Algebras: Homology vs. Resolution as a dg-algebra

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My question is what is the relation (if any) between the following two notions.



  1. 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.


  2. 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







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  • 1




    You mean the Hochschild homology of $A$?
    – Najib Idrissi
    3 hours ago














up vote
2
down vote

favorite












My question is what is the relation (if any) between the following two notions.



  1. 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.


  2. 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







share|cite|improve this question

















  • 1




    You mean the Hochschild homology of $A$?
    – Najib Idrissi
    3 hours ago












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.



  1. 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.


  2. 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







share|cite|improve this question













My question is what is the relation (if any) between the following two notions.



  1. 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.


  2. 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






share|cite|improve this question













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asked 4 hours ago









Maxime Lucas

388113




388113







  • 1




    You mean the Hochschild homology of $A$?
    – Najib Idrissi
    3 hours ago












  • 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










1 Answer
1






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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 $A$.






share|cite|improve this answer




















  • 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










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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 $A$.






share|cite|improve this answer




















  • 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














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 $A$.






share|cite|improve this answer




















  • 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












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 $A$.






share|cite|improve this answer












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 $A$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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
















  • 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

















 

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