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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with product $(a, b) \\mapsto ab$.\nLet $A[\\![t]\\!]$ be the $k[\\![t]\\!]$-module of [[formal power series|033WGN00BF]] with coefficients in $A$.\nA *formal deformation* of $A$ is a $k[\\![t]\\!]$-bilinear associative product $*$ on $A[\\![t]\\!]$ of the form\n$$a * b = ab + t\\,\\mu_1(a, b) + t^2 \\mu_2(a, b) + \\cdots \\quad (a, b \\in A),$$\nwhere each $\\mu_i \\colon A \\times A \\to A$ is $k$-bilinear.\nTwo formal deformations $*$ and $*'$ are *equivalent* if there is a $k[\\![t]\\!]$-linear map $T = \\id + t T_1 + t^2 T_2 + \\cdots$, with each $T_i \\colon A \\to A$ $k$-linear, such that $T(a * b) = T(a) *' T(b)$ for all $a, b$.\n\n## Remarks\n\nComparing the coefficients of $t$ in $(a * b) * c = a * (b * c)$ gives\n$$a \\mu_1(b, c) - \\mu_1(ab, c) + \\mu_1(a, bc) - \\mu_1(a, b) c = 0,$$\nso $\\mu_1$ is a Hochschild $2$-cocycle, and equivalent deformations have cohomologous $\\mu_1$.\nExample: for any $T = \\id + t T_1 + \\cdots$, the product $a * b = T^{-1}(T(a) T(b))$ is a formal deformation, since it is the original product carried over by $T$; it is called trivial.\nNon-example: on $A = k[x]$, let $\\mu_1(f, g) = f(0) g(0)$ and $a * b = ab + t \\mu_1(a, b)$.\nFor $a = x$, $b = c = 1$ the left side of the cocycle identity is $x \\cdot 1 - 0 + 0 - 0 = x \\ne 0$, so $*$ is not associative.\nIf the Hochschild cohomology $HH^2(A, A)$ is zero, then every formal deformation of $A$ is trivial, and $A$ is called rigid.\n",
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