Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108183
Kensuke Arakawa
We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus–Sagave.
{"title":"Monoidal relative categories model monoidal ∞-categories","authors":"Kensuke Arakawa","doi":"10.1016/j.jpaa.2026.108183","DOIUrl":"10.1016/j.jpaa.2026.108183","url":null,"abstract":"<div><div>We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus–Sagave.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108183"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108181
H. Ananthnarayan , Omkar Javadekar , Rajiv Kumar
Let R be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded R-modules and the Koszul property of R. As an application of this, we show that the existence of an R-module of finite regularity and infinite projective dimension forces R to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded R-modules, and show that when , they are spanned by the Betti tables of pure R-modules if and only if R is Cohen–Macaulay with minimal multiplicity.
{"title":"Betti cones over fibre products","authors":"H. Ananthnarayan , Omkar Javadekar , Rajiv Kumar","doi":"10.1016/j.jpaa.2026.108181","DOIUrl":"10.1016/j.jpaa.2026.108181","url":null,"abstract":"<div><div>Let <em>R</em> be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded <em>R</em>-modules and the Koszul property of <em>R</em>. As an application of this, we show that the existence of an <em>R</em>-module of finite regularity and infinite projective dimension forces <em>R</em> to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded <em>R</em>-modules, and show that when <span><math><mtext>depth</mtext><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, they are spanned by the Betti tables of pure <em>R</em>-modules if and only if <em>R</em> is Cohen–Macaulay with minimal multiplicity.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108181"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108186
Elad Paran , Tran Nam Son
We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if are elements of the free associative algebra generated by variables over an algebraically closed division ring D of finite dimension over its center F, and if the induced map is injective, then f must be surjective. With no condition on the dimension over the center, our second result is that if p is either an element in with zero constant term such that , or a nonconstant polynomial in . Furthermore, we also establish some Waring type results. For instance, for any integer , we prove that every matrix in can be expressed as a difference of pairs of multiplicative commutators of elements from , provided again that D is finite-dimensional over F.
{"title":"Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings","authors":"Elad Paran , Tran Nam Son","doi":"10.1016/j.jpaa.2026.108186","DOIUrl":"10.1016/j.jpaa.2026.108186","url":null,"abstract":"<div><div>We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are elements of the free associative algebra <span><math><mi>D</mi><mo>〈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>〉</mo></math></span> generated by <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span> variables over an algebraically closed division ring <em>D</em> of finite dimension over its center <em>F</em>, and if the induced map <span><math><mi>f</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>:</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>→</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> is injective, then <em>f</em> must be surjective. With no condition on the dimension over the center, our second result is that <span><math><mi>p</mi><mo>(</mo><mi>D</mi><mo>)</mo><mo>=</mo><mi>D</mi></math></span> if <em>p</em> is either an element in <span><math><mi>F</mi><mo>〈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>〉</mo></math></span> with zero constant term such that <span><math><mi>p</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≠</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>, or a nonconstant polynomial in <span><math><mi>F</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. Furthermore, we also establish some Waring type results. For instance, for any integer <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span>, we prove that every matrix in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo></math></span> can be expressed as a difference of pairs of multiplicative commutators of elements from <span><math><mi>p</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo><mo>)</mo></math></span>, provided again that <em>D</em> is finite-dimensional over <em>F</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108186"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108182
Petter Andreas Bergh , David A. Jorgensen , Peder Thompson
Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
{"title":"Asymptotic vanishing of cohomology in triangulated categories","authors":"Petter Andreas Bergh , David A. Jorgensen , Peder Thompson","doi":"10.1016/j.jpaa.2026.108182","DOIUrl":"10.1016/j.jpaa.2026.108182","url":null,"abstract":"<div><div>Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108182"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108185
Tomasz Ciborski
The aim of this paper is to calculate entropy in the sense of Dimitrov–Haiden–Katzarkov–Kontsevich and polynomial entropy as defined by Fan–Fu–Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field.
{"title":"Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras","authors":"Tomasz Ciborski","doi":"10.1016/j.jpaa.2026.108185","DOIUrl":"10.1016/j.jpaa.2026.108185","url":null,"abstract":"<div><div>The aim of this paper is to calculate entropy in the sense of Dimitrov–Haiden–Katzarkov–Kontsevich and polynomial entropy as defined by Fan–Fu–Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108185"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108187
Srishti Singh, Hema Srinivasan
Consider a numerical semigroup minimally generated by a subset of the interval with multiplicity e and width . Such numerical semigroups are called Sally type semigroups. We show that the defining ideals of these semigroup rings, when the embedding dimension is , generically have the structure of the sum of two determinantal ideals. More generally, Sally type numerical semigroups with multiplicity e and embedding dimension are obtained by introducing k gaps in the interval . It is known that for , there is precisely one such semigroup that is Gorenstein, and it happens when one deletes consecutive integers. Let denote the Sally type numerical semigroup of multiplcity e, embedding dimension obtained by deleting the k consecutive integers . We prove that for any , the semigroup is Gorenstein if and only if . We construct an explicit minimal free resolution of the semigroup ring of and compute the Betti numbers. In general, we characterize when are symmetric and construct minimal resolutions for these Gorenstein semigroup rings.
