The Cremona groups are the groups of all birational equivalences of projective spaces and, equivalently, the automorphism groups of the rational function fields. We construct highly connected spaces on which these groups act in a way that allows us to deduce that their abelianisations, and more generally, the homologies of these groups, stabilise as the dimension increases.
{"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}
This is the first in a series of papers where scissor congruence and K-theoretical invariants are related to cobordism groups of foams in various dimensions. A model example is provided where the cobordism group of weighted one-foams is identified, via the Sah-Arnoux-Fathi invariant, with the first homology of the group of interval exchange automorphisms and with the Zakharevich first K-group of the corresponding assembler. Several variations on this cobordism group are computed as well.
这是剪刀全等和 K 理论不变式与不同维度泡沫的共线性群相关的一系列论文中的第一篇。本文提供了一个模型示例，通过萨-阿努-法蒂不变式，加权一泡沫的协整群与区间交换自形群的第一同调以及相应装配体的扎卡雷维奇第一 K 群相吻合。我们还计算了这个共线性群的几种变化。
{"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}
Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K), H_2,SL(2,K))$ contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in $SL(2,K)$ it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an $SL(2,{bf R})$-bundle over a surface of genus $g$ admits a flat structure if and only if its Euler number is $leq (g-1)$. We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.
{"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}
We consider slice filtrations in logarithmic motivic homotopy theory. Our main results establish conjectured compatibilities with the Beilinson, BMS, and HKR filtrations on (topological, log) Hochschild homology and related invariants. In the case of perfect fields admitting resolution of singularities, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic Atiyah-Hirzerbruch spectral sequence to the BMS spectral sequence. Finally, we consider the Kummer 'etale hypersheafification of logarithmic $K$-theory and show that its very effective slices compute Lichtenbaum 'etale motivic cohomology.
{"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}
We show that every reciprocity sheaf gives rise to a cycle (pre)module in the sense of Rost over a perfect field, under mild additional hypotheses. Over a perfect field of positive characteristic, we show that the first cohomology group of a logarithmic de Rham-Witt sheaf has a partial cycle module structure. As a consequence, we show that Kato complexes of logarithmic de Rham-Witt sheaves satisfy functoriality properties similar to Rost's cycle complexes.
{"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}
We show that the algebraic K-theory of semi-valuation rings with stably coherent regular semi-fraction ring satisfies homotopy invariance. Moreover, we show that these rings are regular if their valuation is non-trivial. Thus they yield examples of regular rings which are not homotopy invariant for algebraic K-theory. On the other hand, they are not necessarily coherent, so that they provide a class of possibly non-coherent examples for homotopy invariance of algebraic K-theory. As an application, we show that Temkin's relative Riemann-Zariski spaces also satisfy homotopy invariance for K-theory under some finiteness assumption.
我们证明，具有稳定相干正则半分数环的半估值环的代数 K 理论满足同调不变性。此外，我们还证明，如果这些环的估值是非三维的，那么它们就是正则环。因此，它们给出了对代数 K 理论来说不具有同调不变性的正则环的例子。另一方面，它们不一定是相干的，因此它们为代数 K 理论的同调不变性提供了一类可能是非相干的例子。作为一个应用，我们证明了滕金的相对黎曼-扎里斯基空间在某种有限性假设下也满足 K 理论的同调不变性。
{"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}
We prove that Atiyah duality holds in the $infty$-category of non-$mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $mathbb A^1$-colocalization, which transforms any module over the $mathbb A^1$-invariant sphere into an $mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.
{"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}
Flux- and charge-quantization laws for higher gauge fields of Maxwell type -- e.g. the common electromagnetic field (the "A-field") but also the B-, RR-, and C-fields considered in string/M-theory -- specify non-perturbative completions of these fields by encoding their solitonic behaviour and hence by specifying the discrete charges carried by the individual branes (higher-dimensional monopoles or solitons) that source the field fluxes. This article surveys the general (rational-)homotopy theoretic understanding of flux- and charge-quantization via the Chern-Dold character map generalized to the non-linear (self-sourcing) Bianchi identities that appear in higher-dimensional supergravity theories, notably for B&RR-fields in D=10, for the C-field in D=11 supergravity, and for the B-field on fivebrane worldvolumes.
麦克斯韦类型的高规场--例如普通电磁场（"A场"），以及弦/M理论中考虑的B场、RR场和C场--的通量和电荷量化定律通过编码这些场的孤子行为，从而通过指定场通量来源的单个支链（高维单极子或孤子）所携带的离散电荷，来指定这些场的非微扰完备性。这篇文章探讨了一般（有理）同调理论对通量和电荷量化的理解，通量和电荷量化通过切尔诺-多尔德特性图泛化到非线性（自源）比安奇特性，这些特性出现在高维超引力理论中，特别是 D=10 的 B&RR 场、D=11 超引力中的 C 场和五布兰世界卷上的 B 场。
{"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}
We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies d'evissage and A^1-invariance over a regular Noetherian base of finite Krull dimension, as well as a projective bundle formula. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of D'eglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory.
我们在不假定 2 在基本方案上是可逆的情况下，建立了关于全息 K 理论的基本动机结果。特别是，我们证明了二次格罗thendieck-维特理论和对称格罗thendieck-维特理论都满足尼斯内维奇后裔，对称格罗thendieck-维特理论在有限克鲁尔维度的正则诺特基上进一步满足d（'evissage）和A^1不变性，以及投影束公式。我们利用这一点证明，在正则诺特基上，对称格罗滕迪克-维特理论是由一个动机E-无限环谱所代表的，然后我们证明了这是一个绝对纯谱，从而回答了D'eglise的一个问题。与代数K理论一样，我们证明在一般基上，我们也可以构造一个后羿K理论动机谱，这次代表的是格罗登第克-维特理论的一个合适的同调不变和卡鲁比定位版本。
{"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}
We give a conceptual explanation for the somewhat mysterious origin of Suslin matrices. This enables us to generalize the construction of Suslin matrices and to give more conceptual proofs of some well-known results.
{"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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