{"title":"Homological stability for the Cremona groups","authors":"Markus Szymik","doi":"arxiv-2403.07546","DOIUrl":"https://doi.org/arxiv-2403.07546","url":null,"abstract":"The Cremona groups are the groups of all birational equivalences of\u0000projective spaces and, equivalently, the automorphism groups of the rational\u0000function fields. We construct highly connected spaces on which these groups act\u0000in a way that allows us to deduce that their abelianisations, and more\u0000generally, the homologies of these groups, stabilise as the dimension\u0000increases.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140115556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Foam cobordism and the Sah-Arnoux-Fathi invariant","authors":"Mee Seong Im, Mikhail Khovanov","doi":"arxiv-2403.06030","DOIUrl":"https://doi.org/arxiv-2403.06030","url":null,"abstract":"This is the first in a series of papers where scissor congruence and\u0000K-theoretical invariants are related to cobordism groups of foams in various\u0000dimensions. A model example is provided where the cobordism group of weighted\u0000one-foams is identified, via the Sah-Arnoux-Fathi invariant, with the first\u0000homology of the group of interval exchange automorphisms and with the\u0000Zakharevich first K-group of the corresponding assembler. Several variations on\u0000this cobordism group are computed as well.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tautological characteristic classes II: the Witt class","authors":"Jan Dymara, Tadeusz Januszkiewicz","doi":"arxiv-2403.05255","DOIUrl":"https://doi.org/arxiv-2403.05255","url":null,"abstract":"Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K),\u0000H_2,SL(2,K))$ contains the class of the universal central extension. When\u0000studying representations of fundamental groups of surfaces in $SL(2,K)$ it is\u0000useful to have classes stable under deformations (Fenchel--Nielsen twists) of\u0000representations. We identify the maximal quotient of the universal class which\u0000is stable under twists as the Witt class of Nekovar. The Milnor--Wood\u0000inequality asserts that an $SL(2,{bf R})$-bundle over a surface of genus $g$\u0000admits a flat structure if and only if its Euler number is $leq (g-1)$. We\u0000establish an analog of this inequality, and a saturation result for the Witt\u0000class. The result is sharp for the field of rationals, but not sharp in\u0000general.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140098086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the logarithmic slice filtration","authors":"Federico Binda, Doosung Park, Paul Arne Østvær","doi":"arxiv-2403.03056","DOIUrl":"https://doi.org/arxiv-2403.03056","url":null,"abstract":"We consider slice filtrations in logarithmic motivic homotopy theory. Our\u0000main results establish conjectured compatibilities with the Beilinson, BMS, and\u0000HKR filtrations on (topological, log) Hochschild homology and related\u0000invariants. In the case of perfect fields admitting resolution of\u0000singularities, the motivic trace map is compatible with the slice and BMS\u0000filtrations, yielding a natural morphism from the motivic Atiyah-Hirzerbruch\u0000spectral sequence to the BMS spectral sequence. Finally, we consider the Kummer\u0000'etale hypersheafification of logarithmic $K$-theory and show that its very\u0000effective slices compute Lichtenbaum 'etale motivic cohomology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Kato complexes of reciprocity sheaves and applications","authors":"Sandeep S, Anand Sawant","doi":"arxiv-2403.01735","DOIUrl":"https://doi.org/arxiv-2403.01735","url":null,"abstract":"We show that every reciprocity sheaf gives rise to a cycle (pre)module in the\u0000sense of Rost over a perfect field, under mild additional hypotheses. Over a\u0000perfect field of positive characteristic, we show that the first cohomology\u0000group of a logarithmic de Rham-Witt sheaf has a partial cycle module structure.\u0000As a consequence, we show that Kato complexes of logarithmic de Rham-Witt\u0000sheaves satisfy functoriality properties similar to Rost's cycle complexes.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory","authors":"Christian Dahlhausen","doi":"arxiv-2403.02413","DOIUrl":"https://doi.org/arxiv-2403.02413","url":null,"abstract":"We show that the algebraic K-theory of semi-valuation rings with stably\u0000coherent regular semi-fraction ring satisfies homotopy invariance. Moreover, we\u0000show that these rings are regular if their valuation is non-trivial. Thus they\u0000yield examples of regular rings which are not homotopy invariant for algebraic\u0000K-theory. On the other hand, they are not necessarily coherent, so that they\u0000provide a class of possibly non-coherent examples for homotopy invariance of\u0000algebraic K-theory. As an application, we show that Temkin's relative\u0000Riemann-Zariski spaces also satisfy homotopy invariance for K-theory under some\u0000finiteness assumption.