{"title":"Structure and symmetry of sally type semigroup rings","authors":"Srishti Singh, Hema Srinivasan","doi":"10.1016/j.jpaa.2026.108187","DOIUrl":"10.1016/j.jpaa.2026.108187","url":null,"abstract":"<div><div>Consider a numerical semigroup minimally generated by a subset of the interval <span><math><mo>[</mo><mi>e</mi><mo>,</mo><mn>2</mn><mi>e</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span> with multiplicity <em>e</em> and width <span><math><mi>e</mi><mo>−</mo><mn>1</mn></math></span>. Such numerical semigroups are called Sally type semigroups. We show that the defining ideals of these semigroup rings, when the embedding dimension is <span><math><mi>e</mi><mo>−</mo><mn>2</mn></math></span>, generically have the structure of the sum of two determinantal ideals. More generally, Sally type numerical semigroups with multiplicity <em>e</em> and embedding dimension <span><math><mi>d</mi><mo>=</mo><mi>e</mi><mo>−</mo><mi>k</mi></math></span> are obtained by introducing <em>k</em> gaps in the interval <span><math><mo>[</mo><mi>e</mi><mo>,</mo><mn>2</mn><mi>e</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>. It is known that for <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, there is precisely one such semigroup that is Gorenstein, and it happens when one deletes consecutive integers. Let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>(</mo><mi>j</mi><mo>)</mo></math></span> denote the Sally type numerical semigroup of multiplcity <em>e</em>, embedding dimension <span><math><mi>e</mi><mo>−</mo><mi>k</mi></math></span> obtained by deleting the <em>k</em> consecutive integers <span><math><mi>j</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>j</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. We prove that for any <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>e</mi><mo>/</mo><mn>2</mn></math></span>, the semigroup <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>(</mo><mi>j</mi><mo>)</mo></math></span> is Gorenstein if and only if <span><math><mi>j</mi><mo>=</mo><mi>k</mi></math></span>. We construct an explicit minimal free resolution of the semigroup ring of <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>)</mo></math></span> and compute the Betti numbers. In general, we characterize when <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>(</mo><mi>j</mi><mo>)</mo></math></span> are symmetric and construct minimal resolutions for these Gorenstein semigroup rings.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108187"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108184
Enric Nart , Josnei Novacoski
The depth of a simple algebraic extension of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on determined by the choice of different generators of the extension. In [11], we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on , we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of K have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.
有值域的简单代数扩展（L/K,v）的深度是K[x]上的赋值的Mac lane - vaqui链的最小长度，该长度由该扩展的不同生成器的选择决定。在[11]中，我们刻画了深度1的无缺陷无分支扩展。本文分析了Artin-Schreier缺陷扩展塔的这一问题。在（K,v）上的一定条件下，证明了由K的线性不相交缺陷Artin-Schreier扩展复合得到的塔深度为1。我们推测这些是唯一深度的阿汀-施赖尔缺陷塔，我们提出了一些例子来支持这一猜想。
{"title":"Depth of Artin-Schreier defect towers","authors":"Enric Nart , Josnei Novacoski","doi":"10.1016/j.jpaa.2026.108184","DOIUrl":"10.1016/j.jpaa.2026.108184","url":null,"abstract":"<div><div>The depth of a simple algebraic extension <span><math><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on <span><math><mi>K</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> determined by the choice of different generators of the extension. In <span><span>[11]</span></span>, we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>, we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of <em>K</em> have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108184"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.jpaa.2026.108172
Kaiyue He
We introduce a new numerical invariant associated to a finite-length R-module M and an ideal I in an Artinian local ring R. This invariant measures the ratio between and . We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of I-free modules — those modules M for which is isomorphic to a direct sum of copies of . Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.
{"title":"Betti numbers for modules over Artinian local rings","authors":"Kaiyue He","doi":"10.1016/j.jpaa.2026.108172","DOIUrl":"10.1016/j.jpaa.2026.108172","url":null,"abstract":"<div><div>We introduce a new numerical invariant <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> associated to a finite-length <em>R</em>-module <em>M</em> and an ideal <em>I</em> in an Artinian local ring <em>R</em>. This invariant measures the ratio between <span><math><mi>λ</mi><mo>(</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span>. We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of <em>I</em>-free modules — those modules <em>M</em> for which <span><math><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi></math></span> is isomorphic to a direct sum of copies of <span><math><mi>R</mi><mo>/</mo><mi>I</mi></math></span>. Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108172"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.jpaa.2026.108171
Donald M. Davis , Douglas C. Ravenel , W. Stephen Wilson
We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective n-th Morava K-theory of the second mod p Eilenberg-MacLane space.
{"title":"The connective Morava K-theory of the second mod p Eilenberg-MacLane space","authors":"Donald M. Davis , Douglas C. Ravenel , W. Stephen Wilson","doi":"10.1016/j.jpaa.2026.108171","DOIUrl":"10.1016/j.jpaa.2026.108171","url":null,"abstract":"<div><div>We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective <em>n</em>-th Morava <em>K</em>-theory of the second mod <em>p</em> Eilenberg-MacLane space.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108171"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.jpaa.2026.108173
Shengding Sun , Aljaž Zalar
The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In [28] this was extended to the characterization on arbitrary closed semialgebraic sets by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when K is the whole line, an unbounded interval, a union of two unbounded intervals, and according to a conjecture of [28] also when K is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem on a union of a bounded interval and a point. The presented technique for solving the corresponding moment problem can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets K[28, Theorem C].
{"title":"Matrix Fejér-Riesz type theorem for a union of an interval and a point","authors":"Shengding Sun , Aljaž Zalar","doi":"10.1016/j.jpaa.2026.108173","DOIUrl":"10.1016/j.jpaa.2026.108173","url":null,"abstract":"<div><div>The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In <span><span>[28]</span></span> this was extended to the characterization on arbitrary closed semialgebraic sets <span><math><mi>K</mi><mo>⊆</mo><mi>R</mi></math></span> by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when <em>K</em> is the whole line, an unbounded interval, a union of two unbounded intervals, and according to a conjecture of <span><span>[28]</span></span> also when <em>K</em> is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem on a union of a bounded interval and a point. The presented technique for solving the corresponding moment problem can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets <em>K</em> <span><span>[28, Theorem C]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108173"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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