Riemann–Roch-type theorem

In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al.

Formulation due to Baum, Fulton and MacPherson

Let $G_{*}$ and $A_{*}$ be functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that

$G_{*}(X)$ is the Grothendieck group of coherent sheaves on X,
$A_{*}(X)$ is the rational Chow group of X,
for each proper morphism f, $G_{*}(f),A_{*}(f)$ are the direct images (or push-forwards) along f.

Also, if $f:X\to Y$ is a (global) local complete intersection morphism; i.e., it factors as a closed regular embedding $X\hookrightarrow P$ into a smooth scheme P followed by a smooth morphism $P\to Y$ , then let

T_{f}=[T_{P/Y}|_{X}]-[N_{X/P}]

be the class in the Grothendieck group of vector bundles on X; it is independent of the factorization and is called the virtual tangent bundle of f.

Then the Riemann–Roch theorem then amounts to the construction of a unique natural transformation:^[1]

\tau :G_{*}\to A_{*}

between the two functors such that for each scheme X in C, the homomorphism $\tau _{X}:G(X)\to A(X)$ satisfies: for a local complete intersection morphism $f:X\to Y$ , when there are closed embeddings $X\subset M,Y\subset P$ into smooth schemes,

\tau _{X}f^{*}=\operatorname {td} (T_{f})\cdot f^{*}\tau _{Y}

where $\operatorname {td}$ refers to the Todd class.

Moreover, it has the properties:

$\tau _{X}(\beta \otimes \alpha )=\operatorname {ch} (\beta )\tau (\alpha )$ for each $\alpha \in G_{*}(X)$ and the Chern class $\operatorname {ch} (\beta )$ (or the action of it) of the $\beta$ in the Grothendieck group of vector bundles on X.
it X is a closed subscheme of a smooth scheme M, then the theorem is (roughly) the restriction of the theorem in the smooth case and can be written down in terms of a localized Chern class.

The equivariant Riemann–Roch theorem

This section needs expansion. You can help by adding to it. (November 2019)

Over the complex numbers, the theorem is (or can be interpreted as) a special case of the equivariant index theorem.

The Riemann–Roch theorem for Deligne–Mumford stacks

Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case (Kawasaki's Riemann–Roch).

The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for quotient stacks by finite groups.

One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class.

Notes

^ Fulton 1998, Theorem 18.3.

References

Edidin, Dan (2012-05-21). "Riemann-Roch for Deligne-Mumford stacks". arXiv:1205.4742 [math.AG].
Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
Toen, B. (1998-03-17). "Riemann-Roch Theorems for Deligne-Mumford Stacks". arXiv:math/9803076.
Toen, Bertrand (1999-08-18). "K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems". arXiv:math/9908097.
Lowrey, Parker; Schürg, Timo (2012-08-30). "Grothendieck-Riemann-Roch for derived schemes". arXiv:1208.6325 [math.AG].
Vakil, Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry

External links

https://mathoverflow.net/questions/25218/why-is-riemann-roch-for-stacks-so-hard

This algebraic geometry–related article is a stub. You can help Wikipedia by expanding it.

Read other articles:

Kihachiro Uemura

Kihachiro Uemura di tahun 2020 Kihachiro Uemura (植村喜八郎code: ja is deprecated , Uemura Kihachirō, lahir 5 September 1964) adalah seorang aktor dan pengisi suara asal Jepang. Dia mulai berkarier di dunia artis sejak tahun 1986. Kihachiro Uemura sekarang berada di bawah manajemen artis Gekidan Seinenza. Biografi Pada tahun 1980-an, setelah lulus dari kursus latihan Gekidan Seinenza, Kihachiro Uemura menjadi aktor dan menjadi terkenal akan peran-peran pahlawannya di televisi dan film....

كولفاكس (إلينوي)

لمعانٍ أخرى، طالع كولفاكس (توضيح). كولفاكس الإحداثيات 40°33′59″N 88°36′55″W / 40.5664°N 88.6153°W / 40.5664; -88.6153 [1] تاريخ التأسيس 1880 تقسيم إداري البلد الولايات المتحدة[2] التقسيم الأعلى مقاطعة مكلين خصائص جغرافية المساحة 0.55 ميل مربع ار�...

