Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to $mathsf{WKL}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to $mathsf{ACA}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $mathsf{ATR}_0$ (modulo some induction).
{"title":"Cousin’s lemma in second-order arithmetic","authors":"Jordan Barrett, R. Downey, Noam Greenberg","doi":"10.1090/bproc/111","DOIUrl":"https://doi.org/10.1090/bproc/111","url":null,"abstract":"Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $mathsf{RCA}_0$: \u0000(i) Cousin's lemma for continuous functions is equivalent to $mathsf{WKL}_0$; \u0000(ii) Cousin's lemma for Baire class 1 functions is equivalent to $mathsf{ACA}_0$; \u0000(iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $mathsf{ATR}_0$ (modulo some induction).","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120956940","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}
Green’s general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over an infinite field K K modulo a general linear form. We strengthen Green’s result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O’Carroll on the Eakin-Sathaye theorem for complete and joint reductions.
{"title":"A hyperplane restriction theorem and applications to reductions of ideals","authors":"G. Caviglia","doi":"10.1090/bproc/103","DOIUrl":"https://doi.org/10.1090/bproc/103","url":null,"abstract":"Green’s general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over an infinite field \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000 modulo a general linear form. We strengthen Green’s result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O’Carroll on the Eakin-Sathaye theorem for complete and joint reductions.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133815953","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 if GG is of type Fnmathcal {F}_n, and GG splits as a finite graph of groups, then the vertex groups are of type Fnmathcal {F}_n if the edge groups are of type Fnmathcal {F}_n.
我们证明了如果G G是F n mathcal {F}_n型，并且G G分裂为群的有限图，那么如果边群是F n mathcal {F}_n型，则顶点群是F n mathcal {F}_n型。
{"title":"A note on finiteness properties of graphs of groups","authors":"Frédéric Haglund, D. Wise","doi":"10.1090/BPROC/81","DOIUrl":"https://doi.org/10.1090/BPROC/81","url":null,"abstract":"<p>We show that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is of type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {F}_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> splits as a finite graph of groups, then the vertex groups are of type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {F}_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> if the edge groups are of type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {F}_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123686905","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}
Gowers [Combin. Probab. Comput. 17 (2008), pp. 363–387] elegantly characterized the finite groups GG in which A1A2A3=GA_1A_2A_3=G for any positive density subsets A1,A2,A3A_1,A_2,A_3. This property, quasi-randomness, holds if and only if GG does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of tensor quasi-random groups in which multiplication of subsets is replaced by tensor product of representations.
{"title":"Tensor quasi-random groups","authors":"Mark Sellke","doi":"10.1090/bproc/86","DOIUrl":"https://doi.org/10.1090/bproc/86","url":null,"abstract":"<p>Gowers [Combin. Probab. Comput. 17 (2008), pp. 363–387] elegantly characterized the finite groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in which <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 1 upper A 2 upper A 3 equals upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A_1A_2A_3=G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for any positive density subsets <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 1 comma upper A 2 comma upper A 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A_1,A_2,A_3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This property, <italic>quasi-randomness</italic>, holds if and only if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of <italic>tensor quasi-random</italic> groups in which multiplication of subsets is replaced by tensor product of representations.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117062357","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}
In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.
{"title":"Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions","authors":"Maximilian Engel, Felix Hummel, C. Kuehn","doi":"10.1090/bproc/92","DOIUrl":"https://doi.org/10.1090/bproc/92","url":null,"abstract":"In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124096978","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 extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in L 1 L^1 .
{"title":"A Koopman-von Neumann type theorem on the convergence of Cesàro means in Riesz spaces","authors":"Jonathan Homann, Wen-Chi Kuo, B. Watson","doi":"10.1090/BPROC/75","DOIUrl":"https://doi.org/10.1090/BPROC/75","url":null,"abstract":"We extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in \u0000\u0000 \u0000 \u0000 L\u0000 1\u0000 \u0000 L^1\u0000 \u0000\u0000.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116850117","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}
Myers shows that every compact, connected, orientable 3 3 -manifold with no 2 2 -sphere boundary components contains a hyperbolic knot. We use work of Ikeda with an observation of Adams-Reid to show that every 3 3 -manifold subject to the above conditions contains a hyperbolic knot which admits a non-trivial non-hyperbolic surgery, a toroidal surgery in particular. We conclude with a question and a conjecture about reducible surgeries.
