A differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.

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Confusion about representing elements as pure tensors in the definition of the tensor product of dg-algebras

I'm refamiliarizing myself with the tensor product of dg-algebras and struggling to reconcile some of the basic definitions. Let $(A, d_A)$ and $(B, d_B)$ be dg-algebras over a field $k$, and assume ...
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Cohomology of a quotient of a commutative graded differential algebra over GF(2)

Let $P$ be a graded commutative differential algebra over the field $GF(2)$ (hence also actually commutative), and let $d$ be the differential. Let $x$ be a homogeneous element of $P$, which is also a ...
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Gröbner basis and dg structures

Gröbner basis is typically defined for ideals of polynomial rings over a field and there are several generalizations/extensions of this notion for non-commutative structures or differential algebras. ...
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Tensor product preserves quasi-isomorphisms

Let $A, B$ be dg Lie algebras. I'm trying to prove that if $f: A \rightarrow B$ is a quasi-isomorphism then $\otimes^n f: A^{\otimes n} \rightarrow B^{\otimes n}$ is also a quasi-isomorphism. I'm ...
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Do Chevalley-Eilenberg homology functor and taking the cohomology commute?

I've stumbled upon some ideas from homological algebra that I'm trying to piece together from a talk I heard. I don't have much background in this area, so I'm not sure if this is a reasonable thing ...
1 vote
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Why is the bar construction of a DG algebra a coalgebra?

Let $A$ be a differentially graded augmented algebra. Then $\mathbf{B}A$ can be equipped with the structure of a coalgebra. This is proved in, for example, Loday and Vallette's book on Algebraic ...
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Signs involved in suspension of algebra

I have a question that seems almost too trivial to ask, yet it confuses me often. Suppose now that we have a dg-algebra $(A,m,d)$ with multiplication $m: A \otimes A \longrightarrow A$. If we look at ...
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Definition of module of Kähler differentials for a DG algebra

Let $R$ be a $DG$ algebra over $A$, i.e, a $\mathbb{Z}$-graded $A$- algebra with a derivation $d$. For example, if $R$ is an $A$-algebra, then any chain complex $C^{\bullet}$ of $R$-modules with a ...
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Natural transformation between sheaves in homotopy theory

Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself! ...
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What is the (co)homology of a free (graded) Lie algebra?

In characteristic $0$, what is the Chevalley-Eilenberg (co)homology of a free (graded) Lie algebra? Not the definition, but $H^i =$ ??
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1 vote