Algebraic realization of chain maps in differential graded algebras over a principal ideal domain

It is shown how simple principal ideal domains can be obtained from any principal right ideal domain by localization.When no localization is needed one can, under favourable conditions, obtain a simple principal right (but not left) ideal domain, and an easy example is given.In their recent Cambridge tract [4] Cozzens and Faith describe a number of simple Noetherian domains and, in particular, give a sufficient condition for a skew polynomial ring to be simple, taken from [3].But it is not hard to see that any skew polynomial ring is either itself simple or it leads to a simple (left and right) principal ideal domain by localization, provided that (essentially) no power of the defining endomorphism is inner.This generalization of Cozzen's result follows rather easily from an analysis of the ideal structure proved in Theorem 4. We also give a simple example where no localization is necessary; this leads to a simple principal right, but not left, ideal domain.I am indebted to the referee for drawing my attention to some errors in an earlier version.1. We begin by recalling some well-known facts, mainly to fix the terminology.Let bea principal right ideal domain; each two-sided ideal of R is of the form cR, where Re cR, i.e. c is right invariant.The set / of all right invariant elements (^ 0) is easily seen to be a right denominator set (cf. [2, p. 21]) and the localization R, is simple, for every ideal of R, is of the form 2l/\/; where 91 is an ideal of R, but the generator of 31 (= 0) becomes a unit in R" whence 9l/?7 = R,.Clearly the right ideals of R are again principal, so we may state the result as Theorem 1.Let R be a principal right ideal domain and I the set of its right invariant elements (=^0); then I is a right denominator set in R and the localization R is a simple principal right ideal domain.When R is a (left and right) principal ideal domain, it can be shown that R is a skew field precisely when R is not primitive.Theorem 1 provides us with a recipe for obtaining simple principal right ideal domains.For skew polynomial rings this programme is particularly easy

On the cohomology and deformations of differential graded algebras

In this paper we establish a faithfulness result, in a homotopical sense, between a subcategory of the model category of augmented differential graded commutative algebras CDGA and a subcategory of the model category of augmented differential graded algebras DGA over the field of rational numbers $\mathbb{Q}$.

This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally $\mathbb {Z}$ -graded and viewed as a differential graded algebra with trivial differential.

For a finite quiverQwithout sources, we consider the corresponding radical square zero algebraA. We construct an explicit compact generator for the homotopy category of acyclic complexes of projectiveA-modules. We call such a generator the projective Leavitt complex ofQ. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex ofQis quasi-isomorphic to the Leavitt path algebra ofQop. Here,Qopis the opposite quiver ofQ, and the Leavitt path algebra ofQopis naturally${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

In this paper, we interpret Massey products in terms of realizations (twitsting cochains) of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.

In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology.

We generalise two facts about finite‐dimensional algebras to finite‐dimensional differential graded algebras. The first is the Nakayama lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality, respectively. As an application, we prove a corepresentability result for finite‐dimensional differential graded algebras.

In the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A〈x〉 by setting the differential values d: x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these notions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate’s process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod-Shafarevich theorem, growth measures for graded algebras, characterizations of algebras of low homological dimension, and a homological description of Grobner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras.

Disconnected rational homotopy theory

In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology.

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme G, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define G to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group (mathbb Z+theta mathbb Z)^{2}.

Chapter 16 - Differential Graded Algebras in Topology