Alireza Vahidi, Ahmad Khaksari, Mohammad Shirazipour
Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $mathfrak{a}$ an ideal of $R$, $M$ and $N$ two finitely generated $R$-modules, and $X$ an arbitrary $R$-module. In this paper, we study cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules. In some cases, we show that $operatorname{H}^{i}_{mathfrak{a}}(M,X)$ is an $(operatorname{FD}_{
{"title":"Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules","authors":"Alireza Vahidi, Ahmad Khaksari, Mohammad Shirazipour","doi":"arxiv-2409.05090","DOIUrl":"https://doi.org/arxiv-2409.05090","url":null,"abstract":"Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring,\u0000$mathfrak{a}$ an ideal of $R$, $M$ and $N$ two finitely generated $R$-modules,\u0000and $X$ an arbitrary $R$-module. In this paper, we study cofiniteness and\u0000finiteness of associated prime ideals of generalized local cohomology modules.\u0000In some cases, we show that $operatorname{H}^{i}_{mathfrak{a}}(M,X)$ is an\u0000$(operatorname{FD}_{<n},mathfrak{a})$-cofinite $R$-module and\u0000${mathfrak{p}inoperatorname{Ass}_R(operatorname{H}^{i}_{mathfrak{a}}(M,X)):dim(R/mathfrak{p})geq{n}}$\u0000is a finite set for all $i$. If $R$ is semi-local, we observe that\u0000$operatorname{Ass}_R(operatorname{H}^{i}_{mathfrak{a}}(M,N))$ is finite for\u0000all $i$ when $dim_R(M)leq{3}$ or $dim_R(N)leq{3}$. Also, in some\u0000situations, we prove that $operatorname{H}^{i}_{mathfrak{a}}(M,X)$ is an\u0000$mathfrak{a}$-cofinite $R$-module for all $i$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185587","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 $(R,mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $omega_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects. Using these, we study a generalization introduced by Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in particular whether the vanishing of $mathfrak{m} operatorname{Ext}_R^i(omega_R,R)$ should force the trace ideal of $omega_R$ to contain $mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this question has an affirmative answer for numerical semigroup rings of minimal multiplicity, but that the answer is negative in general. Our proofs involve a technical analysis of homogeneous ideals in a numerical semigroup ring, and exploit the behavior of Ulrich modules in this setting.
{"title":"Annihilators of (co)homology and their influence on the trace Ideal","authors":"Justin Lyle, Sarasij Maitra","doi":"arxiv-2409.04686","DOIUrl":"https://doi.org/arxiv-2409.04686","url":null,"abstract":"Let $(R,mathfrak{m})$ be a commutative Noetherian local ring, and suppose\u0000$R$ is Cohen-Macaulay with canonical module $omega_R$. We develop new tools\u0000for analyzing questions involving annihilators of several homologically defined\u0000objects. Using these, we study a generalization introduced by\u0000Dao-Kobayashi-Takahashi of the famous Tachikawa conjecture, asking in\u0000particular whether the vanishing of $mathfrak{m}\u0000operatorname{Ext}_R^i(omega_R,R)$ should force the trace ideal of $omega_R$\u0000to contain $mathfrak{m}$, i.e., for $R$ to be nearly Gorenstein. We show this\u0000question has an affirmative answer for numerical semigroup rings of minimal\u0000multiplicity, but that the answer is negative in general. Our proofs involve a\u0000technical analysis of homogeneous ideals in a numerical semigroup ring, and\u0000exploit the behavior of Ulrich modules in this setting.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185588","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 study edge rings and their $h$-polynomials. We investigate when edge rings are pseudo-Gorenstein, which means that the leading coefficients of the $h$-polynomials of edge rings are equal to $1$. Moreover, we compute the $h$-polynomials of a special family of edge rings and show that some of them are almost Gorenstein.
