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  "body": "\n## Definition\n\nLet $X$ be a [[Noetherian|ND6XEY68P4]] [[integral|17NVR4XBHK]] [[separated|X30G4V7VAJ]] scheme that is regular in codimension one.\nThis means that the local ring of $X$ at every point of codimension one is a regular local ring of dimension one.\nA *prime divisor* on $X$ is a closed integral subscheme $Y \\subseteq X$ of codimension one.\nA *Weil divisor* on $X$ is an element of the free [[abelian group|082GHNPJ53]] $\\operatorname{Div}(X)$ generated by the prime divisors.\nSo it is a finite formal sum $D = \\sum_i n_i Y_i$ with $n_i \\in \\Z$ and prime divisors $Y_i$.\n\n## Remarks\n\nLet $Y$ be a prime divisor with generic point $\\eta_Y$.\nThe local ring $\\mathcal{O}_{X, \\eta_Y}$ is a [[discrete valuation ring|MGM3CNJW9V]], with valuation $v_Y$ on the function field $K$ of $X$.\nFor $f \\in K^\\times$, the *principal divisor* of $f$ is $\\operatorname{div}(f) = \\sum_Y v_Y(f) Y$, which is a finite sum.\nOn $\\mathbb{P}^1_k$ over a field $k$, the function $f = x^2/(x - 1)$ has $\\operatorname{div}(f) = 2[0] - [1] - [\\infty]$.\nA closed point of $\\mathbb{A}^2_k$ is not a prime divisor, since it has codimension two, not one.\nThe quotient of $\\operatorname{Div}(X)$ by the principal divisors is the divisor class group $\\operatorname{Cl}(X)$.\nFor example, $\\operatorname{Cl}(\\mathbb{P}^n_k) \\cong \\Z$ by the degree, for $n \\ge 1$.\n",
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  "name": "Weil divisor",
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