# Cartier divisor

On a scheme $X$, a Cartier divisor is a global section of the sheaf $\mathcal{K}^{*}/\mathcal{O}^{*}$, where $\mathcal{K}^{*}$ is the multiplicative sheaf of meromorphic functions, and $\mathcal{O}^{*}$ the multiplicative sheaf of invertible regular functions (the units of the structure sheaf).

More explicitly, a Cartier divisor is a choice of open cover $U_{i}$ of $X$, and meromorphic functions $f_{i}\in\mathcal{K}^{*}(U_{i})$, such that $f_{i}/f_{j}\in\mathcal{O}^{*}(U_{i}\cap U_{j})$, along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if $f_{i}$ is replaced by $gf_{i}$ with $g\in\mathcal{O}_{*}$.

Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.

Title Cartier divisor CartierDivisor 2013-03-22 13:52:29 2013-03-22 13:52:29 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 14A99
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 11, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9733219742774963, "perplexity": 464.71881942908476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039560245.87/warc/CC-MAIN-20210422013104-20210422043104-00241.warc.gz"}