{"abstracts":[{"sha1":"c98f8eea2466f81d29239fdc4b3c28a8a4c07339","content":"Let K be the field of fractions of a Henselian discrete valuation ring O_K.\nLet X_K/K be a smooth proper geometrically connected scheme admitting a regular\nmodel X/O_K. We show that the index \\delta(X_K/K) of X_K/K can be explicitly\ncomputed using data pertaining only to the special fiber X_k/k of the model X.\n We give two proofs of this theorem, using two moving lemmas. One moving lemma\npertains to horizontal 1-cycles on a regular projective scheme X over the\nspectrum of a semi-local Dedekind domain, and the second moving lemma can be\napplied to 0-cycles on an FA-scheme X which need not be regular.\n The study of the local algebra needed to prove these moving lemmas led us to\nintroduce an invariant \\gamma(A) of a singular local ring (A, \\m): the greatest\ncommon divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all\n\\m-primary ideals Q in \\m. We relate this invariant \\gamma(A) to the index of\nthe exceptional divisor in a resolution of the singularity of Spec(A), and we\ngive a new way of computing the index of a smooth subvariety X_K/K of P^n_K\nover any field K, using the invariant \\gamma of the local ring at the vertex of\na cone over X.","mimetype":"text/plain","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"Ofer Gabber","role":"author"},{"index":1,"raw_name":"Qing Liu","role":"author"},{"index":2,"raw_name":"Dino Lorenzini","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v1","ext_ids":{"arxiv":"1209.2828v1"},"release_year":2012,"release_date":"2012-09-13","release_stage":"submitted","release_type":"article","webcaptures":[],"filesets":[],"files":[{"release_ids":["kibpey66zzc3toswdl3f37rh3u"],"mimetype":"application/pdf","urls":[{"url":"https://archive.org/download/arxiv-1209.2828/1209.2828.pdf","rel":"archive"}],"sha256":"ee1c02d2c2f674249ff816b5aac30fe5f3a5a445b5aaac65196a55ebe98056ce","sha1":"58f1c5bf5a7ad959a3279a087df6f2af30bd2cd1","md5":"7f7b64895bfcd5a8ea937c9f437fcd3c","size":624553,"revision":"de3cabcc-1cb8-4cc1-9440-18af33d14680","ident":"fyw6ljpzffddjlf2xwz4dsrvxq","state":"active"}],"work_id":"wgac7tv6kbdghkvg2qoxbmpthi","title":"The index of an algebraic variety","state":"active","ident":"kibpey66zzc3toswdl3f37rh3u","revision":"63fbf8e8-64ff-48ad-8e7f-ae3e6028fad0","extra":{"arxiv":{"base_id":"1209.2828","categories":["math.AG","math.AC","math.NT"],"comments":"To appear in Invent. Math","journal_ref":"Invent. Math., 192 (2013), 567-626"}}}