We study Neron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the well-known representation-theoretic criteria for good and semiabelian reduction due to Neron-Ogg-Shafarevich and Grothendieck carry over to the pseudo-Abelian case, and give examples to show that our results are the best possible in most cases. Finally, we study the order of the group scheme of connected components of the Neron model in the pseudo-Abelian case. Our method is able to control the $ell$-part (for $ellnot=p$) of this order completely, and we study the $p$-part in a particular (but still reasonably general) situation.
{"title":"Néron models of pseudo-Abelian varieties","authors":"Otto Overkamp","doi":"10.4171/rsmup/145","DOIUrl":"https://doi.org/10.4171/rsmup/145","url":null,"abstract":"We study Neron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the well-known representation-theoretic criteria for good and semiabelian reduction due to Neron-Ogg-Shafarevich and Grothendieck carry over to the pseudo-Abelian case, and give examples to show that our results are the best possible in most cases. Finally, we study the order of the group scheme of connected components of the Neron model in the pseudo-Abelian case. Our method is able to control the $ell$-part (for $ellnot=p$) of this order completely, and we study the $p$-part in a particular (but still reasonably general) situation.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":" 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141224221","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}
A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.
{"title":"A generalization of the total mean curvature","authors":"Katarzyna Charytanowicz, W. Cieslak, W. Mozgawa","doi":"10.4171/rsmup/87","DOIUrl":"https://doi.org/10.4171/rsmup/87","url":null,"abstract":"A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84709328","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 $T=biggl(begin{matrix} A&0 U&B end{matrix}biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $biggl(begin{matrix} M_1 M_2end{matrix}biggr)_{varphi^M}$ is Ding projective if and only if $M_1$ and $M_2/{rm im}(varphi^M)$ are Ding projective and the morphism $varphi^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $ker(widetilde{varphi_{W}})$ are Ding injective and the morphism $widetilde{varphi_{W}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.
设$T=biggl(begin{matrix} a &0 U&B end{matrix}biggr)$是一个形式三角矩阵环，其中$ a $和$B$是环，$U$是一个$(B, a)$-双模。证明了(1)如果$U_A$和$_B U$具有有限的平坦维数，则左$T$-模$biggl(begin{matrix} M_1 M_2end{matrix}biggr)_{varphi^M}$是Ding投影当且仅当$M_1$和$M_2/{rm im}(varphi^M)$是Ding投影且态射$varphi^M$是单态。(2)如果$T$是一个右相干环，$_{B}U$具有有限的平面维数，$U_{a}$是有限的投影维数或$FP$-内射维数，则一个右$T$-模$(W_{1}， W_{2}) $ {varphi_{W}}$是Ding内射当且仅当$W_{1}$和$ker( widdetilde {varphi_{W}})$是Ding内射且态射$ widdetilde {varphi_{W}}$是上射。因此，我们描述了$T$-模的Ding投影维和Ding内射维。
{"title":"Ding modules and dimensions over formal triangular matrix rings","authors":"L. Mao","doi":"10.4171/rsmup/100","DOIUrl":"https://doi.org/10.4171/rsmup/100","url":null,"abstract":"Let $T=biggl(begin{matrix} A&0 U&B end{matrix}biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $biggl(begin{matrix} M_1 M_2end{matrix}biggr)_{varphi^M}$ is Ding projective if and only if $M_1$ and $M_2/{rm im}(varphi^M)$ are Ding projective and the morphism $varphi^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $ker(widetilde{varphi_{W}})$ are Ding injective and the morphism $widetilde{varphi_{W}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"183 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74652160","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}
Baer's Criterion of injectivity implies that injectivity of a module is a factorization property w.r.t. a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring $R$ and on additional set-theoretic hypotheses. For $R$ commutative noetherian of Krull dimension $0 |R|$, then the category of all projective modules is accessible.
{"title":"Test sets for factorization properties of modules","authors":"Jan vSaroch, J. Trlifaj","doi":"10.4171/rsmup/66","DOIUrl":"https://doi.org/10.4171/rsmup/66","url":null,"abstract":"Baer's Criterion of injectivity implies that injectivity of a module is a factorization property w.r.t. a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring $R$ and on additional set-theoretic hypotheses. For $R$ commutative noetherian of Krull dimension $0 |R|$, then the category of all projective modules is accessible.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"148 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73738869","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}
Recently, G. Mason has produced a counterexample of order 128 to a conjecture in conformal field theory and tensor category theory in [Ma]. Here we easily produce an infinite family of counterexamples, the smallest of which has order 72.