{"title":"Pseudo-Gorenstein edge rings and a new family of almost Gorenstein edge rings","authors":"Yuta Hatasa, Nobukazu Kowaki, Koji Matsushita","doi":"arxiv-2409.03176","DOIUrl":"https://doi.org/arxiv-2409.03176","url":null,"abstract":"In this paper, we study edge rings and their $h$-polynomials. We investigate\u0000when edge rings are pseudo-Gorenstein, which means that the leading\u0000coefficients of the $h$-polynomials of edge rings are equal to $1$. Moreover,\u0000we compute the $h$-polynomials of a special family of edge rings and show that\u0000some of them are almost Gorenstein.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185590","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}
This note presents a generalisation of Tressl's structure theorem for differentially finitely generated algebras over differential rings of characteristic 0 to the case of separable algebras over differential rings of arbitrary characteristic.
{"title":"Tressl's Structure Theorem for Separable Algebras","authors":"Gabriel Ng","doi":"arxiv-2409.03654","DOIUrl":"https://doi.org/arxiv-2409.03654","url":null,"abstract":"This note presents a generalisation of Tressl's structure theorem for\u0000differentially finitely generated algebras over differential rings of\u0000characteristic 0 to the case of separable algebras over differential rings of\u0000arbitrary characteristic.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185589","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}
This article presents a heuristic view that shows that the inner states of consciousness experienced by every human being have a physical but imaginary hypercomplex basis. The hypercomplex description is necessary because certain processes of consciousness cannot be physically measured in principle, but nevertheless exist. Based on theoretical considerations, it could be possible - as a result of mathematical investigations into a so-called bicomplex algebra - to generate and use hypercomplex system states on machines in a targeted manner. The hypothesis of the existence of hypercomplex system states on machines is already supported by the surprising performance of highly complex AI systems. However, this has yet to be proven. In particular, there is a lack of experimental data that distinguishes such systems from other systems, which is why this question will be addressed in later articles. This paper describes the developed bicomplex algebra and possible applications of these findings to generate hypercomplex energy states on machines. In the literature, such system states are often referred to as machine consciousness. The article uses mathematical considerations to explain how artificial consciousness could be generated and what advantages this would have for such AI systems.
{"title":"On a heuristic approach to the description of consciousness as a hypercomplex system state and the possibility of machine consciousness (German edition)","authors":"Ralf Otte","doi":"arxiv-2409.02100","DOIUrl":"https://doi.org/arxiv-2409.02100","url":null,"abstract":"This article presents a heuristic view that shows that the inner states of\u0000consciousness experienced by every human being have a physical but imaginary\u0000hypercomplex basis. The hypercomplex description is necessary because certain\u0000processes of consciousness cannot be physically measured in principle, but\u0000nevertheless exist. Based on theoretical considerations, it could be possible -\u0000as a result of mathematical investigations into a so-called bicomplex algebra -\u0000to generate and use hypercomplex system states on machines in a targeted\u0000manner. The hypothesis of the existence of hypercomplex system states on\u0000machines is already supported by the surprising performance of highly complex\u0000AI systems. However, this has yet to be proven. In particular, there is a lack\u0000of experimental data that distinguishes such systems from other systems, which\u0000is why this question will be addressed in later articles. This paper describes\u0000the developed bicomplex algebra and possible applications of these findings to\u0000generate hypercomplex energy states on machines. In the literature, such system\u0000states are often referred to as machine consciousness. The article uses\u0000mathematical considerations to explain how artificial consciousness could be\u0000generated and what advantages this would have for such AI systems.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185595","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}
Some recent investigations indicate that for the classification of Cohen-Macaulay binomial edge ideals, it suffices to consider biconnected graphs with some whiskers attached (in short, `block with whiskers'). This paper provides explicit combinatorial formulae for the Castelnuovo-Mumford regularity of two specific classes of Cohen-Macaulay binomial edge ideals: (i) chain of cycles with whiskers and (ii) $r$-regular $r$-connected block with whiskers. For the first type, we introduce a new invariant of graphs in terms of the number of blocks in certain induced block graphs, and this invariant may help determine the regularity of other classes of binomial edge ideals. For the second type, we present the formula as a linear function of $r$.
