Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107902
Jan O. Kleppe , Rosa M. Miró-Roig
Let be a graded morphism between free R-modules of rank t and , respectively, and let be the ideal generated by the minors of a matrix representing φ. In this paper: (1) We show that the canonical module of is up to twist equal to a suitable Schur power of ; thus equal to if in which case we find a minimal free R-resolution of for any j, (2) For , we construct a free R-resolution of which starts almost minimally (i.e. the first three terms are minimal up to a precise summand), and (3) For , we construct under a certain depth condition the first three terms of a free R-resolution of which are minimal up to a precise summand. As a byproduct we answer the first open case of a question posed by Buchsbaum and Eisenbud in [2, pg. 299].
{"title":"Schur powers of the cokernel of a graded morphism","authors":"Jan O. Kleppe , Rosa M. Miró-Roig","doi":"10.1016/j.jpaa.2025.107902","DOIUrl":"10.1016/j.jpaa.2025.107902","url":null,"abstract":"<div><div>Let <span><math><mi>φ</mi><mo>:</mo><mi>F</mi><mo>⟶</mo><mi>G</mi></math></span> be a graded morphism between free <em>R</em>-modules of rank <em>t</em> and <span><math><mi>t</mi><mo>+</mo><mi>c</mi><mo>−</mo><mn>1</mn></math></span>, respectively, and let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo></math></span> be the ideal generated by the <span><math><mi>j</mi><mo>×</mo><mi>j</mi></math></span> minors of a matrix representing <em>φ</em>. In this paper: (1) We show that the canonical module of <span><math><mi>R</mi><mo>/</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo></math></span> is up to twist equal to a suitable Schur power <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>I</mi></mrow></msup><mi>M</mi></math></span> of <span><math><mi>M</mi><mo>=</mo><mi>coker</mi><mo>(</mo><msup><mrow><mi>φ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>; thus equal to <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>j</mi></mrow></msup><mi>M</mi></math></span> if <span><math><mi>c</mi><mo>=</mo><mn>2</mn></math></span> in which case we find a minimal free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>j</mi></mrow></msup><mi>M</mi></math></span> for any <em>j</em>, (2) For <span><math><mi>c</mi><mo>=</mo><mn>3</mn></math></span>, we construct a free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>M</mi></math></span> which starts almost minimally (i.e. the first three terms are minimal up to a precise summand), and (3) For <span><math><mi>c</mi><mo>≥</mo><mn>4</mn></math></span>, we construct under a certain depth condition the first three terms of a free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>M</mi></math></span> which are minimal up to a precise summand. As a byproduct we answer the first open case of a question posed by Buchsbaum and Eisenbud in <span><span>[2, pg. 299]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107902"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420767","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107852
Grigore Călugăreanu , Horia F. Pop , Adrian Vasiu
A unimodular matrix with entries in a commutative R is called extendable (resp. simply extendable) if it extends to an invertible matrix (resp. invertible matrix whose entry is 0). We obtain necessary and sufficient conditions for a unimodular matrix to be extendable (resp. simply extendable) and use them to study the class (resp. ) of rings R with the property that all unimodular matrices with entries in R are extendable (resp. simply extendable). We also study the larger class of rings R with the property that all unimodular matrices of determinant 0 and with entries in R are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over and Hermite rings, only the EDRs belong to the class or . If R has stable range at most 2 (e.g., R is a Hermite ring or ), then R is an ring iff it is an ring.
{"title":"Matrix invertible extensions over commutative rings. Part I: General theory","authors":"Grigore Călugăreanu , Horia F. Pop , Adrian Vasiu","doi":"10.1016/j.jpaa.2024.107852","DOIUrl":"10.1016/j.jpaa.2024.107852","url":null,"abstract":"<div><div>A unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix with entries in a commutative <em>R</em> is called extendable (resp. simply extendable) if it extends to an invertible <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> matrix (resp. invertible <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> matrix whose <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> entry is 0). We obtain necessary and sufficient conditions for a unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix to be extendable (resp. simply extendable) and use them to study the class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (resp. <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>) of rings <em>R</em> with the property that all unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices with entries in <em>R</em> are extendable (resp. simply extendable). We also study the larger class <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of rings <em>R</em> with the property that all unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices of determinant 0 and with entries in <em>R</em> are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over <span><math><mi>Z</mi></math></span> and Hermite rings, only the EDRs belong to the class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> or <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. If <em>R</em> has stable range at most 2 (e.g., <em>R</em> is a Hermite ring or <span><math><mi>dim</mi><mo></mo><mo>(</mo><mi>R</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span>), then <em>R</em> is an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ring iff it is an <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107852"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094014","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107853
Edoardo Ballico
We study the dimensions of Hadamard products of varieties if we allow to modify of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate hyperplane with a “nice” curve always has the expected dimension.
