To the existing dynamic algorithm FactorizationsUpToElement for factorization sets of elements in a numerical semigroup, we add lexicographic and parallel behavior. To the existing parallel lexicographic algorithm for the same, we add dynamic behavior. The (dimensionwise) dynamic algorithm is parallelized either elementwise or factorizationwise, while the parallel lexicographic algorithm is made dynamic with low-dimension tabulation. The tabulation for the parallel lexicographic algorithm can itself be performed using the dynamic algorithm. We provide reference CUDA implementations with measured runtimes.
针对数值半群中元素因式分解集的现有动态算法 FactorizationsUpToElement，我们增加了词法和并行行为。对于现有的并行词法算法，我们增加了动态行为。这种（维度）动态算法是以元素或因式分解的方式并行化的，而并行词法算法则是通过低维度制表使其动态化的。并行词法算法的制表本身可以使用动态算法进行。我们提供了具有实测运行时间的 CUDA 实现参考。
{"title":"Two parallel dynamic lexicographic algorithms for factorization sets in numerical semigroups","authors":"Thomas Barron","doi":"arxiv-2407.20474","DOIUrl":"https://doi.org/arxiv-2407.20474","url":null,"abstract":"To the existing dynamic algorithm FactorizationsUpToElement for factorization\u0000sets of elements in a numerical semigroup, we add lexicographic and parallel\u0000behavior. To the existing parallel lexicographic algorithm for the same, we add\u0000dynamic behavior. The (dimensionwise) dynamic algorithm is parallelized either\u0000elementwise or factorizationwise, while the parallel lexicographic algorithm is\u0000made dynamic with low-dimension tabulation. The tabulation for the parallel\u0000lexicographic algorithm can itself be performed using the dynamic algorithm. We\u0000provide reference CUDA implementations with measured runtimes.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.
{"title":"Galois theory of differential schemes","authors":"Ivan Tomašić, Behrang Noohi","doi":"arxiv-2407.21147","DOIUrl":"https://doi.org/arxiv-2407.21147","url":null,"abstract":"Since 1883, Picard-Vessiot theory had been developed as the Galois theory of\u0000differential field extensions associated to linear differential equations.\u0000Inspired by categorical Galois theory of Janelidze, and by using novel methods\u0000of precategorical descent applied to algebraic-geometric situations, we develop\u0000a Galois theory that applies to morphisms of differential schemes, and vastly\u0000generalises the linear Picard-Vessiot theory, as well as the strongly normal\u0000theory of Kolchin.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R=K[x_1,ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I$ be a polymatroidal ideal of $R$. In this paper, we provide a comprehensive classification of all unmixed polymatroidal ideals. This work addresses a question raised by Herzog and Hibi in [10]
{"title":"Unmixed polymatroidal ideals","authors":"Mozghan Koolani, Amir Mafi, Hero Saremi","doi":"arxiv-2407.20527","DOIUrl":"https://doi.org/arxiv-2407.20527","url":null,"abstract":"Let $R=K[x_1,ldots,x_n]$ denote the polynomial ring in $n$ variables over a\u0000field $K$ and $I$ be a polymatroidal ideal of $R$. In this paper, we provide a\u0000comprehensive classification of all unmixed polymatroidal ideals. This work\u0000addresses a question raised by Herzog and Hibi in [10]","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe the shape of the Lyubeznik table of either rings in positive characteristic or Stanley-Reisner rings in any characteristic when they satisfy Serre's condition $S_r$ or they are Cohen-Macaulay in a given codimension, condition denoted by $CM_r$. Moreover we show that these results are sharp.
{"title":"Lyubeznik tables of $S_r$ and $CM_r$ rings","authors":"Josep Àlvarez Montaner, Siamak Yassemi","doi":"arxiv-2407.20129","DOIUrl":"https://doi.org/arxiv-2407.20129","url":null,"abstract":"We describe the shape of the Lyubeznik table of either rings in positive\u0000characteristic or Stanley-Reisner rings in any characteristic when they satisfy\u0000Serre's condition $S_r$ or they are Cohen-Macaulay in a given codimension,\u0000condition denoted by $CM_r$. Moreover we show that these results are sharp.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these Ext-modules and formulate a generalisation of the Auslander-Reiten Conjecture. We prove this general version for high syzygies of modules over several classes of rings including analytically unramified Arf rings, 2-dimensional local normal domains with rational singularities, Gorenstein isolated singularities of Krull dimension at least 2 and more. We also prove results for the special case of the canonical module of a Cohen-Macaulay local ring. These results both generalise and also provide evidence for a version of Tachikawa Conjecture that was considered by Dao-Kobayashi-Takahashi.
