BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:James Myer (The CUNY Graduate Center) DTSTART:20210305T180000Z DTEND:20210305T193000Z DTSTAMP:20260422T220453Z UID:STAGE/23 DESCRIPTION:Title: I ntroduction to prismatic cohomology\nby James Myer (The CUNY Graduate Center) as part of STAGE\n\n\nAbstract\nThe study of the cohomology of alg ebraic varieties is depicted by Peter Scholze as a “plane worth” of pa irs of primes $(p\,\\ell)$\, each indexing a cohomology theory for varieti es over $\\mathbb{F}_p$ with coefficients in $\\mathbb{F}_{\\ell}$. The si ngular cohomology occupies a vertical line over $\\infty$\; the étale coh omology dances around\, avoiding the pairs $(p\,p)$\; the analytic de Rham cohomology occupies the top right corner\, intersecting the singular coho mology @ $(\\infty\,\\infty)$\, symbolizing the classical de Rham comparis on theorem\, while the diagonal is picked off by the algebraic de Rham coh omology. Zooming in on a point on the diagonal\, we begin to wonder whethe r there is a cohomology theory interpolating between the étale to the cry stalline (and de Rham). In fact\, the depiction of the plane of pairs of p rimes is striated by lines from each of the various cohomology theories\, but no cohomology theory seems to “wash over” any 2-dimensional part o f the picture and “phase in and out” between any one or the other. The prismatic cohomology theory is this “2-dimensional” theory interpolat ing between the étale and crystalline (and de Rham) theories.\n\nThe clas sical de Rham comparison theorem between the (dual of the) analytic de Rha m cohomology and the singular homology offers a geometric interpretation o f a (co)homology class as an obstruction to (globally) integrating a diffe rential form. This geometric interpretation loses steam when faced with to rsion classes because the integral over a torsion class is always zero. It is also worthwhile to note the relative ease with which we may calculate the de Rham cohomology of a variety (this can be done by machine\, e.g. Ma caulay2) as opposed to the singular cohomology of a variety. So\, how do w e detect these torsion cycles algebraically? We will see via a calculation applying the universal coefficients theorem that the hypothesis of equali ty of dimensions of the analytic and algebraic de Rham cohomology groups o f a variety implies lack of torsion in singular cohomology. Somewhat conve rsely\, we’ll see that the presence of torsion in the singular cohomolog y of the analytic space associated to a variety forces the algebraic de Rh am cohomology group to be larger than expected. This interplay between the various cohomology theories for varieties\, e.g. singular\, étale\, anal ytic de Rham\, algebraic de Rham\, crystalline\, is facilitated by a (spec ialization of a sequence of) remarkable theorem(s) whose proof depends on the existence of\, and motivates the construction of\, the prismatic cohom ology theory. \n\nFollowing this introduction\, we will venture into some detail\, set up some notation for the next speaker\, and elaborate a bit m ore on the story to come.\n LOCATION:https://researchseminars.org/talk/STAGE/23/ END:VEVENT END:VCALENDAR