Mixing
quasi isomorphism of two dg algebras
$begingroup$
I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is isomorphic to $H^*(B)$ as a graded algebra. Here $H^*(A)$ is the graded homology algebra associated with $A$. Now the Kadeishvili Theorem show that we have $H^*(A)rightarrow A$ and $H^*(B)rightarrow B$ are quasi-isomorphism of $A_{infty}-$algebras. Can I deduce that $H^*(A)rightarrow H^*(B)$ is a morphism of $A_{infty}-$algebras?
Added after edit: Now suppose that we have an equivalence of categories $mathcal D(A)rightarrow mathcal D(B)$ which sends A to B. Then is the above question true? We know from A infinity structure on Ext algebras that the higher multiplications of $H^*(A) $ are essentially the same with the Massey products. Now the question is whether such an equivalence ‘preserves’ Massey products.
homological-algebra algebras graded-algebras
asked Jan 6 at 8:16
user12580user12580
490313
$endgroup$
add a comment |
$begingroup$
I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is isomorphic to $H^*(B)$ as a graded algebra. Here $H^*(A)$ is the graded homology algebra associated with $A$. Now the Kadeishvili Theorem show that we have $H^*(A)rightarrow A$ and $H^*(B)rightarrow B$ are quasi-isomorphism of $A_{infty}-$algebras. Can I deduce that $H^*(A)rightarrow H^*(B)$ is a morphism of $A_{infty}-$algebras?
Added after edit: Now suppose that we have an equivalence of categories $mathcal D(A)rightarrow mathcal D(B)$ which sends A to B. Then is the above question true? We know from A infinity structure on Ext algebras that the higher multiplications of $H^*(A) $ are essentially the same with the Massey products. Now the question is whether such an equivalence ‘preserves’ Massey products.
homological-algebra algebras graded-algebras
asked Jan 6 at 8:16
user12580user12580
490313
$endgroup$
$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46
add a comment |
$begingroup$
I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is isomorphic to $H^*(B)$ as a graded algebra. Here $H^*(A)$ is the graded homology algebra associated with $A$. Now the Kadeishvili Theorem show that we have $H^*(A)rightarrow A$ and $H^*(B)rightarrow B$ are quasi-isomorphism of $A_{infty}-$algebras. Can I deduce that $H^*(A)rightarrow H^*(B)$ is a morphism of $A_{infty}-$algebras?
Added after edit: Now suppose that we have an equivalence of categories $mathcal D(A)rightarrow mathcal D(B)$ which sends A to B. Then is the above question true? We know from A infinity structure on Ext algebras that the higher multiplications of $H^*(A) $ are essentially the same with the Massey products. Now the question is whether such an equivalence ‘preserves’ Massey products.
homological-algebra algebras graded-algebras
asked Jan 6 at 8:16
user12580user12580
490313
$endgroup$
I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is isomorphic to $H^*(B)$ as a graded algebra. Here $H^*(A)$ is the graded homology algebra associated with $A$. Now the Kadeishvili Theorem show that we have $H^*(A)rightarrow A$ and $H^*(B)rightarrow B$ are quasi-isomorphism of $A_{infty}-$algebras. Can I deduce that $H^*(A)rightarrow H^*(B)$ is a morphism of $A_{infty}-$algebras?
Added after edit: Now suppose that we have an equivalence of categories $mathcal D(A)rightarrow mathcal D(B)$ which sends A to B. Then is the above question true? We know from A infinity structure on Ext algebras that the higher multiplications of $H^*(A) $ are essentially the same with the Massey products. Now the question is whether such an equivalence ‘preserves’ Massey products.
homological-algebra algebras graded-algebras
homological-algebra algebras graded-algebras
asked Jan 6 at 8:16
user12580user12580
490313
asked Jan 6 at 8:16
user12580user12580
490313
edited Jan 9 at 1:46
user12580
asked Jan 6 at 8:16
user12580user12580
490313
asked Jan 6 at 8:16
user12580user12580
490313
asked Jan 6 at 8:16
user12580user12580
490313
490313
$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46
add a comment |
$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46
$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46
add a comment |
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$begingroup$
The problem addressed in the preprint arxiv.org/pdf/math/0401007.pdf seems to be helpful.
$endgroup$
– 3 A's
Jan 6 at 16:20
$begingroup$
It is generally not true that A and B are linked by quasi isomorphisms. There are counter examples (though I haven’t checked yet and don’t know a concrete example) where A and B are graded fields.
$endgroup$
– user12580
Jan 9 at 1:46