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  "body": "\n## Definition\n\nLet $k$ be a commutative ring and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.\nLet $M$ be an $A$-[[bimodule|EM7FHAFD3N]] on which $k$ acts the same way on both sides.\nAll [[tensor products|F0HQXQ0HCD]] are over $k$.\nThe *Hochschild complex* is the [[chain complex|WQ82C0M3QB]] with $C_n(A, M) = M \\otimes A^{\\otimes n}$ for $n \\ge 0$ and boundary $b \\colon C_n \\to C_{n-1}$ given by\n$$b(m \\otimes a_1 \\otimes \\cdots \\otimes a_n) = m a_1 \\otimes a_2 \\otimes \\cdots \\otimes a_n + \\sum_{i=1}^{n-1} (-1)^i m \\otimes a_1 \\otimes \\cdots \\otimes a_i a_{i+1} \\otimes \\cdots \\otimes a_n + (-1)^n a_n m \\otimes a_1 \\otimes \\cdots \\otimes a_{n-1}.$$\nThe *Hochschild homology* of $A$ with coefficients in $M$ is $HH_n(A, M) = H_n(C_\\bullet(A, M), b)$.\nOne writes $HH_n(A) = HH_n(A, A)$.\n\n## Remarks\n\nOne checks that $b \\circ b = 0$, so the homology is defined.\nIn degree $0$, $b(m \\otimes a) = ma - am$, so $HH_0(A, M)$ is the quotient of $M$ by the $k$-submodule spanned by all $ma - am$, and $HH_0(A) = A/[A, A]$.\nExample: for $A = M_n(k)$ with $k$ a field, $[A, A]$ is the space of trace-zero matrices, so the [[trace|0XC1RDBY7A]] gives $HH_0(A) \\cong k$.\nIf $A$ is projective as a $k$-module, then $HH_n(A, M) \\cong \\operatorname{Tor}_n^{A^e}(A, M)$, where $A^e = A \\otimes A^{\\mathrm{op}}$ uses the [[opposite ring|M8Q24XB0AF]]; see the [[Tor functor|DDQRERAQCV]].\nWithout such a flatness hypothesis the complex still makes sense, but this description by Tor can fail.\nHochschild cohomology $HH^n(A, M)$ is defined dually from $\\Hom_k(A^{\\otimes n}, M)$, and it equals $\\operatorname{Ext}^n_{A^e}(A, M)$ when $A$ is projective over $k$.\n",
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