q: We prove that the derived category of A is triangulated equivalent to the derived category of kQ; which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.

Decorated marked surfaces (part B): topological realizations

We study categories associated to a decorated marked surface $${\mathbf {S}}_\bigtriangleup $$ , which is obtained from an unpunctured marked surface $$\mathbf {S}$$ by adding a set of decorating points. For any triangulation $$\mathbf {T}$$ of $${\mathbf {S}}_\bigtriangleup $$ , let $$\Gamma _\mathbf {T}$$ be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in $${\mathbf {S}}_\bigtriangleup $$ and the reachable rigid indecomposables in the perfect derived category $${\text {per}}\,\Gamma _\mathbf {T}$$ . This is the dual of the bijection, between simple closed arcs in $${\mathbf {S}}_\bigtriangleup $$ and reachable spherical objects in the 3-Calabi-Yau category $${\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$$ , constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient $${\text {per}}\,\Gamma _\mathbf {T}/{\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$$ that defines the generalized cluster categories corresponds to the forgetful map $${\mathbf {S}}_\bigtriangleup \rightarrow \mathbf {S}$$ (forgetting the decorating points) in a suitable sense.

𝔸1-homotopy invariance of algebraic K-theory with coefficients and du Val singularities

C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen fundamental theorem as well as Stienstra’s foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic K-theory (without coefficients).

FlatZ-graded Connections and Loop Spaces

The pull back of a flat bundle E → X along the evaluation map π : LX → X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space Ω∗X of X provides an A∞ quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C•(Ω∗X), the dg algebra of singular chains on Ω∗X. Dedicated to James Dillon Stasheff, with gratitude.

Connected cochain DG algebras of Calabi-Yau dimension 0

Let A A be a connected cochain differential graded (DG, for short) algebra. This note shows that A A is a 0 0 -Calabi-Yau DG algebra if and only if A A is a Koszul DG algebra and T o r A 0 ( k A , A k ) \mathrm {Tor}_A^0(\Bbbk _A,{}_A\Bbbk ) is a symmetric coalgebra. Let V V be a finite dimensional vector space and w w a potential in T ( V ) T(V) . Then the minimal subcoalgebra of T ( V ) T(V) containing w w is a symmetric coalgebra, which implies that a locally finite connected cochain DG algebra is 0 0 -CY if and only if it is defined by a potential w w .

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Given an acyclic twisting cochain π:C→A, from a curved dg coalgebra C to a dg algebra A, we show that the associated twisted hom complex Homkπ(C,A) has cohomology equal to the Hochschild cohomology of A, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra A, along with its cup product, is a subquotient of the tensor product algebra A!⊗A of A with its Koszul dual.

B ∞-Algebra Structure in Homology of a Homotopy Gerstenhaber Algebra

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.

On the growth of deviations

The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.

Relative singularity categories I: Auslander resolutions

Let R be an isolated Gorenstein singularity with a non-commutative resolution A=EndR(R⊕M). In this paper, we show that the relative singularity category ΔR(A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category Dsg(R) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that Dsg(R) determines ΔR(Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

Godement resolution and operad sheaf homotopy theory

We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric \(\Omega \)-spectra, filtered dg algebras, operads and operad algebras.

Semidualizing DG modules over tensor products

We study the existence of nontrivial semidualizing DG modules over tensor products of DG algebras over a field. In particular, this gives a lower bound on the number of semidualizing DG modules over the tensor product.

Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Let A be a dg algebra over \mathbb F_2 and let M be a dg A -bimodule. We show that under certain technical hypotheses on A , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product M \otimes_A^L M and converges to the Hochschild homology of M . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

Poisson bivectors and Poisson brackets on affine derived stacks

Let A be a commutative dg algebra concentrated in degrees (−∞,m], and let SpecA be the associated derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of [16]) on SpecA to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second is more computational and involves an explicit resolution of the Poisson operad.

New enhancements of derived categories of coherent sheaves and applications

We introduce new enhancements for the bounded derived category Db(Coh(X)) of coherent sheaves on a suitable scheme X and for its subcategory Perf(X) of perfect complexes. They are used for translating Fourier–Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of Perf(X) to geometric smoothness of X, and for proving homological smoothness of Db(Coh(X)). Moreover, we characterize properness of Perf(X) and Db(Coh(X)) geometrically.