{"title":"An infinite family of counterexamples to a conjecture on positivity","authors":"J. Mart'inez","doi":"10.4171/RSMUP/79","DOIUrl":"https://doi.org/10.4171/RSMUP/79","url":null,"abstract":"Recently, G. Mason has produced a counterexample of order 128 to a conjecture in conformal field theory and tensor category theory in [Ma]. Here we easily produce an infinite family of counterexamples, the smallest of which has order 72.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88700128","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}
Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of confluent hypergeometric series with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove that the coefficients of the asymptotic expansion of a confluent hypergeometric series with rational parameters are in H. Finally, we prove a similar result for G-functions.
{"title":"On Siegel’s problem for $E$-functions","authors":"S. Fischler, T. Rivoal","doi":"10.4171/rsmup/107","DOIUrl":"https://doi.org/10.4171/rsmup/107","url":null,"abstract":"Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of confluent hypergeometric series with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively \"small\" ring H generated by algebraic numbers, $1/pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove that the coefficients of the asymptotic expansion of a confluent hypergeometric series with rational parameters are in H. Finally, we prove a similar result for G-functions.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83446960","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}
Etant donne un anneau de valuation $V$, de corps residuel $F$ et de groupe des valeurs $Gamma$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $Gamma$. Lorsque $V$ contient un corps $k$, ce resultat est applique a la construction d'un anneau de valuation contenant $V$ et une extension donnee $k'$ de $k$, de groupe $Gamma$ et de corps residuel engendre par $k'$ et $F$. Cela s'avere possible, notamment, lorsque $k'$ ou $F$ est separable sur $k$.
{"title":"Une construction d’extensions faiblement non ramifiées d’un anneau de valuation","authors":"Laurent Moret-Bailly","doi":"10.4171/rsmup/94","DOIUrl":"https://doi.org/10.4171/rsmup/94","url":null,"abstract":"Etant donne un anneau de valuation $V$, de corps residuel $F$ et de groupe des valeurs $Gamma$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $Gamma$. Lorsque $V$ contient un corps $k$, ce resultat est applique a la construction d'un anneau de valuation contenant $V$ et une extension donnee $k'$ de $k$, de groupe $Gamma$ et de corps residuel engendre par $k'$ et $F$. Cela s'avere possible, notamment, lorsque $k'$ ou $F$ est separable sur $k$.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75682671","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 prove in ZFC that an abelian group $C$ is cotorsion if and only if $operatorname{Ext}(F,C) = 0$ for every $aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $barlambda$-Black Box for $aleph_k$-free constructions in ZFC.
{"title":"$aleph_k$-free cogenerators","authors":"M. Dugas, D. Herden, S. Shelah","doi":"10.4171/rsmup/58","DOIUrl":"https://doi.org/10.4171/rsmup/58","url":null,"abstract":"We prove in ZFC that an abelian group $C$ is cotorsion if and only if $operatorname{Ext}(F,C) = 0$ for every $aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $barlambda$-Black Box for $aleph_k$-free constructions in ZFC.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91543332","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}
A criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves is presented. A new example of a plane curve with non-collinear Galois points as an application is described. Furthermore, a new characterization of the Fermat curve in terms of non-collinear Galois points is presented.
{"title":"Algebraic curves admitting non-collinear Galois points","authors":"Satoru Fukasawa","doi":"10.4171/rsmup/114","DOIUrl":"https://doi.org/10.4171/rsmup/114","url":null,"abstract":"A criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves is presented. A new example of a plane curve with non-collinear Galois points as an application is described. Furthermore, a new characterization of the Fermat curve in terms of non-collinear Galois points is presented.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87401528","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 sextic del Pezzo surface fibrations via root stacks.
我们通过根堆研究了del Pezzo的表面振动。
{"title":"Fibrations in sextic del Pezzo surfaces with mild singularities","authors":"A. Kresch, Y. Tschinkel","doi":"10.4171/rsmup/109","DOIUrl":"https://doi.org/10.4171/rsmup/109","url":null,"abstract":"We study sextic del Pezzo surface fibrations via root stacks.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75501150","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：