BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Arvid Siqveland (Universitetet i Sørøst-Norge) DTSTART:20251210T200000Z DTEND:20251210T210000Z DTSTAMP:20260420T053237Z UID:NYC-NCG/217 DESCRIPTION:Title: Localization in Associative Rings and Associative Schemes\nby Arvid Siqveland (Universitetet i Sørøst-Norge) as part of Noncommutative geome try in NYC\n\n\nAbstract\nWe start with the argument for doing associative algebraic geometry: We need schemes of associative algebras to parametriz e (find moduli of) noncommutative objects.\n\nLet $A$ be a commutative rin g. Then we can define the sheaf of rings on $X=Spec ~A$ by letting $O_X(U) =im A\\subseteq %\\underset{m\\in U\\text{ maximal}}\n\\prod A_m\,$ and to generalize this to rings that are not necessarily commutative\, we need a replacement for the local rings $A_m.$\nWe change our view: The interesti ng point about $A_m$ is not that it is local\, but rather that it is local ly representing\, i.e. that in the category of pointed rings\, $mor( m\\su bset A\,-)$ is represented by $A_m.$\n\nLet $A$ be an Associative (not nec essarily commutative) ring. and let $M$ be a simple right $A$-module. We p rove that in the category of pointed associative rings there is a pointed associative ring $A_M$ representing $mor((A\,M)\,-).$ Moreover\, we prove that for any set of $r>0$ simple modules $M=\\{M_i\\}_{i=1}^r\,$ the cate gorical product $A_M=\\prod_{i=1}^r A_{M_i}$ exists. (When $A$ is noncommu tative\, this is certainly not the Cartesian product). Given this\, we can define $aspec ~A$ as the set of simple right $A$-modules\, together with the contractions of such\, and we give $X=aspec ~A$ the topology generated by $\\{D(f)\\}_{f\\in A}\,$ defined in such a way that if $A$ is commutat ive\, this is the ordinary Zariski topology. Then $O_X(U)=im A\\subseteq%\ \underset{M\\in\\simp A\\cap U}\n\\prod A_M$ is a sheaf\, and an associati ve scheme is a ringed space $X$ covered by affine open sets.\n\nWe end by defining A Noncommutative Geometry. Let $Y=\\mathbb R^3\\times\\mathbb R^3 =\\{(\\text{observer}\,\\text{observed})\\}.$ We let $\\mathbb U$ be the n oncommutative blowup of $\\Delta\\subseteq\\mathbb R^3\\times\\mathbb R^3$ which is adding a tangent direction to each $(x\,x)\\in\\Delta.$ Choose a Riemannian metric on $\\mathbb R^3.$ Then the maximal velocity is the len gth of the tangent vector on one side of the diagonal\, and we also get a n opposite tangent vector on the dark side of the diagonal.\n\nEverything in this lecture are Turing computable\, and so everything can be computed by infinitesimally deformation theory. See O.A. Laudal's book [2] for th e study of this model.\n\n\nBibliography\n\n\n\n1. E. Eriksen\, O. A. Laud al\, A. Siqveland\,\nNoncommutative Deformation Theory. Monographs and Res earch Notes in Mathematics. CRC Press\, Boka Raton\, FL\, 2017\n\n\n\n2. O . A. Laudal\, Mathematical models in science\, World Scientific Publishing Co. Pte. Ltd.\, Hackensack\, NJ\, 2021\n\n\n\n3. A. Siqveland\, \nAssocia tive Algebraic Geometry\,\nWorld Scientific Publishing Co. Pte. Ltd.\, Hac kensack\, NJ\, 2023\nISBN: 977-1-80061-354-6\n\n\n4. A. Siqveland\,\nAssoc iative Schemes\,\\\\\nhttps://doi.org/10.48550/arXiv.2302.13843\,\n2024\n\ n\n5. A. Siqveland\,\nCountably Generated Matrix Algebras\,\\\\\nhttps://d oi.org/10.48550/arXiv.2408.01034\,\n2024\n\n\n6. A. Siqveland\,\nShemes of Associative Algebras\,\\\\\nhttps://doi.org/10.48550/arXiv.2410.17703\,\n 2024\n\n\n7. A. Siqveland\,\nAssociative Local Function Rings\,\\\\\nhttps ://doi.org/10.48550/arXiv.2410.16819\,\n2024\n\n\n8. A. Siqveland\,\nCateg orical Construction of Schemes\,\\\\\nhttps://arxiv.org/abs/2511.03433\,\n 2025\n\n\n9. A. Siqveland\,\nSchemes of Object in abelian Categories\,\\\\ \nhttp://arxiv.org/abs/2511.04191\,\n2025\n\n\n10. A. Siqveland\,\nLocaliz ation in Associative Rings\,\\\\\nhttp://arxiv.org/abs/2511.07900\,\n2025\ n\n\n11. A. Siqveland\,\nAssociative Schemes and Subschemes\,\\\\\nhttp:// arxiv.org/abs/2511.09176\,\n2025\n LOCATION:https://researchseminars.org/talk/NYC-NCG/217/ END:VEVENT END:VCALENDAR