BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:James Borger DTSTART:20251203T090000Z DTEND:20251203T095000Z DTSTAMP:20260422T102722Z UID:FGC-IPM/61 DESCRIPTION:Title: Scheme Theory over Semirings\nby James Borger as part of FGC-HRI-IPM Number Theory Webinars\n\n\nAbstract\nUsual scheme theory can be viewed as the syntactic theory of\npolynomial equations with coefficients in a ring \, most importantly the\nring of integers. But none of its most fundamenta l ingredients\, such\nas\nfaithfully flat descent\, require subtraction. S o we can set up a\nscheme theory over semirings (``rings but possibly with out additive\ninverses’’\, such as the non-negative integers or reals) \, thus\nbringing positivity in to the foundations of scheme theory. It is then\nreasonable to view non-negativity as integrality at the infinite\np lace\, the Boolean semiring as the residue field there\, and the non-negat ive\nreals as the completion.\n\n In this talk\, I'll discuss some recent developments in module theory\n over semirings. While the classical defini tions of ``vector bundle''\n are\nnot all equivalent over semirings\, the classical definitions of ``line\nbundle'' are all equivalent\, which allow s us to define Picard groups\nand\nPicard stacks. The narrow class group o f a number field can be\nrecovered\nas the reflexive class group of the se miring of its totally\n nonnegative\nintegers\, i.e. the arithmetic compac tification of the spectrum of the\nring of integers. This gives a scheme-t heoretic definition of the\nnarrow\nclass group\, as was done for the ordi nary class group a long time ago.\n\nThis is based mostly on arXiv:2405.18 645\, which is joint work with\n Jaiung Jun\, and also on forthcoming pape r with Johan de Jong and Ivan\nZelich.\n LOCATION:https://researchseminars.org/talk/FGC-IPM/61/ END:VEVENT END:VCALENDAR