Commutative Differential Graded Algebras Research Articles

Cartan calculi on the free loop spaces

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André–Quillen cohomology of a commutative differential graded algebra A on the Hochschild homology of A in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space M to the derivation ring on the loop cohomology of M. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the Γ1 map due to Félix and Thomas.

On the behavior of Massey products under field extension

We show that global vanishing of Massey products on a commutative differential graded algebra is not invariant under field extension. Non-vanishing triple Massey products remain non-vanishing upon field extension, while higher Massey products can generally vanish. If the field being extended is algebraically closed, all non-vanishing Massey products remain non-vanishing on a finite type commutative differential graded algebra.

Some refinements of the Deligne–Illusie theorem

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, and Li-Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz. The main technical result used in the case $p>2$ belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us "abstract Koszul complexes", of which the de Rham complex in characteristic $p$ is an example. In the appendix, we use the aforementioned stronger decomposition result to prove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both hold for $F$-split $(p+1)$-folds.

On the Hochschild cohomology of a map between commutative differential graded algebras

AbstractWe consider a surjective morphism between commutative differential graded algebrasϕ: (∧V,d) → (B,d), whereVis finite dimensional, and (B,d) is a module over ∧Vvia the mappingϕ. We show that the Hochschild cohomologyHH*(∧V;B) can be computed in terms of the graded vector space of positiveϕ-derivations. Moreover, ifVis finite dimensional, thenHH*(∧V;B) contains a polynomial algebra.

Homological identities and dualizing complexes of commutative differential graded algebras

In this paper we study a connective commutative differential graded algebra (CDGA) which is piecewise Noetherian. The principal aim is to analyze a dualizing complex of CDGA’s and investigate a Gorenstein CDGA. We also establish many other results which can be expected to play basic roles in the study of a CDGA, such as the Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolutions for ordinary modules. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules.

Windows for cdgas

We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the category of perfect complexes on the two GIT quotients. More generally, we obtain semi-orthogonal decompositions which show that these categories differ by a certain number of copies of the derived category of the derived fixed locus. The derived equivalence for the Mukai flop is recovered as a very special case.

A strictly commutative model for the cochain algebra of a space

AbstractThe commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

Diagram Complexes, Formality, and Configuration Space Integrals for Spaces of Braids

AbstractWe use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.

A differential graded model for derived analytic geometry

We give a formulation for derived analytic geometry built from commutative differential graded algebras equipped with entire functional calculus on their degree 0 part, a theory well-suited to developing shifted Poisson structures and quantisations. In the complex setting, we show that this formulation recovers equivalent derived analytic spaces and stacks to those coming from Lurie's structured topoi. In non-Archimedean settings, there is a similar comparison, but for derived dagger analytic spaces and stacks, based on overconvergent functions.

Realisability problem in arrow categories

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal C = \mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained.

Naturality properties and comparison results for topological and infinitesimal embedded jump loci

We use augmented commutative differential graded algebra (▪) models to study G-representation varieties of fundamental groups π=π1(M) and their embedded cohomology jump loci, around the trivial representation 1. When the space M admits a finite family of maps, uniformly modeled by ▪ morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of Hom(π,G) and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when M is either a compact Kähler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group G is a non-abelian subalgebra of sl2(C).

Generalized Euler classes, differential forms and commutative DGAs

In the context of commutative differential graded algebras over [Formula: see text], we show that an iteration of “odd spherical fibration” creates a “total space” commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its “fibrations” with “fiber” of finite cohomological dimension, the induced map on cohomology is injective.

Fibrations and higher products in cohomology

This paper is a continuation of [2]. Working in the context of commutative differential graded algebras, we study the ideal of the cohomology classes which can be annihilated by fibrations whose fiber has finite homological dimension. In this paper we identify these classes with certain higher products in cohomology.

The Lambrechts\u2013Stanley model of configuration spaces

We prove the validity over {mathbb {R}} of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold. We moreover prove, if the dimension of the manifold is at least 4, that our model is compatible with the action of the Fulton–MacPherson operad (weakly equivalent to the little disks operad) when the manifold is framed. We use this more precise result to get a complex computing factorization homology of framed manifolds. Our proofs use the same ideas as Kontsevich’s proof of the formality of the little disks operads.

The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras

Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant $$L_\infty $$ -algebras. More precisely, for a given GBV algebra $$(\mathcal {A}=\oplus _{m\ge 0}\mathcal {A}_m, \delta , \ell _2^\delta )$$ , we provide another explicit GBV algebra $$(\widetilde{\mathcal {A}}=\oplus _{m\ge 0}\widetilde{\mathcal {A}}_m, \widetilde{\delta }, \ell _2^{\widetilde{\delta }})$$ such that its total homology is the same as the degree zero part of the homology $$H_0(\mathcal {A}, \delta )$$ of the given GBV algebra $$(\mathcal {A}, \delta , \ell _2^\delta )$$ .

The Beilinson regulator is a map of ring spectra

We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E∞-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E∞-refinement and prove a uniqueness result for these refinements.

Motives for an elliptic curve

In this paper we describe the rigid tensor triangulated subcategory of Voevodsky’s triangulated category of motives generated by the motive of an elliptic curve as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.

A new lower bound for LS-category

Let X be a simply connected CW-complex of finite type and {mathbb {K}} an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of C_*(Omega (X), mathbb K) to introduce a new homotopical invariant textsc {r}(X, {mathbb {K}}). If X is a Gorenstein space with nonzero evaluation map, then textsc {r}(X, {mathbb {K}}) turns out to interpolate mathrm {depth}(H_*(Omega (X), {mathbb {K}})) and mathrm {e}_{{mathbb {K}}}(X). We also define for any minimal Sullivan algebra (Lambda V,d) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant textsc {r}(A,d). When (Lambda V,d) is a minimal Sullivan model of X, this invariant fulfills the relation textsc {r}(X, {mathbb {K}}) = textsc {r}(Lambda V,d).

Koszul duality and homotopy theory of curved Lie algebras

This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic zero---Quillen equivalent to a model category of pseudo-compact unital commutative differential graded algebras; this extends known results regarding the Koszul duality of unital commutative differential graded algebras and differential graded Lie algebras. As an application of the theory developed within this paper, algebraic deformation theory is extended to functors on pseudo-compact, not necessarily local, commutative differential graded algebras.

Extension groups for DG modules

ABSTRACTLet M and N be differential graded (DG) modules over a positively graded commutative DG algebra A. We show that the Ext-groups defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups given in terms of equivalence classes of extensions. On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective.