Journal of Pure and Applied Algebra最新文献

{"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}

{"title":"Coideal subalgebras of quantum SL2 at roots of unity","authors":"Kenichi Shimizu ,&nbsp;Rei Sugitani","doi":"10.1016/j.jpaa.2025.107923","DOIUrl":"10.1016/j.jpaa.2025.107923","url":null,"abstract":"<div><div>We classify right coideal subalgebras of the finite-dimensional quotient of the quantized enveloping algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and that of the quantized coordinate algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> at a root of unity <em>q</em> of odd order. All those coideal subalgebras are described by generators and relations.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107923"},"PeriodicalIF":0.7,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552061","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":"An algorithm for g-invariant on unary Hermitian lattices over imaginary quadratic fields","authors":"Jingbo Liu","doi":"10.1016/j.jpaa.2025.107916","DOIUrl":"10.1016/j.jpaa.2025.107916","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>d</mi></mrow></msqrt><mo>)</mo></math></span> be an imaginary quadratic field for a square-free positive integer <em>d</em>, and let <span><math><mi>O</mi></math></span> be its ring of integers. For every positive integer <em>m</em>, let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be the free Hermitian lattice over <span><math><mi>O</mi></math></span> with an orthonormal basis, let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> be the set consisting of all the positive definite integral unary Hermitian lattices over <span><math><mi>O</mi></math></span> which can be represented by some <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, and let <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> be the smallest positive integer such that all the lattices in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> can be uniformly represented by <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msub></math></span>. In this work, I provide an algorithm to compute the explicit form of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> and the exact value of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> for every imaginary quadratic field <em>E</em>, which may be viewed as a natural extension of the Pythagoras number in the lattice setting.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107916"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143610148","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":"Fibrations by plane quartic curves with a canonical moving singularity","authors":"Cesar Hilario ,&nbsp;Karl-Otto Stöhr","doi":"10.1016/j.jpaa.2025.107918","DOIUrl":"10.1016/j.jpaa.2025.107918","url":null,"abstract":"<div><div>We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behavior of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107918"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487223","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 4-intersection unprojection format","authors":"Vasiliki Petrotou","doi":"10.1016/j.jpaa.2025.107915","DOIUrl":"10.1016/j.jpaa.2025.107915","url":null,"abstract":"<div><div>Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In this present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal <em>I</em> which is contained in four codimension 3 complete intersection ideals <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space which correspond to the entries with identifier numbers 29376, 9176 and 24198 respectively in the Graded Ring Database.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107915"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478534","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":"Spaces of generators for Azumaya algebras with unitary involution","authors":"Omer Cantor,&nbsp;Uriya A. First","doi":"10.1016/j.jpaa.2025.107919","DOIUrl":"10.1016/j.jpaa.2025.107919","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;A&lt;/em&gt; be a finite dimensional algebra (possibly with some extra structure) over an infinite field &lt;em&gt;K&lt;/em&gt; and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;em&gt;r&lt;/em&gt;-tuples &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; which fail to generate &lt;em&gt;A&lt;/em&gt; are the &lt;em&gt;K&lt;/em&gt;-points of a closed subvariety &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of the affine space underlying &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, the codimension of which may be thought of as quantifying how well a generic &lt;em&gt;r&lt;/em&gt;-tuple in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; generates &lt;em&gt;A&lt;/em&gt;. Taking this intuition one step further, the second author, Reichstein and Williams showed that lower bounds on the codimension of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (for every &lt;em&gt;r&lt;/em&gt;) imply upper bounds on the number of generators of &lt;em&gt;forms&lt;/em&gt; of the &lt;em&gt;K&lt;/em&gt;-algebra &lt;em&gt;A&lt;/em&gt; over finitely generated &lt;em&gt;K&lt;/em&gt;-rings. That work also demonstrates how finer information on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; may be used to construct forms of &lt;em&gt;A&lt;/em&gt; which require many elements to generate.&lt;/div&gt;&lt;div&gt;The dimension and irreducible components of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are known in a few cases, which in particular lead to upper bounds on the number of generators of Azumaya algebras and Azumaya algebras with involution of the first kind (orthogonal or symplectic). This paper treats the case of Azumaya algebras with a unitary involution by finding the dimension and irreducible components of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; when &lt;em&gt;A&lt;/em&gt; is the &lt;em&gt;K&lt;/em&gt;-algebra with involution &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Our analysis implies that every degree-&lt;em&gt;n&lt;/em&gt; Azumaya algebra with a unitary involution over a finitely generated &lt;em&gt;K&lt;/em&gt;-ring of Krull dimension &lt;em&gt;d&lt;/em&gt; can be generated by &lt;span&gt;&lt;math&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107919"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479280","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}