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  "body": "\n## Definition\n\nLet $X$ be a [[scheme|5A35TYZD02]] with structure sheaf $\\mathcal{O}_X$.\nFor an open set $U \\subseteq X$, let $S(U)$ be the set of $s \\in \\mathcal{O}_X(U)$ whose germ $s_x$ is a non-zero-divisor in $\\mathcal{O}_{X,x}$ for every $x \\in U$.\nThe *sheaf of total quotient rings* $\\mathcal{K}_X$ is the [[sheaf|TB6TM6R4MD]] associated to the presheaf $U \\mapsto S(U)^{-1}\\mathcal{O}_X(U)$.\nFor an integral scheme it is the constant sheaf of the function field; in general it is built from the [[total ring of fractions|K5Z67HR9ET]].\nLet $\\mathcal{K}_X^*$ and $\\mathcal{O}_X^*$ be the sheaves of [[units|07C6ECNJET]] of $\\mathcal{K}_X$ and of $\\mathcal{O}_X$.\nA *Cartier divisor* on $X$ is a global section of the quotient sheaf $\\mathcal{K}_X^* / \\mathcal{O}_X^*$.\n\nEquivalently, a Cartier divisor is given by an [[open cover|049PXYHW1T]] $(U_i)$ of $X$ and sections $f_i \\in \\mathcal{K}_X^*(U_i)$ such that $f_i / f_j \\in \\mathcal{O}_X^*(U_i \\cap U_j)$ for all $i, j$.\nTwo such families $(U_i, f_i)$ and $(V_k, g_k)$ define the same divisor when $f_i / g_k \\in \\mathcal{O}_X^*(U_i \\cap V_k)$ for all $i, k$.\n\n## Remarks\n\nThe Cartier divisors form an abelian group, written additively, with $(U_i, f_i) + (U_i, g_i) = (U_i, f_i g_i)$.\nA divisor is *principal* if it is given by one global $f \\in \\mathcal{K}_X^*(X)$, and *effective* if all $f_i$ lie in $\\mathcal{O}_X(U_i)$.\n\nExample: on $X = \\mathbb{A}^1_k = \\operatorname{Spec} k[t]$, the single pair $(X, t)$ is an effective principal Cartier divisor, the point $0$.\nThe compatibility condition holds because there is only one open set, and $t/t = 1$ is a unit.\n\nNon-example: on $\\mathbb{A}^1_k$ take $U = X$ with $f_U = t$ and $V = X \\setminus \\set{0}$ with $f_V = t - 1$.\nOn $U \\cap V = V$, the ratio $t/(t-1)$ is not a unit, since it has a pole at $t = 1$, so the compatibility condition fails.\n\nThe divisor $D = (U_i, f_i)$ defines the invertible subsheaf $\\mathcal{O}_X(D) \\subseteq \\mathcal{K}_X$ generated by $f_i^{-1}$ on $U_i$.\n",
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