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  "body": "\n## Summary\n\nThe Rees algebra of an [[ideal|096Z9MP2ME]] $I$ of a [[commutative ring|0310REJJY8]] $R$ is the [[graded ring|8KB4SQGC0C]] $R[It] = \\bigoplus_{n \\ge 0} I^n t^n$, a subring of the polynomial ring $R[t]$.\nIt packs all the powers of $I$ into one ring, so that statements about all $I^n$ become statements about a single ring.\nIts quotient by $I R[It]$ is the [[associated graded ring|8HFAEGFQT1]], and in geometry it gives the blowing up along $I$.\n\n## Explanations\n\n### L1 human\n\nThe powers $I, I^2, I^3, \\dots$ of an ideal form an infinite list.\nThe Rees algebra stores the whole list in one ring, using a new variable $t$ to label which power each element comes from.\nFor $R = k[x]$ and $I = (x)$, the Rees algebra is $k[x, xt]$, a polynomial ring in the two elements $x$ and $xt$.\nThe [[definition|6YCKECNR3T]] makes this precise.\n",
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