{"title":"Projective surfaces not as Hadamard products and the dimensions of the Hadamard joins","authors":"Edoardo Ballico","doi":"10.1016/j.jpaa.2024.107853","DOIUrl":"10.1016/j.jpaa.2024.107853","url":null,"abstract":"<div><div>We study the dimensions of Hadamard products of <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> varieties if we allow to modify <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate hyperplane with a “nice” curve always has the expected dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107853"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094017","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107854
Akihiro Higashitani, Koichiro Tani
The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous -subalgebra of a polynomial ring . Clearly, any affine monoid can be realized since the initial algebra of the affine monoid -algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous -algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous -algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous -algebra.
{"title":"Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras","authors":"Akihiro Higashitani, Koichiro Tani","doi":"10.1016/j.jpaa.2024.107854","DOIUrl":"10.1016/j.jpaa.2024.107854","url":null,"abstract":"<div><div>The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous <span><math><mi>k</mi></math></span>-subalgebra of a polynomial ring <span><math><mi>k</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>n</mi></mrow></msub><mo>]</mo></math></span>. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid <span><math><mi>k</mi></math></span>-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107854"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094023","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}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107844
Sophie Chemla
We study when left (op)Hopf algebroids in the sense of Takeuchi-Schauenburg give rise to a Frobenius or quasi-Frobenius extension. The case of Hopf algebroids in the sense of Böhm was treated by G. Böhm ([4]). Contrary to Hopf algebroids, (op)Hopf left algebroids don't necessarily have an antipode but their Hopf-Galois map is invertible. We make use of recent results about left Hopf algebroids ([18], [35], [25]). Our results are applied to the restricted enveloping algebra of a restricted Lie-Rinehart algebra.
{"title":"Frobenius and quasi-Frobenius left Hopf algebroids","authors":"Sophie Chemla","doi":"10.1016/j.jpaa.2024.107844","DOIUrl":"10.1016/j.jpaa.2024.107844","url":null,"abstract":"<div><div>We study when left (op)Hopf algebroids in the sense of Takeuchi-Schauenburg give rise to a Frobenius or quasi-Frobenius extension. The case of Hopf algebroids in the sense of Böhm was treated by G. Böhm (<span><span>[4]</span></span>). Contrary to Hopf algebroids, (op)Hopf left algebroids don't necessarily have an antipode but their Hopf-Galois map is invertible. We make use of recent results about left Hopf algebroids (<span><span>[18]</span></span>, <span><span>[35]</span></span>, <span><span>[25]</span></span>). Our results are applied to the restricted enveloping algebra of a restricted Lie-Rinehart algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107844"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094013","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}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107855
Luca Mesiti
We generalize to dimension 2 the well-known fact that a colimit in a 1-dimensional slice is precisely the map from the colimit of the domains of the diagram that is induced by the universal property. For this, we find the need to reduce weighted 2-colimits to cartesian-marked oplax conical ones, and as a consequence the need to consider lax slices.
We prove results of preservation, reflection and lifting of 2-colimits for the domain 2-functor from a lax slice. We thus generalize to dimension 2 the whole fruitful calculus of colimits in 1-dimensional slices. We achieve this within the framework of enhanced (or -)category theory. The preservation result assumes products in the base 2-category and uses an original general theorem which states that a lax left adjoint preserves appropriate 2-colimits if the adjunction is strict on one side and suitably -categorical.
Finally, we apply the same general theorem of preservation of 2-colimits to the 2-functor of change of base along a split Grothendieck opfibration between lax slices. We prove that this change of base 2-functor is indeed a left adjoint of the kind described above by laxifying the proof that Conduché functors are exponentiable. We conclude extending the result of preservation of 2-colimits for the change of base 2-functor to any finitely complete 2-category with a dense generator.
