In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we introduce the concept of monomial ideals of well-nearly normally torsion-free type, and show that these ideals are normal. After that, we present some results on the existence of embedded associated prime ideals in the associated primes set of powers of monomial ideals. Further, we employ them in investigating the edge and cover ideals of cones of graphs. Next, we present counterexamples to several questions concerning the relations between relevant algebraic properties of the edge ideals of clutters and complement clutters. We conclude by providing counterexamples to questions on the possible connections between normally torsion-freeness and normality of monomial ideals under polarization.
{"title":"Normally torsion-freeness and normality criteria for monomial ideals","authors":"M. Nasernejad, V. Crispin Quinonez, J. Toledo","doi":"arxiv-2408.05561","DOIUrl":"https://doi.org/arxiv-2408.05561","url":null,"abstract":"In this paper, we focus on the associated primes of powers of monomial ideals\u0000and asymptotic behavior properties such as normally torsion-freeness,\u0000normality, the strong persistence property, and the persistence property. In\u0000particular, we introduce the concept of monomial ideals of well-nearly normally\u0000torsion-free type, and show that these ideals are normal. After that, we\u0000present some results on the existence of embedded associated prime ideals in\u0000the associated primes set of powers of monomial ideals. Further, we employ them\u0000in investigating the edge and cover ideals of cones of graphs. Next, we present\u0000counterexamples to several questions concerning the relations between relevant\u0000algebraic properties of the edge ideals of clutters and complement clutters. We\u0000conclude by providing counterexamples to questions on the possible connections\u0000between normally torsion-freeness and normality of monomial ideals under\u0000polarization.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185617","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}
The goal of this paper is to study the Rees algebra $mathfrak{R}(I)$and the special fiber ring $mathfrak{F}(I)$ for a family of ideals. Let $R=mathbb{K}[x_1, ldots, x_d]$ with $dgeq 4$ be a polynomial ring with homogeneous maximal ideal $mathfrak{m}$. We study the $R$-ideals $I$, which are $mathfrak{m}$-primary, Gorenstein, generated in degree 2, and have a Gorenstein linear resolution. In the smallest case, $d=4$, this family includes the ideals of $2times 2$ minors of a general $3times 3$ matrix of linear forms in $R$. We show that the defining ideal of the Rees algebra will be of fiber type. That is, the defining ideal of the Rees algebra is generated by the defining ideals of the special fiber ring and of the symmetric algebra. We use the fact that these ideals differ from $mathfrak{m}^2$ by exactly one minimal generator to describe the defining ideal $mathfrak{F}(I)$ as a sub-ideal of the defining ideal of $mathfrak{F}(mathfrak{m}^2)$, which is well known to be the ideal of $2times 2$ minors of a symmetric matrix of variables.
{"title":"Rees algebras of ideals submaximally generated by quadrics","authors":"Whitney Liske","doi":"arxiv-2408.05199","DOIUrl":"https://doi.org/arxiv-2408.05199","url":null,"abstract":"The goal of this paper is to study the Rees algebra $mathfrak{R}(I)$and the\u0000special fiber ring $mathfrak{F}(I)$ for a family of ideals. Let\u0000$R=mathbb{K}[x_1, ldots, x_d]$ with $dgeq 4$ be a polynomial ring with\u0000homogeneous maximal ideal $mathfrak{m}$. We study the $R$-ideals $I$, which\u0000are $mathfrak{m}$-primary, Gorenstein, generated in degree 2, and have a\u0000Gorenstein linear resolution. In the smallest case, $d=4$, this family includes\u0000the ideals of $2times 2$ minors of a general $3times 3$ matrix of linear\u0000forms in $R$. We show that the defining ideal of the Rees algebra will be of\u0000fiber type. That is, the defining ideal of the Rees algebra is generated by the\u0000defining ideals of the special fiber ring and of the symmetric algebra. We use\u0000the fact that these ideals differ from $mathfrak{m}^2$ by exactly one minimal\u0000generator to describe the defining ideal $mathfrak{F}(I)$ as a sub-ideal of\u0000the defining ideal of $mathfrak{F}(mathfrak{m}^2)$, which is well known to be\u0000the ideal of $2times 2$ minors of a symmetric matrix of variables.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938326","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}
Strongly stable ideals are a class of monomial ideals which correspond to generic initial ideals in characteristic zero and can be described completely by their Borel generators, a subset of the minimal monomial generators of the ideal. Francisco, Mermin, and Schweig developed formulas for the Hilbert series and Betti numbers of strongly stable ideals in terms of their Borel generators. In this work, a specialization of strongly stable ideals is presented which further restricts the subset of relevant generators. A choice of weight vector $winmathbb{N}_{> 0}^n$ restricts the set of strongly stable ideals to a subset designated as $w$-stable ideals. This restriction further compresses the Borel generators to a subset termed the weighted Borel generators of the ideal. A new Macaulay2 package wStableIdeals.m2 has been developed alongside this paper and segments of code support computations within.
