Let $Rto A$ be a homomorphism of associative rings, and let $(mathcal F,mathcal C)$ be a hereditary complete cotorsion pair in $Rmathsf{-Mod}$. Let $(mathcal F_A,mathcal C_A)$ be the cotorsion pair in $Amathsf{-Mod}$ in which $mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $mathcal F$. Assuming that the $mathcal F$-resolution dimension of every left $R$-module is finite and the class $mathcal F$ is preserved by the coinduction functor $operatorname{Hom}_R(A,-)$, we show that $mathcal C_A$ is the class of all direct summands of left $A$-modules finitely filtered by $A$-modules coinduced from $R$-modules from $mathcal C$. Assuming that the class $mathcal F$ is closed under countable products and preserved by the functor $operatorname{Hom}_R(A,-)$, we prove that $mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $mathcal F$ have finite $mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $omega+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of the classical Bongartz lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.
{"title":"An explicit self-dual construction of complete cotorsion pairs in the relative context","authors":"L. Positselski","doi":"10.4171/RSMUP/118","DOIUrl":"https://doi.org/10.4171/RSMUP/118","url":null,"abstract":"Let $Rto A$ be a homomorphism of associative rings, and let $(mathcal F,mathcal C)$ be a hereditary complete cotorsion pair in $Rmathsf{-Mod}$. Let $(mathcal F_A,mathcal C_A)$ be the cotorsion pair in $Amathsf{-Mod}$ in which $mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $mathcal F$. Assuming that the $mathcal F$-resolution dimension of every left $R$-module is finite and the class $mathcal F$ is preserved by the coinduction functor $operatorname{Hom}_R(A,-)$, we show that $mathcal C_A$ is the class of all direct summands of left $A$-modules finitely filtered by $A$-modules coinduced from $R$-modules from $mathcal C$. Assuming that the class $mathcal F$ is closed under countable products and preserved by the functor $operatorname{Hom}_R(A,-)$, we prove that $mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $mathcal F$ have finite $mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $omega+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of the classical Bongartz lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85124162","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}
It is well known that all of the prime cyclotomic polynomials and binary cyclotomic polynomials are flat, and the flatness of ternary cyclotomic polynomials is much more complicated. Let p < q < r be odd primes such that zr ≡ ±1 (mod pq), where z is a positive integer. So far, the classification of flat ternary cyclotomic polynomials for 1 ≤ z ≤ 5 has been given. In this paper, for z = 6 and q ≡ ±1 (mod p), we give the necessary and sufficient conditions for ternary cyclotomic polynomials Φpqr(x) to be flat. Mathematics Subject Classification (2010). 11B83, 11C08, 11N56.
{"title":"The flatness of ternary cyclotomic polynomials","authors":"Bin Zhang","doi":"10.4171/rsmup/47","DOIUrl":"https://doi.org/10.4171/rsmup/47","url":null,"abstract":"It is well known that all of the prime cyclotomic polynomials and binary cyclotomic polynomials are flat, and the flatness of ternary cyclotomic polynomials is much more complicated. Let p < q < r be odd primes such that zr ≡ ±1 (mod pq), where z is a positive integer. So far, the classification of flat ternary cyclotomic polynomials for 1 ≤ z ≤ 5 has been given. In this paper, for z = 6 and q ≡ ±1 (mod p), we give the necessary and sufficient conditions for ternary cyclotomic polynomials Φpqr(x) to be flat. Mathematics Subject Classification (2010). 11B83, 11C08, 11N56.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"463 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75124336","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 give a criterion for the good reduction of semistable K3 surfaces over p-adic fields. We use neither p-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or K3 surfaces. We achieve our goal by realizing the special fiber Xs of a semistable model X of a K3 surface over the p-adic field K, as a special fiber of a log-family in characteristic p and use an arithmetic version of the Clemens-Schmid exact sequence in order to obtain a Kulikov-Persson-Pinkham classification theorem in characteristic p. Mathematics Subject Classification (2010). 14F30, 11G25, 14F35
{"title":"A monodromy criterion for the good reduction of $K3$ surfaces","authors":"Genaro Hernandez-Mada","doi":"10.4171/rsmup/50","DOIUrl":"https://doi.org/10.4171/rsmup/50","url":null,"abstract":"We give a criterion for the good reduction of semistable K3 surfaces over p-adic fields. We use neither p-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or K3 surfaces. We achieve our goal by realizing the special fiber Xs of a semistable model X of a K3 surface over the p-adic field K, as a special fiber of a log-family in characteristic p and use an arithmetic version of the Clemens-Schmid exact sequence in order to obtain a Kulikov-Persson-Pinkham classification theorem in characteristic p. Mathematics Subject Classification (2010). 14F30, 11G25, 14F35","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79051361","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}
For a completely regular frame L, the ring RL of real-valued continuous functions on L is equipped with the uniform topology. The closed ideals of RL in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of RL, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame L is finite if and only if every ideal of RL is closed. Finally, we prove that every closed ideal in RL is an intersection of maximal ideals. Mathematics Subject Classification (2010). Primary: 06D22; Secondary: 54C40, 54C50, 16D25, 13J20.
