Journal of Pure and Applied Algebra最新文献

{"title":"Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class","authors":"E. Javier Elizondo ,&nbsp;Alex Fink ,&nbsp;Cristhian Garay López","doi":"10.1016/j.jpaa.2025.107930","DOIUrl":"10.1016/j.jpaa.2025.107930","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>(</mo><mi>d</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> be the complex Grassmannian of affine <em>d</em>-planes in <em>n</em>-space. We study the problem of characterizing the set of algebraic subvarieties of <span><math><mi>G</mi><mo>(</mo><mi>d</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> invariant under the action of the maximal torus <em>T</em> and having given homology class <em>λ</em>. We give a complete answer for the case where <em>λ</em> is the class of a <em>T</em>-orbit, and partial results for other cases, using techniques inspired by matroid theory. This problem has applications to the computation of the Euler-Chow series for Grassmannians of projective lines.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107930"},"PeriodicalIF":0.7,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143610149","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}

{"title":"A negative answer to a Bahturin-Regev conjecture about regular algebras in positive characteristic","authors":"Lucio Centrone ,&nbsp;Plamen Koshlukov ,&nbsp;Kauê Pereira","doi":"10.1016/j.jpaa.2025.107933","DOIUrl":"10.1016/j.jpaa.2025.107933","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊕</mo><mo>⋯</mo><mo>⊕</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> be a decomposition of the algebra <em>A</em> as a direct sum of vector subspaces. If for every choice of the indices <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≤</mo><mi>r</mi></math></span> there exist <span><math><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub></math></span> such that the product <span><math><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>⋯</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>≠</mo><mn>0</mn></math></span>, and for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>r</mi></math></span> there is a constant <span><math><mi>β</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo><mo>≠</mo><mn>0</mn></math></span> with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mi>β</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>, the above decomposition is <em>regular</em>. Bahturin and Regev raised the following conjecture: suppose that the regular decomposition comes from a group grading on <em>A</em>, and form the <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span> matrix whose <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math></span>th entry equals <span><math><mi>β</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math></span>. Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of <em>A</em> by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107933"},"PeriodicalIF":0.7,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143610147","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}

{"title":"The category of π-finite spaces","authors":"Mathieu Anel","doi":"10.1016/j.jpaa.2025.107925","DOIUrl":"10.1016/j.jpaa.2025.107925","url":null,"abstract":"<div><div>We show that the category of truncated spaces with finite homotopy invariants (<em>π</em>-finite spaces) has many of the features expected of an elementary ∞-topos. It should be thought of as the natural higher analogue of the elementary 1-topos of finite sets, with which it shares several initiality properties. The paper has also an appendix about univalent families in ∞-pretopoi.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107925"},"PeriodicalIF":0.7,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The spectrum of the real line","authors":"Jan-Paul Lerch","doi":"10.1016/j.jpaa.2025.107928","DOIUrl":"10.1016/j.jpaa.2025.107928","url":null,"abstract":"<div><div>Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective objects up to isomorphism. These classes form the spectrum, which we show to be homeomorphic to an ordered space. Moreover, as the spectral category turns out to be discrete, the spectrum parametrises all injective objects.</div><div>Finally, for the case of the real line we show that this topology refines the topology induced by the interleaving distance, which is known from persistence homology.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107928"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143610150","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}