{"title":"Weighted Borel Generators","authors":"Seth Ireland","doi":"arxiv-2408.04120","DOIUrl":"https://doi.org/arxiv-2408.04120","url":null,"abstract":"Strongly stable ideals are a class of monomial ideals which correspond to\u0000generic initial ideals in characteristic zero and can be described completely\u0000by their Borel generators, a subset of the minimal monomial generators of the\u0000ideal. Francisco, Mermin, and Schweig developed formulas for the Hilbert series\u0000and Betti numbers of strongly stable ideals in terms of their Borel generators.\u0000In this work, a specialization of strongly stable ideals is presented which\u0000further restricts the subset of relevant generators. A choice of weight vector\u0000$winmathbb{N}_{> 0}^n$ restricts the set of strongly stable ideals to a\u0000subset designated as $w$-stable ideals. This restriction further compresses the\u0000Borel generators to a subset termed the weighted Borel generators of the ideal.\u0000A new Macaulay2 package wStableIdeals.m2 has been developed alongside this\u0000paper and segments of code support computations within.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938367","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 use the inclusion of injectives into the canonical heart as a replacement for tilting objects in computations of the characteristic morphism. Then we apply this construction to proofs of non-liftability of candidate non-Fourier-Mukai functors, i.e. functors that do not admit an $mathcal{A}_infty$/$mathrm{dg}$-lift. This approach allows explicit computation of the obstruction against an $mathcal{A}_infty$-lift. We in particular observe that this computation gives for smooth degree $d>2$ hypersurfaces an abundance of non-Fourier-Mukai functors.
{"title":"Injectives obstruct Fourier-Mukai functors","authors":"Felix Küng","doi":"arxiv-2408.03027","DOIUrl":"https://doi.org/arxiv-2408.03027","url":null,"abstract":"We use the inclusion of injectives into the canonical heart as a replacement\u0000for tilting objects in computations of the characteristic morphism. Then we\u0000apply this construction to proofs of non-liftability of candidate\u0000non-Fourier-Mukai functors, i.e. functors that do not admit an\u0000$mathcal{A}_infty$/$mathrm{dg}$-lift. This approach allows explicit\u0000computation of the obstruction against an $mathcal{A}_infty$-lift. We in\u0000particular observe that this computation gives for smooth degree $d>2$\u0000hypersurfaces an abundance of non-Fourier-Mukai functors.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938368","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 $X$ be an integral affine scheme of characteristic $p>0$, and $sigma $ a non-identity automorphism of $X$. If $sigma $ is $textit{exponential}$, i.e., induced from a ${bf G}_a$-action on $X$, then $sigma $ is obviously of order $p$. It is easy to see that the converse is not true in general. In fact, there exists $X$ which admits an automorphism of order $p$, but admits no non-trivial ${bf G}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${bf A}_R^n$, because ${bf A}_R^n$ admits various ${bf G}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${bf A}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${bf G}_a$-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ${bf A}_R^n$ of order $p$ are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${bf A}_R^n$ of order $p$ for each $nge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${bf G}_a$-actions inducing an elementary automorphism of ${bf A}_R^n$.
{"title":"On exponentiality of automorphisms of ${bf A}^n$ of order $p$ in characteristic $p>0$","authors":"Shigeru Kuroda","doi":"arxiv-2408.02204","DOIUrl":"https://doi.org/arxiv-2408.02204","url":null,"abstract":"Let $X$ be an integral affine scheme of characteristic $p>0$, and $sigma $ a\u0000non-identity automorphism of $X$. If $sigma $ is $textit{exponential}$, i.e.,\u0000induced from a ${bf G}_a$-action on $X$, then $sigma $ is obviously of order\u0000$p$. It is easy to see that the converse is not true in general. In fact, there\u0000exists $X$ which admits an automorphism of order $p$, but admits no non-trivial\u0000${bf G}_a$-actions. However, the situation is not clear in the case where $X$\u0000is the affine space ${bf A}_R^n$, because ${bf A}_R^n$ admits various ${bf\u0000G}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${bf A}_R^n$ of\u0000order $p$, where the difficulty stems from the non-uniqueness of ${bf\u0000G}_a$-actions inducing an exponential automorphism. Our main results are as\u0000follows. (1) We show that the triangular automorphisms of ${bf A}_R^n$ of order $p$\u0000are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${bf A}_R^n$ of order $p$\u0000for each $nge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${bf G}_a$-actions inducing an elementary\u0000automorphism of ${bf A}_R^n$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938372","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 calculate the determinant of the bilinear form in middle degree of the generic artinian reduction of the Stanley-Reisner ring of an odd-dimensional simplicial sphere. This proves the odd multiplicity conjecture of Papadakis and Petrotou and implies that this determinant is a complete invariant of the simplicial sphere. We extend this result to odd-dimensional connected oriented simplicial homology manifolds, and we conjecture a generalization to the Hodge-Riemann forms of any connected oriented simplicial homology manifold. We show that our conjecture follows from the strong Lefschetz property for certain quotients of the Stanley-Reisner rings.