{"title":"Auslander-Reiten annihilators","authors":"Özgür Esentepe","doi":"arxiv-2407.19999","DOIUrl":"https://doi.org/arxiv-2407.19999","url":null,"abstract":"Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a\u0000finitely generated module is projective when certain Ext-modules vanish. But\u0000what if those Ext-modules do not vanish? We study the annihilators of these\u0000Ext-modules and formulate a generalisation of the Auslander-Reiten Conjecture.\u0000We prove this general version for high syzygies of modules over several classes\u0000of rings including analytically unramified Arf rings, 2-dimensional local\u0000normal domains with rational singularities, Gorenstein isolated singularities\u0000of Krull dimension at least 2 and more. We also prove results for the special\u0000case of the canonical module of a Cohen-Macaulay local ring. These results both\u0000generalise and also provide evidence for a version of Tachikawa Conjecture that\u0000was considered by Dao-Kobayashi-Takahashi.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $S=k[x_1,cdots,x_n]$ be a polynomial ring over an arbitrary field $k$. We construct a new symmetric polytopal minimal resolution of $(x_1,cdots,x_n)^n$. Using this resolution, we also obtain a symmetric polytopal minimal resolution of the ideal obtained by removing $x_1cdots x_n$ from the generators of $(x_1,cdots,x_n)^n$.
{"title":"A new symmetric resolution for $(x_{1},dots, x_{n})^{n}$","authors":"Hoài Đào, Jeff Mermin","doi":"arxiv-2407.20365","DOIUrl":"https://doi.org/arxiv-2407.20365","url":null,"abstract":"Let $S=k[x_1,cdots,x_n]$ be a polynomial ring over an arbitrary field $k$.\u0000We construct a new symmetric polytopal minimal resolution of\u0000$(x_1,cdots,x_n)^n$. Using this resolution, we also obtain a symmetric\u0000polytopal minimal resolution of the ideal obtained by removing $x_1cdots x_n$\u0000from the generators of $(x_1,cdots,x_n)^n$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal $I$ in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of $R/overline{I^n}$ for some $n$. We show in Theorem 1.6 that McAdam's theorem holds for $mathbb Q$-divisorial filtrations in an equidimensional local ring which is essentially of finite type over a field. This generalizes an earlier result for $mathbb Q$-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations. We consider the question of the asymptotic behavior of the function $nmapsto lambda_R(R/I_n)$ for a $mathbb Q$-divisorial filtration $mathcal I={I_n}$ of $m_R$-primary ideals on a $d$-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity $$ e(mathcal I)=d! lim_{nrightarrowinfty}frac{lambda_R(R/I_n)}{n^d} $$ can be irrational. We show in Lemma 4.1 that the limsup of the first difference function $$ limsup_{nrightarrowinfty}frac{lambda_R(I_n/I_{n+1})}{n^{d-1}} $$ is always finite for a $mathbb Q$-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit. In the final section, we give an example of a symbolic filtration ${P^{(n)}}$ of a prime ideal $P$ in a normal two dimensional excellent local ring which has the property that the set of Rees valuations of all the symbolic powers $P^{(n)}$ of $P$ is infinite.
{"title":"The Rees algebra and analytic spread of a divisorial filtration","authors":"Steven Dale Cutkosky","doi":"arxiv-2407.19585","DOIUrl":"https://doi.org/arxiv-2407.19585","url":null,"abstract":"In this paper we investigate some properties of Rees algebras of divisorial\u0000filtrations and their analytic spread. A classical theorem of McAdam shows that\u0000the analytic spread of an ideal $I$ in a formally equidimensional local ring is\u0000equal to the dimension of the ring if and only if the maximal ideal is an\u0000associated prime of $R/overline{I^n}$ for some $n$. We show in Theorem 1.6\u0000that McAdam's theorem holds for $mathbb Q$-divisorial filtrations in an\u0000equidimensional local ring which is essentially of finite type over a field.\u0000This generalizes an earlier result for $mathbb Q$-divisorial filtrations in an\u0000equicharacteristic zero excellent local domain by the author. This theorem does\u0000not hold for more general filtrations. We consider the question of the asymptotic behavior of the function $nmapsto\u0000lambda_R(R/I_n)$ for a $mathbb Q$-divisorial filtration $mathcal I={I_n}$\u0000of $m_R$-primary ideals on a $d$-dimensional normal excellent local ring. It is\u0000known from earlier work of the author that the multiplicity $$ e(mathcal I)=d!\u0000lim_{nrightarrowinfty}frac{lambda_R(R/I_n)}{n^d} $$ can be irrational. We\u0000show in Lemma 4.1 that the limsup of the first difference function $$\u0000limsup_{nrightarrowinfty}frac{lambda_R(I_n/I_{n+1})}{n^{d-1}} $$ is always\u0000finite for a $mathbb Q$-divisorial filtration. We then give an example in\u0000Section 4 showing that this limsup may not exist as a limit. In the final section, we give an example of a symbolic filtration\u0000${P^{(n)}}$ of a prime ideal $P$ in a normal two dimensional excellent local\u0000ring which has the property that the set of Rees valuations of all the symbolic\u0000powers $P^{(n)}$ of $P$ is infinite.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Edoardo Ballico, Maria Chiara Brambilla, Claudio Fontanari
The Terracini locus $mathbb{T}(n, d; x)$ is the locus of all finite subsets $S subset mathbb{P}^n$ of cardinality $x$ such that $langle S rangle = mathbb{P}^n$, $h^0(mathcal{I}_{2S}(d)) > 0$, and $h^1(mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $dimmathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)=(6,10)$ or $(d,x)=(4,4)$ or $dequiv 1,2 pmod{3}$ and $x=(d+2)(d+1)/6$.
