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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a [[commutative|0310REJJY8]] $k$-algebra.\nFor $a \\in A$, write $a$ also for the map $A \\to A$, $f \\mapsto af$.\nFor a $k$-linear map $\\theta \\colon A \\to A$, write $[\\theta, a] = \\theta \\circ a - a \\circ \\theta$.\nDefine subspaces $D^m(A)$ of the [[endomorphism ring|AWSBZWWMG9]] $\\End_k(A)$ by induction on $m \\ge 0$:\n\n1. $D^0(A)$ is the set of maps $f \\mapsto af$ for $a \\in A$.\n2. For $m \\ge 1$, $D^m(A)$ is the set of $k$-linear maps $\\theta \\colon A \\to A$ with $[\\theta, a] \\in D^{m-1}(A)$ for every $a \\in A$.\n\nThe *ring of differential operators* of $A$ is $D(A) = \\bigcup_{m \\ge 0} D^m(A)$, and the elements of $D^m(A)$ are the differential operators of order at most $m$.\n\n## Remarks\n\n$D(A)$ is a subring of $\\End_k(A)$, because $D^m(A) \\circ D^l(A) \\subseteq D^{m+l}(A)$.\nEvery $k$-linear [[derivation|3J89REV9WQ]] $\\partial$ of $A$ lies in $D^1(A)$, since $[\\partial, a]$ is multiplication by $\\partial(a)$.\nExample: let $k$ have characteristic $0$ and $A = k[x_1, \\dots, x_n]$.\nThen $D(A)$ is generated by the multiplications by $x_i$ and the [[partial derivatives|09EGC6F8PT]] $\\partial/\\partial x_i$, and it is isomorphic to the [[Weyl algebra|98QE7KY55W]] $A_n(k)$.\nNon-example: let $k = \\Q$, $A = \\Q[x]$ and let $\\tau$ be the shift $f(x) \\mapsto f(x + 1)$.\nThen $[\\tau, x] = \\tau$, since $\\tau(xf) = (x+1)\\tau(f)$.\nIf $\\tau$ were in $D^m(A)$, then $\\tau = [\\tau, x]$ would be in $D^{m-1}(A)$, and by induction $\\tau$ would be in $D^0(A)$, a multiplication map; this is false, so $\\tau \\notin D(A)$.\nIn characteristic $p \u003e 0$ the ring $D(k[x])$ is not generated by $x$ and $d/dx$, since $(d/dx)^p = 0$ while the operator $x^p \\mapsto 1$, $x^j \\mapsto \\binom{j}{p} x^{j-p}$ lies in $D^p(k[x])$.\n",
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