# Tagged Questions

A differential graded algebra is a graded algebra endowed with a differential of degree $1$ respecting the graded Leibniz rule.

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### Acyclic extension of free DGA-modules

I want to find some method to do the following: given an DGA-module $M$ over some commutative ring $k$, positively graded ($M_i=0$ if $i<0$), where each component $M_i$ has a free action by the ...
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### Surjections to free commutative dgas

Consider the category $C$ of commutative dgas (unbounded in both degrees) over a ring $R$ (in practice the integers or the integers mod $p$). Let $F$ be the free functor from chain complexes to $C$ ...
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### Cofibrations in the model structures for non-negative graded (commutative) DG algebras

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...
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### If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation): Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
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### Why do some literatures prefer right module to left module when dealing with DG modules?

I've been trying to read some papers on differential graded modules (for example, Keller, Deriving DG categories) In most of literature I found about dg-modules, they define them as right modules (Of ...
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### When does an $E_\infty$ algebra come from a commutative differential graded algebra?

Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
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### Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$ If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
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### Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
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### Quasi-isomorphisms and Subalgebras

Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$. What is if $f$ is ...
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### Gauss Manin connection in algebraic geometry and DG setting

E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ...
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### Complexes of sheaves with locally constant cohomology versus $C_{*}(\Omega M)$-modules

Let $M$ be a nice, connected topological space. Assume it is a manifold, if you like. There are two rather similar looking differential-graded (dg) categories that one can associate to $M$ that ...
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### Semi-free resolutions

Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of $\mathscr{C}$-...
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Let $A_\bullet$ be a dg-algebra over a field $k$. Let $M_\bullet$ (resp. $N_\bullet$) be a right (resp. left) $A_\bullet$-module. There is then a notion of the derived tensor product: $$M_\bullet\... 2answers 522 views ### Higher commutators in E_n algebras and the Maurer--Cartan equation Let A be an associative algebra in dgVect_k. Then the commutator [\cdot,\cdot]:A\otimes A\to A defined by [x,y]=xy-(-1)^{|x||y|}yx gives A the structure of a (dg-)Lie algebra. The Maurer--... 2answers 610 views ### Homological smoothness implies projectivity? Let A be a unital associative algebra over a commutative noetherian ring R. Assume that A is homologically smooth, which means that A\in D_{perf}(A\otimes_R^L A^{op}), which also means that A... 1answer 221 views ### Formality of classifying spaces (for not necessarily connected groups) As should be evident from the title this question has a similar flavor to: Formality of classifying spaces However, unlike Geordie's question, I will be working with torsion free coefficients (say ... 2answers 184 views ### Natural Isomorphism of S(V[1]) and (\bigwedge V)[n] Let V:=\oplus_{j\in\mathbb{Z}}V_j be a graded \mathbb{F}-vector space over the field \mathbb{F}. The graded tensor product of graded vector spaces is given by V \otimes W:= \oplus_{j\in \... 0answers 136 views ### Decalage isomorphism and algebra structure Consider the symmetric monoidal category of graded vector spaces in which the symmetric structure is given by the Koszul sign rule. Assume if necessary that the ground field is of characteristic zero. ... 0answers 236 views ### Coordinate free Koszul-Tate resolution Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ... 1answer 312 views ### Coequalizer in category of dg-algebras It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ... 1answer 166 views ### Contraction of graded vector fields on de Rham complex Given a commutative algebra A smooth over a field k of characteristic zero, the module of K\"ahler differentials \Omega^{1} is projective of finite rank and so the sum of all wedge powers \... 0answers 100 views ### bar construction for algebras with unusual grading of d The bar construction is usually applied to differential graded algebras with differential d of degree +1 or -1. Using multiple (de)suspensions, it also works for d of any degree \neq 0. Is this ... 1answer 403 views ### Do the solutions of the Maurer--Cartan equation form a simplicial set? The Maurer--Cartan equation is the equation:$$d\gamma+\frac 12[\gamma,\gamma]=0 where $\gamma$ represents a degree one element in a differential graded Lie algebra $\mathfrak g^\ast$. Let's denote ...
As in Algebra+Homotopy=Operad by Bruno Vallette, let $A$ with multiplication $\nu$ be a differential graded associative algebra equipped with degree +1 map $h$ and let $H$ be a chain complex such that ...