This paper has two related parts. The first generalizes Hochster's formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley-Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.
{"title":"A COLORFUL HOCHSTER FORMULA AND UNIVERSAL PARAMETERS FOR FACE RINGS","authors":"Ashleigh Adams, V. Reiner","doi":"10.1216/jca.2023.15.151","DOIUrl":"https://doi.org/10.1216/jca.2023.15.151","url":null,"abstract":"This paper has two related parts. The first generalizes Hochster's formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley-Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"46 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87907566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-24DOI: 10.1007/978-3-030-89694-2_8
David A. Cox
{"title":"Stickelberger and the Eigenvalue Theorem","authors":"David A. Cox","doi":"10.1007/978-3-030-89694-2_8","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_8","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89582931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbf{A}_{n, m}$ be the polynomial ring $text{Sym}(mathbf{C}^n otimes mathbf{C}^m)$ with the natural action of $mathbf{GL}_m(mathbf{C})$. We construct a family of $mathbf{GL}_m(mathbf{C})$-stable ideals $J_{n, m}$ in $mathbf{A}_{n, m}$, each equivariantly generated by one homogeneous polynomial of degree $2$. Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.
{"title":"Stillman’s question for twisted commutative algebras","authors":"Karthik Ganapathy","doi":"10.1216/jca.2022.14.315","DOIUrl":"https://doi.org/10.1216/jca.2022.14.315","url":null,"abstract":"Let $mathbf{A}_{n, m}$ be the polynomial ring $text{Sym}(mathbf{C}^n otimes mathbf{C}^m)$ with the natural action of $mathbf{GL}_m(mathbf{C})$. We construct a family of $mathbf{GL}_m(mathbf{C})$-stable ideals $J_{n, m}$ in $mathbf{A}_{n, m}$, each equivariantly generated by one homogeneous polynomial of degree $2$. Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74021233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we construct examples of irrational behavior of multiplicities and mixed multiplicities of divisorial filtrations. The construction makes essential use of anti-positive intersection products.
本文构造了除法过滤的多重性和混合多重性的非理性行为的例子。该构造主要利用了反正交积。
{"title":"Examples of multiplicities and mixed multiplicities of filtrations","authors":"S. Cutkosky","doi":"10.1090/conm/773/15530","DOIUrl":"https://doi.org/10.1090/conm/773/15530","url":null,"abstract":"In this paper we construct examples of irrational behavior of multiplicities and mixed multiplicities of divisorial filtrations. The construction makes essential use of anti-positive intersection products.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83050404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A locally F-finite Noetherian domain that is not F-finite","authors":"T. Dumitrescu, Cristodor Ionescu","doi":"10.1216/jca.2022.14.177","DOIUrl":"https://doi.org/10.1216/jca.2022.14.177","url":null,"abstract":"Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"47 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72480406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ela Celikbas, Olgur Celikbas, Cuatualin Ciupercua, Naoki Endo, S. Goto, Ryotaro Isobe, Naoyuki Matsuoka
We introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.
{"title":"ON THE UBIQUITY OF ARF RINGS","authors":"Ela Celikbas, Olgur Celikbas, Cuatualin Ciupercua, Naoki Endo, S. Goto, Ryotaro Isobe, Naoyuki Matsuoka","doi":"10.1216/jca.2023.15.177","DOIUrl":"https://doi.org/10.1216/jca.2023.15.177","url":null,"abstract":"We introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"110 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85776266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Among other results, we prove the following: (1) A locally Archimedean stable domain satisfies accp. (2) A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of R′ (this result is related to Ohm’s Theorem for Prüfer domains). (3) An Archimedean stable domain R is one-dimensional if and only if R′ is equidimensional (generally, an Archimedean stable local domain is not necessarily onedimensional). (4) An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.
{"title":"On finitely stable domains, II","authors":"Stefania Gabelli, M. Roitman","doi":"10.1216/jca.2020.12.179","DOIUrl":"https://doi.org/10.1216/jca.2020.12.179","url":null,"abstract":"Among other results, we prove the following: (1) A locally Archimedean stable domain satisfies accp. (2) A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of R′ (this result is related to Ohm’s Theorem for Prüfer domains). (3) An Archimedean stable domain R is one-dimensional if and only if R′ is equidimensional (generally, an Archimedean stable local domain is not necessarily onedimensional). (4) An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"33 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76215636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group G , a ring-valued functor W G . This paper gives the first concrete interpretation of any Witt–Burnside rings outside the procyclic cases in terms of known rings. In particular, the rings W Z p 2 ( k ) , where k is a field of characteristic p > 0 have a quotient realized as rings of Lipschitz continuous functions on the p -adic upper half plane P 1 ( Q p ) . As a consequence we show that the Krull dimensions of the rings W Z p d ( k ) are infinite for d ≥ 2 and we show the Teichmuller representatives form an analogue of the van der Put basis for continuous functions on Z p .
Dress和Siebeneicher给出了Witt向量构造的一个有意义的推广，他们对任意无限群G产生了一个环值函子wg。本文首次用已知环对顺环以外的任何威特-伯恩赛德环进行了具体的解释。特别地，环wzp2 (k)，其中k是特征为p > 0的域，其商被实现为p进上半平面p1 (Q p)上的Lipschitz连续函数环。因此，我们证明了环wz p d (k)的Krull维对于d≥2是无限的，并且我们证明了Teichmuller表示形成了zp上连续函数的van der Put基的类似物。
{"title":"Witt–Burnside functor attached to $boldsymbol{Z}_{p}^{2}$ and $p$-adic Lipschitz continuous functions","authors":"L. Miller, B. Steinhurst","doi":"10.1216/jca.2020.12.263","DOIUrl":"https://doi.org/10.1216/jca.2020.12.263","url":null,"abstract":"Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group G , a ring-valued functor W G . This paper gives the first concrete interpretation of any Witt–Burnside rings outside the procyclic cases in terms of known rings. In particular, the rings W Z p 2 ( k ) , where k is a field of characteristic p > 0 have a quotient realized as rings of Lipschitz continuous functions on the p -adic upper half plane P 1 ( Q p ) . As a consequence we show that the Krull dimensions of the rings W Z p d ( k ) are infinite for d ≥ 2 and we show the Teichmuller representatives form an analogue of the van der Put basis for continuous functions on Z p .","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"47 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79012419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we give a few examples of local rings in relation to weak normality and seminormality in mixed characteristic. It is known that two concepts can differ in the equal prime characteristic case, while they coincide in the equal characteristic zero case. No explicit examples seem to be documented in the existing literature in the mixed characteristic case. We also establish the local Bertini theorem for weak normality in mixed characteristic under a certain condition.
{"title":"Weak normality and seminormality in the mixed characteristic case","authors":"Jun Horiuchi, Kazuma Shimomoto","doi":"10.1216/jca.2022.14.351","DOIUrl":"https://doi.org/10.1216/jca.2022.14.351","url":null,"abstract":"In this article, we give a few examples of local rings in relation to weak normality and seminormality in mixed characteristic. It is known that two concepts can differ in the equal prime characteristic case, while they coincide in the equal characteristic zero case. No explicit examples seem to be documented in the existing literature in the mixed characteristic case. We also establish the local Bertini theorem for weak normality in mixed characteristic under a certain condition.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76576349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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