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Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, when $A$ is augmented, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is flat over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is flat over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would i choose one over the other?

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is flat over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.

On the other hand, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?