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Here we read the following

Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a quasi-isomorphism of commutative differential graded algebras $$\phi: (\wedge V,d)\longrightarrow (A,d),$$ where $\wedge V$ denotes the free commutative algebra on the graded vector space of finite type $V$ and $d(V)\subset \wedge^{\geq 2}V$.

1) What is exactly the vector space $V$ here do we know it explicitly or is it part of the existence statement ?

2) Is the ground field for $V$ is the same as the ground field of the algebra $A$ ?

3) Do the $(A,d)$ and $(\wedge V,d)$ have the same differential $d$ or is it simply a notation but they are different differentials ?

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$V$ is part of the existence statement, the ground field is the same, and the $d$'s are different. –  Grumpy Parsnip Feb 6 '13 at 2:47
    
Thank you Jim!! I would accept it as an answer. –  palio Feb 6 '13 at 8:23
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up vote 1 down vote accepted

$V$ is part of the existence statement, the ground field is the same, and the $d$'s are different.

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