{"title":"Terracini loci and a codimension one Alexander-Hirschowitz theorem","authors":"Edoardo Ballico, Maria Chiara Brambilla, Claudio Fontanari","doi":"arxiv-2407.18751","DOIUrl":"https://doi.org/arxiv-2407.18751","url":null,"abstract":"The Terracini locus $mathbb{T}(n, d; x)$ is the locus of all finite subsets\u0000$S subset mathbb{P}^n$ of cardinality $x$ such that $langle S rangle =\u0000mathbb{P}^n$, $h^0(mathcal{I}_{2S}(d)) > 0$, and $h^1(mathcal{I}_{2S}(d)) >\u00000$. The celebrated Alexander-Hirschowitz Theorem classifies the triples\u0000$(n,d,x)$ for which $dimmathbb{T}(n, d; x)=xn$. Here we fully characterize\u0000the next step in the case $n=2$, namely, we prove that $mathbb{T}(2,d;x)$ has\u0000at least one irreducible component of dimension $2x-1$ if and only if either\u0000$(d,x)=(6,10)$ or $(d,x)=(4,4)$ or $dequiv 1,2 pmod{3}$ and $x=(d+2)(d+1)/6$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Buchweitz related the singularity category of a (strongly) Gorenstein ring and the stable category of maximal Cohen-Macaulay modules by a triangle equivalence. We phrase his result in a relative categorical setting based on N-complexes instead of classical 2-complexes. The role of Cohen-Macaulay modules is played by chains of monics in a Frobenius subcategory of an exact category. As a byproduct, we provide foundational results on derived categories of N-complexes over exact categories known from the Abelian case or for 2-complexes.
布赫维茨通过三角等价关系将（强）戈伦斯坦环的奇异性范畴与最大科恩-麦考莱模块的稳定范畴联系起来。我们将他的结果放在一个基于 N 复数而非经典 2 复数的相对分类环境中进行表述。科恩-马科莱模块的作用由精确范畴的弗罗贝尼斯子范畴中的单子链扮演。作为副产品，我们提供了关于在阿贝尔情况下已知的精确范畴上的 N-复数派生范畴或 2-复数的基础性结果。
{"title":"The stabilized bounded N-derived category of an exact category","authors":"Jonas Frank, Mathias Schulze","doi":"arxiv-2407.18708","DOIUrl":"https://doi.org/arxiv-2407.18708","url":null,"abstract":"Buchweitz related the singularity category of a (strongly) Gorenstein ring\u0000and the stable category of maximal Cohen-Macaulay modules by a triangle\u0000equivalence. We phrase his result in a relative categorical setting based on\u0000N-complexes instead of classical 2-complexes. The role of Cohen-Macaulay\u0000modules is played by chains of monics in a Frobenius subcategory of an exact\u0000category. As a byproduct, we provide foundational results on derived categories\u0000of N-complexes over exact categories known from the Abelian case or for\u00002-complexes.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce and study the class of $phi$-$w$-P-flat modules, which can be seen as generalizations of both $phi$-P-flat modules and $w$-P-flat modules. In particular, we obtain that the class of $phi$-$w$-P-flat modules is covering. We also utilize the class of $phi$-$w$-P-flat modules to characterize $phi$-von Neumann regular rings, strong $phi$-rings and $phi$-PvMRs.
{"title":"A new version of P-flat modules and its applications","authors":"Wei Qi, Xiaolei Zhang","doi":"arxiv-2407.17865","DOIUrl":"https://doi.org/arxiv-2407.17865","url":null,"abstract":"In this paper, we introduce and study the class of $phi$-$w$-P-flat modules,\u0000which can be seen as generalizations of both $phi$-P-flat modules and\u0000$w$-P-flat modules. In particular, we obtain that the class of\u0000$phi$-$w$-P-flat modules is covering. We also utilize the class of\u0000$phi$-$w$-P-flat modules to characterize $phi$-von Neumann regular rings,\u0000strong $phi$-rings and $phi$-PvMRs.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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