{"title":"Infinity categories with duality and hermitian multiplicative infinite loop space machines","authors":"Hadrian Heine,&nbsp;Alejo Lopez-Avila,&nbsp;Markus Spitzweck","doi":"10.1016/j.jpaa.2025.107929","DOIUrl":"10.1016/j.jpaa.2025.107929","url":null,"abstract":"<div><div>We show that any preadditive ∞-category with duality gives rise to a direct sum hermitian <em>K</em>-theory spectrum. This assignment is lax symmetric monoidal, thereby producing <span><math><msub><mrow><mi>E</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-ring spectra from preadditive symmetric monoidal ∞-categories with duality. To have examples of preadditive symmetric monoidal ∞-categories with duality we show that any preadditive symmetric monoidal ∞-category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over <span><math><msub><mrow><mi>E</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-ring spectra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107929"},"PeriodicalIF":0.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143610145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Integral Picard group of some stacks of polarized K3 surfaces of low degree","authors":"Andrea Di Lorenzo","doi":"10.1016/j.jpaa.2025.107926","DOIUrl":"10.1016/j.jpaa.2025.107926","url":null,"abstract":"<div><div>We compute the integral Picard group of the stack <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></msub></math></span> of polarized K3 surfaces with at most rational double points of degree <span><math><mn>2</mn><mi>l</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>8</mn></math></span>. We show that in this range the integral Picard group is torsion-free and that a basis is given by certain elliptic Noether-Lefschetz divisors together with the Hodge line bundle.</div><div>To achieve this result, we investigate certain stacks of complete intersections and their Picard groups by means of equivariant geometry.</div><div>In the end we compute an expression of the class of some Noether-Lefschetz divisors, restricted to an open substack of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></msub></math></span>, in terms of the basis mentioned above.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107926"},"PeriodicalIF":0.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563231","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}

{"title":"Classifying smashing ideals in derived categories of valuation domains","authors":"Scott Balchin ,&nbsp;Florian Tecklenburg","doi":"10.1016/j.jpaa.2025.107917","DOIUrl":"10.1016/j.jpaa.2025.107917","url":null,"abstract":"<div><div>Building on results of Bazzoni–Št'ovíček, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107917"},"PeriodicalIF":0.7,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relative higher homology and representation theory","authors":"Rasool Hafezi ,&nbsp;Javad Asadollahi ,&nbsp;Yi Zhang","doi":"10.1016/j.jpaa.2025.107924","DOIUrl":"10.1016/j.jpaa.2025.107924","url":null,"abstract":"<div><div>Higher homological algebra, basically done in the framework of an <em>n</em>-cluster tilting subcategory <span><math><mi>M</mi></math></span> of an abelian category <span><math><mi>A</mi></math></span>, has been the topic of several recent researches. In this paper, we study a relative version, in the sense of Auslander-Solberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor <em>F</em> of <span><math><msubsup><mrow><mi>Ext</mi></mrow><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mo>−</mo><mo>,</mo><mo>−</mo><mo>)</mo></math></span> as the basis of our relative theory. This, in turn, specifies a collection of <em>n</em>-exact sequences in <span><math><mi>M</mi></math></span>, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative <em>n</em>-Auslander-Reiten duality formula, as well as an exploration of relative Grothendieck groups, among other results. As an application, we provide necessary and sufficient conditions for <span><math><mi>M</mi></math></span> to be of finite type.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107924"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552062","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}

{"title":"On the degenerate Whittaker space for GL4(o2)","authors":"Ankita Parashar ,&nbsp;Shiv Prakash Patel","doi":"10.1016/j.jpaa.2025.107921","DOIUrl":"10.1016/j.jpaa.2025.107921","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be a finite principal ideal local ring of length 2. For a representation <em>π</em> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, the degenerate Whittaker space <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> is a representation of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. We describe <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> explicitly for an irreducible strongly cuspidal representation <em>π</em> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. This description verifies a special case of a conjecture of Prasad. We also prove that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> is a multiplicity free representation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107921"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529842","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}

{"title":"On the symbolic F-splitness of binomial edge ideals","authors":"Pedro Ramírez-Moreno","doi":"10.1016/j.jpaa.2025.107922","DOIUrl":"10.1016/j.jpaa.2025.107922","url":null,"abstract":"<div><div>We study the symbolic <em>F</em>-splitness of families of binomial edge ideals. We also study the strong <em>F</em>-regularity of the symbolic blowup algebras of families of binomial edge ideals. We make use of Fedder-like criteria and combinatorial properties of the graphs associated to the binomial edge ideals in order to approach the aforementioned scenarios.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107922"},"PeriodicalIF":0.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552063","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}