{"title":"Exceptional surgeries in 3-manifolds","authors":"K. Baker, Neil R. Hoffman","doi":"10.1090/bproc/105","DOIUrl":"https://doi.org/10.1090/bproc/105","url":null,"abstract":"Myers shows that every compact, connected, orientable \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-manifold with no \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000-sphere boundary components contains a hyperbolic knot. We use work of Ikeda with an observation of Adams-Reid to show that every \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000-manifold subject to the above conditions contains a hyperbolic knot which admits a non-trivial non-hyperbolic surgery, a toroidal surgery in particular. We conclude with a question and a conjecture about reducible surgeries.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125560799","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}
In this paper, we determine which half-space contains a complete translating soliton of the mean curvature flow and it is related to the well-known half-space theorem for minimal surfaces. We prove that a complete translating soliton does not exist with respect to the velocity v{mathrm {v}} in a closed half-space Hv~={x∈Rn+1∣⟨x,v~⟩≤0}mathcal {H}_{widetilde {{mathrm {v}}}}= { x in mathbb {R}^{n+1} mid langle x, widetil
{"title":"Half-space type theorem for translating solitons of the mean curvature flow in Euclidean space","authors":"Daehwan Kim, Juncheol Pyo","doi":"10.1090/BPROC/67","DOIUrl":"https://doi.org/10.1090/BPROC/67","url":null,"abstract":"<p>In this paper, we determine which half-space contains a complete translating soliton of the mean curvature flow and it is related to the well-known half-space theorem for minimal surfaces. We prove that a complete translating soliton does not exist with respect to the velocity <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal v\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{mathrm {v}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a closed half-space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H Subscript normal v overTilde Baseline equals left-brace x element-of double-struck upper R Superscript n plus 1 Baseline bar mathematical left-angle x comma normal v overTilde mathematical right-angle less-than-or-equal-to 0 right-brace\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo>∣<!-- ∣ --></mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟨<!-- ⟨ --></mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟩<!-- ⟩ --></mml:mo>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {H}_{widetilde {{mathrm {v}}}}= { x in mathbb {R}^{n+1} mid langle x, widetil","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116240327","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 kk be a field, let RR be a commutative ring, and assume the exponential characteristic of kk is invertible in RR. In this note, we prove that isomorphisms in Voevodsky’s triangulated category of motives DM(k;R)mathcal {DM}(k;R) are detected by motivic homology groups of base changes to all separable finitely generated field extensions of kk. It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category H(
{"title":"Detecting motivic equivalences with motivic homology","authors":"David Hemminger","doi":"10.1090/bproc/82","DOIUrl":"https://doi.org/10.1090/bproc/82","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a field, let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a commutative ring, and assume the exponential characteristic of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is invertible in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this note, we prove that isomorphisms in Voevodsky’s triangulated category of motives <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D script upper M left-parenthesis k semicolon upper R right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>;</mml:mo>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {DM}(k;R)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are detected by motivic homology groups of base changes to all separable finitely generated field extensions of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H left-parenthesis k right-parenthesis Subscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 ","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123013413","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 develop a theory of Grassmann semialgebra triples using HasseSchmidt derivations, which formally generalizes results such as the CayleyHamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.
{"title":"Grassmann semialgebras and the Cayley-Hamilton theorem","authors":"Letterio Gatto, L. Rowen","doi":"10.1090/bproc/53","DOIUrl":"https://doi.org/10.1090/bproc/53","url":null,"abstract":"We develop a theory of Grassmann semialgebra triples using HasseSchmidt derivations, which formally generalizes results such as the CayleyHamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121239872","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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