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  "body": "\n## Definition\n\nA [[scheme|5A35TYZD02]] $X$ is *reduced* if for every open set $U \\subseteq X$ the ring $\\mathcal{O}_X(U)$ is a [[reduced ring|SZHY975FFX]].\nThat is, $\\mathcal{O}_X(U)$ has no nonzero [[nilpotent elements|0ZB2JG1M9C]].\n\n## Remarks\n\nEquivalently, every [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is a reduced ring.\nAn affine scheme $\\operatorname{Spec} A$ is reduced if and only if $A$ is reduced.\nExample: $\\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced, since $k[x,y]/(xy)$ has no nonzero nilpotents.\nIndeed, a power of a polynomial lies in $(xy)$ only if the polynomial itself is divisible by both $x$ and $y$.\nNon-example: $\\operatorname{Spec} k[x]/(x^2)$ is not reduced, since the class of $x$ is nonzero but $x^2 = 0$.\nEvery scheme $X$ has a reduced scheme $X_{\\mathrm{red}}$ with the same underlying space, obtained by dividing out the nilpotents.\n",
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