{"title":"Determinants of Hodge-Riemann forms and simplicial manifolds","authors":"Matt Larson, Alan Stapledon","doi":"arxiv-2408.02737","DOIUrl":"https://doi.org/arxiv-2408.02737","url":null,"abstract":"We calculate the determinant of the bilinear form in middle degree of the\u0000generic artinian reduction of the Stanley-Reisner ring of an odd-dimensional\u0000simplicial sphere. This proves the odd multiplicity conjecture of Papadakis and\u0000Petrotou and implies that this determinant is a complete invariant of the\u0000simplicial sphere. We extend this result to odd-dimensional connected oriented\u0000simplicial homology manifolds, and we conjecture a generalization to the\u0000Hodge-Riemann forms of any connected oriented simplicial homology manifold. We\u0000show that our conjecture follows from the strong Lefschetz property for certain\u0000quotients of the Stanley-Reisner rings.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969005","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}
Cosimo Flavi, Joachim Jelisiejew, Mateusz Michałek
We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that they are optimal under mild conditions. We give applications to complexity theory.
{"title":"Symmetric powers: structure, smoothability, and applications","authors":"Cosimo Flavi, Joachim Jelisiejew, Mateusz Michałek","doi":"arxiv-2408.02754","DOIUrl":"https://doi.org/arxiv-2408.02754","url":null,"abstract":"We investigate border ranks of twisted powers of polynomials and\u0000smoothability of symmetric powers of algebras. We prove that the latter are\u0000smoothable. For the former, we obtain upper bounds for the border rank in\u0000general and prove that they are optimal under mild conditions. We give\u0000applications to complexity theory.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938371","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 study the canonical Hodge filtration on the sheaf $mathscr{O}_X(*D)$ of meromorphic functions along a divisor. For a germ of an analytic function $f$ whose Bernstein-Sato's polynomial's roots are contained in $(-2,0)$, we: give a simple algebraic formula for the zeroeth piece of the Hodge filtration; bound the first step of the Hodge filtration containing $f^{-1}$. If we additionally require $f$ to be Euler homogeneous and parametrically prime, then we extend our algebraic formula to compute every piece of the canonical Hodge filtration, proving in turn that the Hodge filtration is contained in the induced order filtration. Finally, we compute the Hodge filtration in many examples and identify several large classes of divisors realizing our theorems.
{"title":"The Hodge filtration and parametrically prime divisors","authors":"Daniel Bath, Henry Dakin","doi":"arxiv-2408.02601","DOIUrl":"https://doi.org/arxiv-2408.02601","url":null,"abstract":"We study the canonical Hodge filtration on the sheaf $mathscr{O}_X(*D)$ of\u0000meromorphic functions along a divisor. For a germ of an analytic function $f$\u0000whose Bernstein-Sato's polynomial's roots are contained in $(-2,0)$, we: give a\u0000simple algebraic formula for the zeroeth piece of the Hodge filtration; bound\u0000the first step of the Hodge filtration containing $f^{-1}$. If we additionally\u0000require $f$ to be Euler homogeneous and parametrically prime, then we extend\u0000our algebraic formula to compute every piece of the canonical Hodge filtration,\u0000proving in turn that the Hodge filtration is contained in the induced order\u0000filtration. Finally, we compute the Hodge filtration in many examples and\u0000identify several large classes of divisors realizing our theorems.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938370","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 extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups and finally we introduce the extension of gapsets and prove that the sequence of the number of gapsets of size $g$ is non-decreasing as a weak version of Bras-Amor'os's conjecture.
{"title":"Gapset Extensions, Theory and Computations","authors":"Arman Ataei Kachouei, Farhad Rahmati","doi":"arxiv-2408.02425","DOIUrl":"https://doi.org/arxiv-2408.02425","url":null,"abstract":"In this paper we extend some set theoretic concepts of numerical semigroups\u0000for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets\u0000which leads to a more efficient computational approach towards numerical\u0000semigroups and finally we introduce the extension of gapsets and prove that the\u0000sequence of the number of gapsets of size $g$ is non-decreasing as a weak\u0000version of Bras-Amor'os's conjecture.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938376","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}
Ela Celikbas, Olgur Celikbas, Hiroki Matsui, Ryo Takahashi
In this paper we study rigid modules over commutative Noetherian local rings, establish new freeness criteria for certain periodic rigid modules, and extend several results from the literature. Along the way, we prove general Ext vanishing results over Cohen-Macaulay rings and investigate modules which have zero class in the reduced Grothendieck group with rational coefficients.
{"title":"On the vanishing of self extensions of even-periodic modules","authors":"Ela Celikbas, Olgur Celikbas, Hiroki Matsui, Ryo Takahashi","doi":"arxiv-2408.02820","DOIUrl":"https://doi.org/arxiv-2408.02820","url":null,"abstract":"In this paper we study rigid modules over commutative Noetherian local rings,\u0000establish new freeness criteria for certain periodic rigid modules, and extend\u0000several results from the literature. Along the way, we prove general Ext\u0000vanishing results over Cohen-Macaulay rings and investigate modules which have\u0000zero class in the reduced Grothendieck group with rational coefficients.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938287","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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