Mixing
Algebras: Homology vs. Resolution as a dg-algebra
Clash Royale CLAN TAG#URR8PPP
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My question is what is the relation (if any) between the following two notions.
Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
On the other hand, one could also see $A$ as a (let's say positively graded) dg-algebra (concentrated in degree 0), and take a cofibrant replacement of $A$ in this category, using the projective model structure.
Both notions give some notions homotopical information about $A$, and I would be interested in knowing if it is possible to deduce one from the other (presumably, to recover a resolution of $k$ from a cofibrant replacement of $A$), at least in "good" cases.
The reason I ask is because I have two very similar constructions building these objects for suitable algebras, and I am wondering whether one of these constructions can be deduced from the other one.
homological-algebra dg-algebras
asked 4 hours ago
Maxime Lucas
388113
add a comment |Â
up vote
2
down vote
favorite
My question is what is the relation (if any) between the following two notions.
Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
On the other hand, one could also see $A$ as a (let's say positively graded) dg-algebra (concentrated in degree 0), and take a cofibrant replacement of $A$ in this category, using the projective model structure.
Both notions give some notions homotopical information about $A$, and I would be interested in knowing if it is possible to deduce one from the other (presumably, to recover a resolution of $k$ from a cofibrant replacement of $A$), at least in "good" cases.
The reason I ask is because I have two very similar constructions building these objects for suitable algebras, and I am wondering whether one of these constructions can be deduced from the other one.
homological-algebra dg-algebras
asked 4 hours ago
Maxime Lucas
388113
1
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
My question is what is the relation (if any) between the following two notions.
Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
On the other hand, one could also see $A$ as a (let's say positively graded) dg-algebra (concentrated in degree 0), and take a cofibrant replacement of $A$ in this category, using the projective model structure.
Both notions give some notions homotopical information about $A$, and I would be interested in knowing if it is possible to deduce one from the other (presumably, to recover a resolution of $k$ from a cofibrant replacement of $A$), at least in "good" cases.
The reason I ask is because I have two very similar constructions building these objects for suitable algebras, and I am wondering whether one of these constructions can be deduced from the other one.
homological-algebra dg-algebras
asked 4 hours ago
Maxime Lucas
388113
My question is what is the relation (if any) between the following two notions.
Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective resolution of $k$ as an $A$-module.
On the other hand, one could also see $A$ as a (let's say positively graded) dg-algebra (concentrated in degree 0), and take a cofibrant replacement of $A$ in this category, using the projective model structure.
Both notions give some notions homotopical information about $A$, and I would be interested in knowing if it is possible to deduce one from the other (presumably, to recover a resolution of $k$ from a cofibrant replacement of $A$), at least in "good" cases.
The reason I ask is because I have two very similar constructions building these objects for suitable algebras, and I am wondering whether one of these constructions can be deduced from the other one.
homological-algebra dg-algebras
homological-algebra dg-algebras
asked 4 hours ago
Maxime Lucas
388113
asked 4 hours ago
Maxime Lucas
388113
asked 4 hours ago
Maxime Lucas
388113
asked 4 hours ago
Maxime Lucas
388113
asked 4 hours ago
Maxime Lucas
388113
388113
1
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago
add a comment |Â
1
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago
1
1
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago
add a comment |Â
1 Answer
1
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5
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From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R xrightarrowsim A$ is a projective resolution of $A$. Let $Omega_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k otimes_A otimes A^op (A otimes_R Omega_R).$$
The object $mathbbL_R/A = A otimes_R Omega_R$ is called the cotangent complex of $A$.
answered 3 hours ago
Najib Idrissi
1,3511026
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R xrightarrowsim A$ is a projective resolution of $A$. Let $Omega_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k otimes_A otimes A^op (A otimes_R Omega_R).$$
The object $mathbbL_R/A = A otimes_R Omega_R$ is called the cotangent complex of $A$.
answered 3 hours ago
Najib Idrissi
1,3511026
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
add a comment |Â
up vote
5
down vote
accepted
From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R xrightarrowsim A$ is a projective resolution of $A$. Let $Omega_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k otimes_A otimes A^op (A otimes_R Omega_R).$$
The object $mathbbL_R/A = A otimes_R Omega_R$ is called the cotangent complex of $A$.
answered 3 hours ago
Najib Idrissi
1,3511026
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R xrightarrowsim A$ is a projective resolution of $A$. Let $Omega_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k otimes_A otimes A^op (A otimes_R Omega_R).$$
The object $mathbbL_R/A = A otimes_R Omega_R$ is called the cotangent complex of $A$.
answered 3 hours ago
Najib Idrissi
1,3511026
From your first description, I guess that what you call "the homology of $A$" is in fact the Hochschild homology of $A$ with constant coefficients, i.e. $HH_*(A;k)$. Please tell me if I got this wrong.
Then yes, it's possible to compute it from a projective resolution of $A$. Suppose that $R xrightarrowsim A$ is a projective resolution of $A$. Let $Omega_R$ be the $R$-module of Kähler differentials. Then you can compute the Hochschild homology of $A$ as the André–Quillen homology, i.e. the homology of the complex:
$$k otimes_A otimes A^op (A otimes_R Omega_R).$$
The object $mathbbL_R/A = A otimes_R Omega_R$ is called the cotangent complex of $A$.
answered 3 hours ago
Najib Idrissi
1,3511026
answered 3 hours ago
Najib Idrissi
1,3511026
answered 3 hours ago
Najib Idrissi
1,3511026
answered 3 hours ago
Najib Idrissi
1,3511026
1,3511026
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
add a comment |Â
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
PS: it works for any operad. You can compute the Harrison homology of a commutative algebra, the Chevalley–Eilenberg homology of a Lie algebra, etc, as the analogue of the André–Quillen homology. See e.g. Chapter 12 of the book Algebraic Operads (Loday & Vallette) and the references therein.
– Najib Idrissi
3 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
This seems to be exactly what I was looking for. Thank you very much!
– Maxime Lucas
2 hours ago
add a comment |Â
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1
You mean the Hochschild homology of $A$?
– Najib Idrissi
3 hours ago