{"title":"Closed ideals in the uniform topology on the ring of real-valued continuous functions on a frame","authors":"M. Abedi, A. Estaji","doi":"10.4171/rsmup/43","DOIUrl":"https://doi.org/10.4171/rsmup/43","url":null,"abstract":"For a completely regular frame L, the ring RL of real-valued continuous functions on L is equipped with the uniform topology. The closed ideals of RL in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of RL, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame L is finite if and only if every ideal of RL is closed. Finally, we prove that every closed ideal in RL is an intersection of maximal ideals. Mathematics Subject Classification (2010). Primary: 06D22; Secondary: 54C40, 54C50, 16D25, 13J20.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"15 1","pages":"135-152"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89328385","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 G be a finitely generated relatively free group that is locally graded. We show that either G contains a non-trivial free subsemigroup or G is nilpotent-by-finite.
设G是一个有限生成的局部分级的相对自由群。我们证明了G包含一个非平凡的自由子半群或者G是幂零的。
{"title":"Milnor’s alternative for finitely generated relatively free groups that are locally graded","authors":"W. Hołubowski, O. Macedońska","doi":"10.4171/rsmup/46","DOIUrl":"https://doi.org/10.4171/rsmup/46","url":null,"abstract":"Let G be a finitely generated relatively free group that is locally graded. We show that either G contains a non-trivial free subsemigroup or G is nilpotent-by-finite.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"73 1","pages":"247-250"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84299300","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 (K, |·|) be a local field. In this paper we define an invariant analogous to the discriminant over K for certain transcendental elements over K. Mathematics Subject Classification (2010). Primary: 11S99; Secondary: 13A18, 12J05, 12J10, 12J25.
{"title":"A notion analogous to the discriminant for transcendental elements in certain extensions of local fields","authors":"S. Achimescu, V. Alexandru, C. S. Andronescu","doi":"10.4171/rsmup/41","DOIUrl":"https://doi.org/10.4171/rsmup/41","url":null,"abstract":"Let (K, |·|) be a local field. In this paper we define an invariant analogous to the discriminant over K for certain transcendental elements over K. Mathematics Subject Classification (2010). Primary: 11S99; Secondary: 13A18, 12J05, 12J10, 12J25.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"105-112"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86301991","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}
{"title":"Sign-changing solutions of boundary value problems for semilinear $Delta_{gamma}$-Laplace equations","authors":"D. T. Luyen","doi":"10.4171/rsmup/42","DOIUrl":"https://doi.org/10.4171/rsmup/42","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"16 1","pages":"113-134"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73319214","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 paper investigates the projective dimension of the maximal right ring of quotients Q(R) of a right non-singular ring R. Our discussion addresses the question under which conditions p.d.(Q)) ≤ 1 guarantees that the module Q/R is a direct sum of countably generated modules extending Matlis’ Theorem for integral domains to a non-commutative setting. Mathematics Subject Classification (2010). Primary: 16D10; Secondary: 16D40, 16E30, 16P50, 16P60, 16S85.
{"title":"Divisibility and duo-rings","authors":"U. Albrecht, Bradley McQuaig","doi":"10.4171/rsmup/40","DOIUrl":"https://doi.org/10.4171/rsmup/40","url":null,"abstract":"This paper investigates the projective dimension of the maximal right ring of quotients Q(R) of a right non-singular ring R. Our discussion addresses the question under which conditions p.d.(Q)) ≤ 1 guarantees that the module Q/R is a direct sum of countably generated modules extending Matlis’ Theorem for integral domains to a non-commutative setting. Mathematics Subject Classification (2010). Primary: 16D10; Secondary: 16D40, 16E30, 16P50, 16P60, 16S85.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"90 1","pages":"81-103"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77874172","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 subject of this paper is an algebraic version of the irregular Riemann-Hilbert correspondence which was mentioned in [arXiv:1910.09954] by the author. In particular, we prove an equivalence of categories between the triangulated category of algebraic holonomic D-modules on a smooth algebraic variety and the one of algebraic C-constructible enhanced ind-sheaves. Moreover we show that there exists a t-structure on the triangulated category of algebraic C-constructible enhanced ind-sheaves whose heart is equivalent to the abelian category of algebraic holonomic D-modules. Furthermore we shall consider simple objects of its heart and minimal extensions of objects of its heart.
{"title":"Note on algebraic irregular Riemann–Hilbert correspondence","authors":"Yohei Ito","doi":"10.4171/rsmup/119","DOIUrl":"https://doi.org/10.4171/rsmup/119","url":null,"abstract":"The subject of this paper is an algebraic version of the irregular Riemann-Hilbert correspondence which was mentioned in [arXiv:1910.09954] by the author. In particular, we prove an equivalence of categories between the triangulated category of algebraic holonomic D-modules on a smooth algebraic variety and the one of algebraic C-constructible enhanced ind-sheaves. Moreover we show that there exists a t-structure on the triangulated category of algebraic C-constructible enhanced ind-sheaves whose heart is equivalent to the abelian category of algebraic holonomic D-modules. Furthermore we shall consider simple objects of its heart and minimal extensions of objects of its heart.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85848475","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}
Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we show several convolution identities of poly-Cauchy numbers with level $2$. In particular, that of three poly-Cauchy numbers with level $2$ can be expressed as a simple form. In the sequel, we introduce the Stirling numbers of the first kind with level $2$
{"title":"Convolution identities of poly-Cauchy numbers with level 2","authors":"T. Komatsu","doi":"10.4171/rsmup/106","DOIUrl":"https://doi.org/10.4171/rsmup/106","url":null,"abstract":"Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we show several convolution identities of poly-Cauchy numbers with level $2$. In particular, that of three poly-Cauchy numbers with level $2$ can be expressed as a simple form. In the sequel, we introduce the Stirling numbers of the first kind with level $2$","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76566254","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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