最近的一些研究表明，对于科恩-麦考莱二项式边理想的分类，只需考虑附有一些须的双连图（简言之，"带须块"）即可。本文提供了两类特定科恩-麦考莱二项式边理想的卡斯特努沃-蒙福德正则性的明确组合公式：(i) 带须的循环链和 (ii) $r$-regular $r$-connected block with whiskers。对于第一类，我们根据某些诱导块图中的块数引入了一种新的图不变式，这种不变式可能有助于确定其他类二项式边理想的正则性。对于第二种类型，我们将公式表述为 $r$ 的线性函数。
{"title":"Regularity of two classes of Cohen-Macaulay binomial edge ideals","authors":"Om Prakash Bhardwaj, Kamalesh Saha","doi":"arxiv-2409.01639","DOIUrl":"https://doi.org/arxiv-2409.01639","url":null,"abstract":"Some recent investigations indicate that for the classification of\u0000Cohen-Macaulay binomial edge ideals, it suffices to consider biconnected graphs\u0000with some whiskers attached (in short, `block with whiskers'). This paper\u0000provides explicit combinatorial formulae for the Castelnuovo-Mumford regularity\u0000of two specific classes of Cohen-Macaulay binomial edge ideals: (i) chain of\u0000cycles with whiskers and (ii) $r$-regular $r$-connected block with whiskers.\u0000For the first type, we introduce a new invariant of graphs in terms of the\u0000number of blocks in certain induced block graphs, and this invariant may help\u0000determine the regularity of other classes of binomial edge ideals. For the\u0000second type, we present the formula as a linear function of $r$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185592","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 $Csubsetmathbb{N}^p$ be an integer polyhedral cone. An affine semigroup $Ssubset C$ is a $ C$-semigroup if $| Csetminus S|<+infty$. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to $C$-semigroups.
{"title":"C-semigroups with its induced order","authors":"D. Marín-Aragón, R. Tapia-Ramos","doi":"arxiv-2409.02299","DOIUrl":"https://doi.org/arxiv-2409.02299","url":null,"abstract":"Let $Csubsetmathbb{N}^p$ be an integer polyhedral cone. An affine semigroup\u0000$Ssubset C$ is a $ C$-semigroup if $| Csetminus S|<+infty$. This structure\u0000has always been studied using a monomial order. The main issue is that the\u0000choice of these orders is arbitrary. In the present work we choose the order\u0000given by the semigroup itself, which is a more natural order. This allows us to\u0000generalise some of the definitions and results known from numerical semigroup\u0000theory to $C$-semigroups.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185591","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 2019, Kaveh and Manon introduced Khovanskii bases as a special 'Gr"obner-like' generating system of an algebra. We extend their work by considering an arbitrary grading on the algebra and propose a definition for a 'homogeneous Khovanskii basis' that respects this grading. We generalize Khovanskii bases further by taking multiple valuations into account (MUVAK bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a quotient singularity. Our algorithm is an improvement of an algorithm by Yamagishi in this situation.
{"title":"Homogeneous Khovanskii bases and MUVAK bases","authors":"Johannes Schmitt","doi":"arxiv-2409.01146","DOIUrl":"https://doi.org/arxiv-2409.01146","url":null,"abstract":"In 2019, Kaveh and Manon introduced Khovanskii bases as a special\u0000'Gr\"obner-like' generating system of an algebra. We extend their work by\u0000considering an arbitrary grading on the algebra and propose a definition for a\u0000'homogeneous Khovanskii basis' that respects this grading. We generalize\u0000Khovanskii bases further by taking multiple valuations into account (MUVAK\u0000bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a\u0000quotient singularity. Our algorithm is an improvement of an algorithm by\u0000Yamagishi in this situation.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185593","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}
An integral domain $R$ is called atomic if every nonzero nonunit of $R$ factors into irreducibles, while $R$ satisfies the ascending chain condition on principal ideals if every ascending chain of principal ideals of $R$ stabilizes. It is well known and not hard to verify that if an integral domain satisfies the ACCP, then it must be atomic. The converse does not hold in general, but examples are hard to come by and most of them are the result of crafty and technical constructions. Sporadic constructions of such atomic domains have appeared in the literature in the last five decades, including the first example of a finite-dimensional atomic monoid algebra not satisfying the ACCP recently constructed by the second and third authors. Here we construct the first known one-dimensional monoid algebras satisfying the almost ACCP but not the ACCP (the almost ACCP is a notion weaker than the ACCP but still stronger than atomicity). Although the two constructions we provide here are rather technical, the corresponding monoid algebras are perhaps the most elementary known examples of atomic domains not satisfying the ACCP.
