scholarly journals A LERAY SPECTRAL SEQUENCE FOR NONCOMMUTATIVE DIFFERENTIAL FIBRATIONS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350015 ◽  
Author(s):  
EDWIN BEGGS ◽  
IBTISAM MASMALI
Keyword(s):  
Spectral Sequence ◽  
Sheaf Cohomology ◽  
Left Module ◽  
Graded Algebras ◽  
Serre Spectral Sequence ◽  
Differential Graded Algebras ◽  
Zero Curvature ◽  
Total Differential ◽  
Leray Spectral Sequence ◽  
Special Case

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

scholarly journals DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES

2018 ◽  
Vol 61 (03) ◽  
pp. 557-573
Author(s):  
SHENGYONG PAN ◽  
ZHEN PENG ◽  
JIE ZHANG
Keyword(s):  
Affirmative Answer ◽  
Derived Equivalence ◽  
Graded Algebras ◽  
Cohomology Rings ◽  
Derived Equivalences ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Differential Graded Algebras ◽  
Endomorphism Algebras ◽  
Special Case

AbstractIn 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.

On the simple object associated to a diagram in a closed model category

1986 ◽  
Vol 100 (3) ◽  
pp. 459-474 ◽  
Author(s):  
Pere Pascual-Gainza
Keyword(s):  
Model Category ◽  
General Context ◽  
Algebraic Varieties ◽  
Sheaf Cohomology ◽  
Graded Algebras ◽  
Closed Model ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Differential Graded Algebras ◽  
Closed Model Category

In this paper we develop a descent technique for generalized (co)-homology theories defined in the category of algebraic varieties. By such a theory we mean a functor Sch→C, where C is a closed model category in the sense of Quillen satisfying certain axioms (cf. §4). We have chosen to work in such a general context so as to include two situations for which the results of SGA 4 of Deligne and Saint-Donat are not applicable: descent of multiplicative structures (i.e. of differential graded algebras) and descent for generalized sheaf cohomology (such as the algebraic K-theory of coherent sheaves over a noetherian scheme).

scholarly journals Homological epimorphisms, homotopy epimorphisms and acyclic maps

2020 ◽  
Vol 32 (6) ◽  
pp. 1395-1406
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Loop Spaces ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Algebraic Description ◽  
Acyclic Map ◽  
Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

scholarly journals A strictly commutative model for the cochain algebra of a space

2020 ◽  
Vol 156 (8) ◽  
pp. 1718-1743
Author(s):  
Birgit Richter ◽  
Steffen Sagave
Keyword(s):  
Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Chain Complexes ◽  
Simplicial Set ◽  
New Construction ◽  
Integral Version

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.

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