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Atiyah duality for motivic spectra","authors":"Toni Annala, Marc Hoyois, Ryomei Iwasa","doi":"arxiv-2403.01561","DOIUrl":"https://doi.org/arxiv-2403.01561","url":null,"abstract":"We prove that Atiyah duality holds in the $infty$-category of non-$mathbb\u0000A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth\u0000projective scheme is dualizable with dual given by the Thom spectrum of its\u0000negative tangent bundle. The Gysin maps recently constructed by L. Tang are a\u0000key ingredient in the proof. We then present several applications. First, we\u0000study $mathbb A^1$-colocalization, which transforms any module over the\u0000$mathbb A^1$-invariant sphere into an $mathbb A^1$-invariant motivic spectrum\u0000without changing its values on smooth projective schemes. This can be applied\u0000to all known $p$-adic cohomology theories and gives a new elementary approach\u0000to \"logarithmic\" or \"tame\" cohomology theories; it recovers for instance the\u0000logarithmic crystalline cohomology of strict normal crossings compactifications\u0000over perfect fields and shows that the latter is independent of the choice of\u0000compactification. Second, we prove a motivic Landweber exact functor theorem,\u0000associating a motivic spectrum to any graded formal group law classified by a\u0000flat map to the moduli stack of formal groups. Using this theorem, we compute\u0000the ring of $mathbb P^1$-stable cohomology operations on the algebraic\u0000K-theory of qcqs derived schemes, and we prove that rational motivic cohomology\u0000is an idempotent motivic spectrum.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flux Quantization","authors":"Hisham Sati, Urs Schreiber","doi":"arxiv-2402.18473","DOIUrl":"https://doi.org/arxiv-2402.18473","url":null,"abstract":"Flux- and charge-quantization laws for higher gauge fields of Maxwell type --\u0000e.g. the common electromagnetic field (the \"A-field\") but also the B-, RR-, and\u0000C-fields considered in string/M-theory -- specify non-perturbative completions\u0000of these fields by encoding their solitonic behaviour and hence by specifying\u0000the discrete charges carried by the individual branes (higher-dimensional\u0000monopoles or solitons) that source the field fluxes. This article surveys the general (rational-)homotopy theoretic understanding\u0000of flux- and charge-quantization via the Chern-Dold character map generalized\u0000to the non-linear (self-sourcing) Bianchi identities that appear in\u0000higher-dimensional supergravity theories, notably for B&RR-fields in D=10, for\u0000the C-field in D=11 supergravity, and for the B-field on fivebrane\u0000worldvolumes.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A motivic spectrum representing hermitian K-theory","authors":"Baptiste Calmès, Yonatan Harpaz, Denis Nardin","doi":"arxiv-2402.15136","DOIUrl":"https://doi.org/arxiv-2402.15136","url":null,"abstract":"We establish fundamental motivic results about hermitian K-theory without\u0000assuming that 2 is invertible on the base scheme. In particular, we prove that\u0000both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich\u0000descent, and that symmetric Grothendieck-Witt theory further satisfies\u0000d'evissage and A^1-invariance over a regular Noetherian base of finite Krull\u0000dimension, as well as a projective bundle formula. We use this to show that\u0000over a regular Noetherian base, symmetric Grothendieck-Witt theory is\u0000represented by a motivic E-infinity-ring spectrum, which we then show is an\u0000absolutely pure spectrum, answering a question of D'eglise. As with algebraic\u0000K-theory, we show that over a general base, one can also construct a hermitian\u0000K-theory motivic spectrum, representing this time a suitable homotopy invariant\u0000and Karoubi-localising version of Grothendieck-Witt theory.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139968850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on Suslin matrices and Clifford algebras","authors":"Tariq Syed","doi":"arxiv-2402.15094","DOIUrl":"https://doi.org/arxiv-2402.15094","url":null,"abstract":"We give a conceptual explanation for the somewhat mysterious origin of Suslin\u0000matrices. This enables us to generalize the construction of Suslin matrices and\u0000to give more conceptual proofs of some well-known results.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139968849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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