{"title":"Colimits in 2-dimensional slices","authors":"Luca Mesiti","doi":"10.1016/j.jpaa.2024.107855","DOIUrl":"10.1016/j.jpaa.2024.107855","url":null,"abstract":"<div><div>We generalize to dimension 2 the well-known fact that a colimit in a 1-dimensional slice is precisely the map from the colimit of the domains of the diagram that is induced by the universal property. For this, we find the need to reduce weighted 2-colimits to cartesian-marked oplax conical ones, and as a consequence the need to consider lax slices.</div><div>We prove results of preservation, reflection and lifting of 2-colimits for the domain 2-functor from a lax slice. We thus generalize to dimension 2 the whole fruitful calculus of colimits in 1-dimensional slices. We achieve this within the framework of enhanced (or <span><math><mi>F</mi></math></span>-)category theory. The preservation result assumes products in the base 2-category and uses an original general theorem which states that a lax left adjoint preserves appropriate 2-colimits if the adjunction is strict on one side and suitably <span><math><mi>F</mi></math></span>-categorical.</div><div>Finally, we apply the same general theorem of preservation of 2-colimits to the 2-functor of change of base along a split Grothendieck opfibration between lax slices. We prove that this change of base 2-functor is indeed a left adjoint of the kind described above by laxifying the proof that Conduché functors are exponentiable. We conclude extending the result of preservation of 2-colimits for the change of base 2-functor to any finitely complete 2-category with a dense generator.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107855"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099039","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}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107858
Heath Camphire
Let k be a field of odd characteristic p. Fix an even number and a power of p. For most choices of degree d standard graded hypersurfaces with homogeneous maximal ideal , we can determine the graded Betti numbers of . In fact, given two fixed powers , for most choices of R the graded Betti numbers in high homological degree of and are the same up to a constant shift. This paper shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-q-compressed polynomials in [13]. We show that link-q-compressed polynomials are indeed fairly common in many polynomial rings.
{"title":"Determining the Betti numbers of R/(xpe,ype,zpe) for most even degree hypersurfaces in odd characteristic","authors":"Heath Camphire","doi":"10.1016/j.jpaa.2024.107858","DOIUrl":"10.1016/j.jpaa.2024.107858","url":null,"abstract":"<div><div>Let <em>k</em> be a field of odd characteristic <em>p</em>. Fix an even number <span><math><mi>d</mi><mo><</mo><mi>p</mi><mo>+</mo><mn>1</mn></math></span> and a power <span><math><mi>q</mi><mo>≥</mo><mi>d</mi><mo>+</mo><mn>3</mn></math></span> of <em>p</em>. For most choices of degree <em>d</em> standard graded hypersurfaces <span><math><mi>R</mi><mo>=</mo><mi>k</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo><mo>/</mo><mo>(</mo><mi>f</mi><mo>)</mo></math></span> with homogeneous maximal ideal <span><math><mi>m</mi></math></span>, we can determine the graded Betti numbers of <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><mi>q</mi><mo>]</mo></mrow></msup></math></span>. In fact, given two fixed powers <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mi>d</mi><mo>+</mo><mn>3</mn></math></span>, for most choices of <em>R</em> the graded Betti numbers in high homological degree of <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo></mrow></msup></math></span> and <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow></msup></math></span> are the same up to a constant shift. This paper shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-<em>q</em>-compressed polynomials in <span><span>[13]</span></span>. We show that link-<em>q</em>-compressed polynomials are indeed fairly common in many polynomial rings.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107858"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099041","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107850
Daniela Paiva , Ana Quedo
In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface are induced by a Cremona transformation of . We provide the first steps towards a complete solution of this problem when . In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of , giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface is induced by a Cremona transformation.
{"title":"Automorphisms of quartic surfaces and Cremona transformations","authors":"Daniela Paiva , Ana Quedo","doi":"10.1016/j.jpaa.2024.107850","DOIUrl":"10.1016/j.jpaa.2024.107850","url":null,"abstract":"<div><div>In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface <span><math><mi>S</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> are induced by a Cremona transformation of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. We provide the first steps towards a complete solution of this problem when <span><math><mi>ρ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>. In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface <span><math><mi>S</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> is induced by a Cremona transformation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107850"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094020","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107845
Kenji Hashimoto , Kwangwoo Lee
It is known that the automorphism group of any projective K3 surface is finitely generated. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group . In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.
{"title":"Free automorphism groups of K3 surfaces with Picard number 3","authors":"Kenji Hashimoto , Kwangwoo Lee","doi":"10.1016/j.jpaa.2024.107845","DOIUrl":"10.1016/j.jpaa.2024.107845","url":null,"abstract":"<div><div>It is known that the automorphism group of any projective K3 surface is finitely generated. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group <span><math><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107845"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094022","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 : 2025-01-01DOI: 10.1016/j.jpaa.2024.107856
Nathan Fieldsteel , Uwe Nagel
Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely minimal and width-wise minimal free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say i, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algebra admits a graded minimal free resolution and that it is unique. A width-wise minimal free resolution is a free resolution that provides a minimal free resolution of a module in every width. Such a resolution is necessarily minimal. Its existence is not guaranteed. However, we show that certain monomial OI-ideals do admit width-wise minimal free or, more generally, width-wise minimal flat resolutions. These ideals include families of well-known monomial ideals such as Ferrers ideals and squarefree strongly stable ideals. The arguments rely on the theory of cellular resolutions.
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