Dg Algebra Research Articles

Transferring algebra structures on complexes

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $R$. The weak equivalences are given by: (1) an $R$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers, and (3) $R$-homology equivalences. Analogously, for any field ${\mathbb{F}}$, we construct three model structures on the category of connected simplicial cocommutative ${\mathbb{F}}$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when ${\mathbb{F}}$ is algebraically closed, the simplicial ${\mathbb{F}}$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when ${\mathbb{F}}$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of ${\mathbb{F}}$.

On the semifree resolutions of DG algebras over the enveloping DG algebras

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra B over the enveloping DG algebra B ⊗ A B , where A ⊆ B is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by ( B , D ) , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of ( B , D ) , we provide a new characterization of naïve liftability of semifree DG B-modules.

Homological properties of 3-dimensional DG Sklyanin algebras

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let [Formula: see text] be a [Formula: see text]-dimensional DG Sklyanin algebra with [Formula: see text], where [Formula: see text] and [Formula: see text] We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for [Formula: see text] to be Calabi–Yau, Koszul, Gorenstein and homologically smooth, respectively.

Q-stability conditions on Calabi–Yau-𝕏 categories

We introduce $q$ -stability conditions $(\sigma,s)$ on Calabi–Yau- $\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$ , where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$ , where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$ . When $s=N$ is an integer, we show that $q$ -stability conditions can be identified with the stability conditions on $\mathcal {D}_N$ , provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$ -baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$ -stability conditions on $\mathcal {D}_\mathbb {X}$ . As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$ -stability conditions is a complex manifold of dimension $n+1$ . Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau- $\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$ -equivariant coherent sheaves on canonical bundles provide the Calabi–Yau- $\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster- $\mathbb {X}$ categories for acyclic quivers.

The ST correspondence for proper non-positive dg algebras

Let A be a proper non-positive dg algebra over a field k. For a simple-minded collection of the finite-dimensional derived category , we construct a “dual” silting object of the perfect derived category by using the Koszul duality for dg algebras. This induces a one-to-one correspondence between the equivalence classes of silting objects in and algebraic t-structures of .

Cohomological jump loci and duality in local algebra

In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local complete intersection rings, exterior algebras and certain group algebras in prime characteristic. This family of varieties generalizes the well-studied support varieties in each of these contexts. We show that cohomological jump loci satisfy several interesting properties, including being closed under (Grothendieck) duality. The main application of this support theory is that over a local ring the homological invariants of Betti degree and complexity are preserved under duality for finitely generated modules having finite complete intersection dimension.

Relative cluster categories and Higgs categories

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) [3] and Plamondon (2011) [50] to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011) [17]. Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow us to categorify such cluster algebras.In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n-cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schröer), which is stably n-Calabi–Yau and Hom-finite, arising from a left (n+1)-Calabi–Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n-representation-finite algebras.

Cohomology Algebras of a Family of DG Skew Polynomial Algebras

Let A be a connected cochain DG algebra such that its underlying graded algebra A# is the graded skew polynomial algebra k⟨x1,x2,x3⟩/x1x2+x2x1x2x3+x3x2x3x1+x1x3,|x1|=|x2|=|x3|=1. Then the differential ∂A is determined by ∂A(x1)∂A(x2)∂A(x3)=Mx12x22x32 for some M∈M3(k). When the rank r(M) of M belongs to {1,2,3}, we compute H(A) case by case. The computational results in this paper give substantial support for the research of the various homological properties of such DG algebras. We find some examples, which indicate that the cohomology graded algebras of such kind of DG algebras may be not left (right) Gorenstein.

A comparison of dg algebra resolutions with prime residual characteristic

Abstract In this article, we fix a prime integer p and compare certain dg algebra resolutions over a local ring whose residue field has characteristic p. Namely, we show that given a closed surjective map between such algebras there is a precise description for the minimal model in terms of the acyclic closure and that the latter is a quotient of the former. A first application is that the homotopy Lie algebra of a closed surjective map is abelian. We also use these calculations to show deviations enjoy rigidity properties which detect the (quasi-)complete intersection property.

Smooth DG algebras and twisted tensor product

The twisted tensor product of DG algebras is studied and sufficient conditions for smoothness of such a product are presented. It is shown that in the case of finite-dimensional DG algebras, applying this operation offers great possibilities for constructing new examples of smooth DG algebras and algebras. In particular, examples are given of families of algebras of finite global dimension with two simple modules that have non-trivial moduli spaces. Bibliography: 24 titles.

Smooth DG algebras and twisted tensor product

The twisted tensor product of DG algebras is studied and sufficient conditions for smoothness of such a product are presented. It is shown that in the case of finite-dimensional DG algebras, applying this operation offers great possibilities for constructing new examples of smooth DG algebras and algebras. In particular, examples are given of families of algebras of finite global dimension with two simple modules that have non-trivial moduli spaces. Bibliography: 24 titles.

Hopf Differential Graded Galois Extensions

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that D(R#H)≅D(RH) is equivalent to that R/RH is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions.

On homological properties of strict polynomial functors of degree p

We study the homological algebra in the category Pp of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category Pp has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e>p.

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $${\cal A}$$ , we define a group homomorphism $${\rm{Hdet}}:{\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right) \to {k^ \times }$$ , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $${{\cal A}^G}$$ to be Gorenstein, where $${\cal A}$$ is a homologically smooth DG algebra such that $$H\left( {\cal A} \right)$$ is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of $${\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right)$$ . Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

Proper connective differential graded algebras and their geometric realizations

We show that every proper connective dg algebra A admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. If A is moreover smooth, we compute the noncommutative Chow motive of A. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective dg algebras.

Categorical Koszul duality

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor transforms the Quillen equivalence between quasicategories and simplicial categories into this Koszul duality. This allows us to give a conceptual interpretation of the dg nerve of a dg category and its adjoint. As an application, we prove that the category of representations of a quasicategory K is equivalent to the coderived category of comodules over C⁎(K), the chain coalgebra of K. A corollary of this is a characterization of the category of constructible dg sheaves on a stratified space as the coderived category of a certain dg coalgebra.

Systolic inequalities, Ginzburg dg algebras and Milnor fibers

Systolic inequalities, Ginzburg dg algebras and Milnor fibers

Exact Hochschild extensions and deformed Calabi-Yau completions

We introduce the Hochschild extensions of dg algebras, which are -algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric -algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual.

Hirzebruch-Riemann-Roch and Lefschetz type formulas for finite dimensional algebras

We introduce the dimension matrix of a bimodule complex and the trace matrix of a chain endomorphism of a bimodule complex, and give the Hirzebruch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension. Moreover, we compare our HRR and Lefschetz type formulas with Shklyarov's HRR type formulas and Petit's Lefschetz type formulas for dg algebras.