Selecta Mathematica最新文献_第2页

Rings of differentiable semialgebraic functions 可微半代数函数环

Selecta Mathematica Pub Date : 2024-07-19 DOI: 10.1007/s00029-024-00965-z

E. Baro, José F. Fernando, J. M. Gamboa

{"title":"Rings of differentiable semialgebraic functions","authors":"E. Baro, José F. Fernando, J. M. Gamboa","doi":"10.1007/s00029-024-00965-z","DOIUrl":"https://doi.org/10.1007/s00029-024-00965-z","url":null,"abstract":"In this work we analyze the main properties of the Zariski and maximal spectra of the ring ({{mathcal {S}}}^r(M)) of differentiable semialgebraic functions of class ({{mathcal {C}}}^r) on a semialgebraic set (Msubset {{mathbb {R}}}^m). Denote ({{mathcal {S}}}^0(M)) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in ({text {Cl}}(M)). This ring is the real closure of ({{mathcal {S}}}^r(M)). If M is locally compact, the ring ({{mathcal {S}}}^r(M)) enjoys a Łojasiewicz’s Nullstellensatz, which becomes a crucial tool. Despite ({{mathcal {S}}}^r(M)) is not real closed for (rge 1), the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring ({{mathcal {S}}}^0(M)). In addition, the quotients of ({{mathcal {S}}}^r(M)) by its prime ideals have real closed fields of fractions, so the ring ({{mathcal {S}}}^r(M)) is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of ({{mathcal {S}}}^r(M)) and ({{mathcal {S}}}^0(M)) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring ({{mathcal {S}}}^r(M)) is a Gelfand ring and its Krull dimension is equal to (dim (M)). We also show similar properties for the ring ({{mathcal {S}}}^{r*}(M)) of differentiable bounded semialgebraic functions. In addition, we confront the ring ({mathcal S}^{infty }(M)) of differentiable semialgebraic functions of class ({{mathcal {C}}}^{infty }) with the ring ({{mathcal {N}}}(M)) of Nash functions on M.\u0000","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744435","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}

引用次数: 0

Mutations of noncommutative crepant resolutions in geometric invariant theory 几何不变理论中的非交换褶皱决议的突变

Selecta Mathematica Pub Date : 2024-07-16 DOI: 10.1007/s00029-024-00957-z

Wahei Hara, Yuki Hirano

{"title":"Mutations of noncommutative crepant resolutions in geometric invariant theory","authors":"Wahei Hara, Yuki Hirano","doi":"10.1007/s00029-024-00957-z","DOIUrl":"https://doi.org/10.1007/s00029-024-00957-z","url":null,"abstract":"Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack (mathfrak {X}=[X^text {ss}(ell )/G]) with respect to a generic (ell ) is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category ({{textrm{D}}^{textrm{b}}}({text {coh}}mathfrak {X})) is equivalent to certain subcategories of ({{textrm{D}}^{textrm{b}}}({text {coh}}[X/G])), which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group (pi _1(mathbb {P}^1,backslash {0,1,infty })) on the derived category of a Calabi-Yau complete intersection in a weighted projective space.","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"7 Suppl 8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718513","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}

引用次数: 0

The Manin–Peyre conjecture for smooth spherical Fano threefolds 光滑球面法诺三围的马宁-佩雷猜想

Selecta Mathematica Pub Date : 2024-07-12 DOI: 10.1007/s00029-024-00952-4

Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi

引用次数: 0

Zero-cycles in families of rationally connected varieties 有理连接品种族中的零循环

Selecta Mathematica Pub Date : 2024-07-09 DOI: 10.1007/s00029-024-00963-1

Morten Lüders

引用次数: 0

Coxeter quiver representations in fusion categories and Gabriel’s theorem 融合范畴中的考斯特震颤表示和加布里埃尔定理

Selecta Mathematica Pub Date : 2024-07-06 DOI: 10.1007/s00029-024-00947-1

Edmund Heng