{"title":"One-dimensional monoid algebras and ascending chains of principal ideals","authors":"Alan Bu, Felix Gotti, Bangzheng Li, Alex Zhao","doi":"arxiv-2409.00580","DOIUrl":"https://doi.org/arxiv-2409.00580","url":null,"abstract":"An integral domain $R$ is called atomic if every nonzero nonunit of $R$\u0000factors into irreducibles, while $R$ satisfies the ascending chain condition on\u0000principal ideals if every ascending chain of principal ideals of $R$\u0000stabilizes. It is well known and not hard to verify that if an integral domain\u0000satisfies the ACCP, then it must be atomic. The converse does not hold in\u0000general, but examples are hard to come by and most of them are the result of\u0000crafty and technical constructions. Sporadic constructions of such atomic\u0000domains have appeared in the literature in the last five decades, including the\u0000first example of a finite-dimensional atomic monoid algebra not satisfying the\u0000ACCP recently constructed by the second and third authors. Here we construct\u0000the first known one-dimensional monoid algebras satisfying the almost ACCP but\u0000not the ACCP (the almost ACCP is a notion weaker than the ACCP but still\u0000stronger than atomicity). Although the two constructions we provide here are\u0000rather technical, the corresponding monoid algebras are perhaps the most\u0000elementary known examples of atomic domains not satisfying the ACCP.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185594","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}
Generalizing the strong Lefschetz property for an $mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $mathbb{N}^m$-graded algebra. We show that, for $bm{a} in mathbb{N}^m_+$, the generic $mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $bm{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. A corollary is the inequality $h_{bm{b}} leq h_{bm{c}}$ for $bm{b} leq bm{c} leq bm{a}-bm{b}$ among the flag $h$-numbers of an $bm{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $bm{a}$-balanced homology manifolds and $bm{a}$-balanced simplicial cycles over a field of characteristic $2$.
{"title":"Multigraded strong Lefschetz property for balanced simplicial complexes","authors":"Ryoshun Oba","doi":"arxiv-2408.17110","DOIUrl":"https://doi.org/arxiv-2408.17110","url":null,"abstract":"Generalizing the strong Lefschetz property for an $mathbb{N}$-graded\u0000algebra, we introduce the multigraded strong Lefschetz property for an\u0000$mathbb{N}^m$-graded algebra. We show that, for $bm{a} in mathbb{N}^m_+$,\u0000the generic $mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner\u0000ring of an $bm{a}$-balanced homology sphere over a field of characteristic $2$\u0000satisfies the multigraded strong Lefschetz property. A corollary is the\u0000inequality $h_{bm{b}} leq h_{bm{c}}$ for $bm{b} leq bm{c} leq\u0000bm{a}-bm{b}$ among the flag $h$-numbers of an $bm{a}$-balanced simplicial\u0000sphere. This can be seen as a common generalization of the unimodality of the\u0000$h$-vector of a simplicial sphere by Adiprasito and the balanced generalized\u0000lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize\u0000these results to $bm{a}$-balanced homology manifolds and $bm{a}$-balanced\u0000simplicial cycles over a field of characteristic $2$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185596","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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