{ "142253-21890273": { "datetime_modified":"20251130T142656", "datetime_start":"20251201T160000", "datetime_end":"20251201T170000", "has_end_time":1, "date_start":"2025-12-01", "date_end":"2025-12-01", "time_start":"16:00:00", "time_end":"17:00:00", "time_zone":"America\/Detroit", "event_title":"Generically Artinian modules and duality", "occurrence_title":"", "combined_title":"Generically Artinian modules and duality: Mel Hochster", "event_subtitle":"Mel Hochster", "event_type":"Workshop \/ Seminar", "event_type_id":"21", "description":"Abstract: The talk treats joint work of Yongwei Yao and the speaker that is in progress. Let $R$ be a Noetherian ring, let $P \\in \\Spec(R)$, and let $A:= R\/P$. We discuss a theory of generically Artinian modules for $R$ at $P$ when $P$ is a prime that need not be maximal: results similar to Matlis duality hold on a *Zariski neighborhood of* $P$. We introduce the notion of a *generically Artinian module at* $P$, of a *generically injective module at* $P$, and of a *generically injective hull* $E$ for $R\/P$. When it exists, $E$ turns out to be unique up to non-unique isomorphism after possibly passing to a smaller Zariski neighborhood of $P$. The results are proved under mild conditions on $R$. The key results show that many statements from classical duality theory hold after localizing at just *one* element of $R \\setminus P$. Here is one example. If $H$ is any generically Artinian module at $P$ then, after localizing at one element $g \\in R \\setminus P$, the associated graded module of $H_g$, namely, $\\bigoplus_{t = 0} ^\\infty {{\\Ann_{H_g}P^{t+1}}\\over{\\Ann_{H_g} P^t}}$, is free over $A_g$: in fact, all of its graded components are $A_g$-free. This parallels classical results of Grothendieck on generic freeness in EGA, but the strength of this and several other results is surprising, because the modules considered typically have neither ACC nor DCC. It turns out that under mild conditions on $R$, the local cohomology modules $H^i_P(M)$ for a Noetherian $R$-module $M$ are generically Artinian. These results recover, in a much more general framework, earlier work of the authors, which generalized results of Karen Smith and J\\'anos Koll\\'ar. The authors have used these ideas to settle long standing questions in tight closure theory.", "occurrence_notes":null, "guid":"142253-21890273@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/142253", "building_id":"1000166", "building_name":"East Hall", "campus_maps_id":"53", "room":"3088", "location_name":"East Hall", "has_livestream":0, "cost":"", "tags":["Mathematics","seminar"], "website":"", "sponsors":[ { "group_name":"Commutative Algebra Seminar - Department of Mathematics", "group_id":"4886", "website":"" }, { "group_name":"Department of Mathematics", "group_id":"3791", "website":"" } ], "image_url":"", "styled_images":{ "event_thumb":"", "event_large":"", "event_large_2x":"", "event_large_lightbox":"", "group_thumb":"", "group_thumb_square":"", "group_large":"", "group_large_lightbox":"", "event_large_crop":"", "event_list":"", "event_list_2x":"", "event_grid":"", "event_grid_2x":"", "event_feature_large":"", "event_feature_thumb":"" }, "occurrence_count":1, "first_occurrence":21890273 } , "136341-21878520": { "datetime_modified":"20251130T075111", "datetime_start":"20251201T160000", "datetime_end":"20251201T170000", "has_end_time":1, "date_start":"2025-12-01", "date_end":"2025-12-01", "time_start":"16:00:00", "time_end":"17:00:00", "time_zone":"America\/Detroit", "event_title":"GLNT: Duality and Fourier-Deligne for certain connected commutative unipotent group ind-schemes", "occurrence_title":"", "combined_title":"GLNT: Duality and Fourier-Deligne for certain connected commutative unipotent group ind-schemes: Saniya Wagh (UM)", "event_subtitle":"Saniya Wagh (UM)", "event_type":"Workshop \/ Seminar", "event_type_id":"21", "description":"There is a notion of duality due to Serre on the category of perfect connected commutative unipotent groups (cpu) over an algebraically closed field of positive characteristics k. For the objects in (cpu), there is also an analogue of the Fourier transform known as the Fourier-Deligne transform due to Deligne. \r\n\r\nIn this talk, we introduce the category of Tate objects Tate(cpu), whose objects are certain connected commutative unipotent group ind-schemes. We then extend the aforementioned notions of duality and Fourier-Deligne transform to the category of Tate objects. This talk is based on joint work with Tanmay Deshpande.", "occurrence_notes":null, "guid":"136341-21878520@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/136341", "building_id":"1000166", "building_name":"East Hall", "campus_maps_id":"53", "room":"4096", "location_name":"East Hall", "has_livestream":0, "cost":"", "tags":["Mathematics"], "website":"", "sponsors":[ { "group_name":"Group, Lie and Number Theory Seminar - Department of Mathematics", "group_id":"4892", "website":"" }, { "group_name":"Department of Mathematics", "group_id":"3791", "website":"" } ], "image_url":"", "styled_images":{ "event_thumb":"", "event_large":"", "event_large_2x":"", "event_large_lightbox":"", "group_thumb":"", "group_thumb_square":"", "group_large":"", "group_large_lightbox":"", "event_large_crop":"", "event_list":"", "event_list_2x":"", "event_grid":"", "event_grid_2x":"", "event_feature_large":"", "event_feature_thumb":"" }, "occurrence_count":1, "first_occurrence":21878520 } }