Asensio rekta

Sistem Koordinat Ekuator Asensio rekta atau panjatan tegak (Inggris: Right ascensioncode: en is deprecated (RA), dengan simbol α) adalah istilah astronomi yang dikaitkan dengan sistem koordinat ekuator. RA merupakan salah satu dari dua koordinat bola langit pada sistem koordinat khatulistiwa. Koordinat lainnya adalah deklinasi. RA bisa dibandingkan dengan garis bujur, diukur dari titik nolnya yang berada di titik Aries atau titik vernal ekuinoks. RA diukur dalam jam, menit, dan detik; dengan...

Love is Pink

Love is PinkGenre Drama Roman Ditulis oleh Endik Koeswoyo Novy Pritania Skenario Endik Koeswoyo Novy Pritania SutradaraSony GaokasakPemeran Randy Pangalila Anggika Bölsterli Edwin Syarief Rebecca Tamara Rendi Jhon Pratama Santana Sartana Erlin Sarintan Oka Sugawa Penggubah lagu tema Kibay Virzha Lagu pembukaAku dan Dia — KibayLagu penutupNyaman — VirzhaPenata musikAndhika TriyadiNegara asalIndonesiaBahasa asliBahasa IndonesiaJmlh. musim1Jmlh. episode10 (daftar episode)ProduksiProd...

VIe millénaire av. J.-C.

../.. | VIIe millénaire av. J.-C. | VIe millénaire av. J.-C. | Ve millénaire av. J.-C. | ../.. Liste des millénaires | Liste des siècles Le VIe millénaire av. J.-C. a débuté le 1er janvier 6000 av. J.-C. et s’est achevé le 31 décembre 5001 av. J.-C. dans le calendrier julien proleptique. Évènements 6050-3050 av. J.-C. : altithermal, ou optimum climatique holocène. Climat aride en Amérique du Nord mais plus humide en Europe avec des temp...

Kemono Friends

Kemono FriendsVisual utama Anime musim 1けものフレンズ(Kemono Furenzu)GenrePetualangan[1] PermainanPengembangNexonPenerbitNexonPerancangMine YoshizakiGenreRPGPlatformAndroid, iOSRilisAndroid16 Maret 2015iOS19 Maret 2015 MangaKemono Friends: Welcome to Japari Park!PengarangFuraiPenerbitKadokawa ShotenPenerbit bahasa InggrisNA Yen PressMajalahMonthly Shōnen AceDemografiShōnenTerbitJuli 2015 – Maret 2017Volume2 AnimeSutradaraTatsukiProduserKōji AkioJun FukudaNobuyuki HosoyaY�...

Ernest Istook

American politician (born 1950) Ernie IstookMember of the U.S. House of Representativesfrom Oklahoma's 5th districtIn officeJanuary 3, 1993 – January 3, 2007Preceded byMickey EdwardsSucceeded byMary Fallin Personal detailsBorn (1950-02-11) February 11, 1950 (age 74)Fort Worth, Texas, U.S.Political partyRepublicanSpouseJudy IstookChildren5EducationBaylor University (BA)Oklahoma City University (JD) Ernest Istook's voice Ernest Istook discusses FY2005 transportation,...

Метод золотого сечения

Метод золотого сечения — метод поиска экстремума действительной функции одной переменной на заданном отрезке. В основе метода лежит принцип деления отрезка в пропорциях золотого сечения. Является одним из простейших вычислительных методов решения задач оптимизаци...

Corbin Bernsen

American actor and director (born 1954) Corbin BernsenBernsen in 2016BornCorbin Dean Bernsen (1954-09-07) September 7, 1954 (age 69)Los Angeles, California, U.S.EducationUniversity of California, Los Angeles (BFA, MFA)OccupationsActorfilm directorYears active1967–presentKnown forLA LawLA Law: The MovieMajor LeaguePsychPsych: The MovieSpouses Brenda Cooper (m. 1983⁠–⁠1987) Amanda Pays (m. 1988) C...

Northam Traincare Facility

Traction maintenance depot in Southampton, England Northam Traincare FacilityA South West Trains Class 444 inside the facility in 2010LocationLocationNortham, Southampton, Hampshire, EnglandCoordinates50°54′43″N 1°23′29″W / 50.9119°N 1.3913°W / 50.9119; -1.3913OS gridSU428127CharacteristicsOwnerSiemens MobilityDepot codeNT[1]TypeEMUHistoryOpened2002[2] Northam Traincare Facility, also known as Northam Carriage Servicing Depot, is an electric...

コルク

この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。（このテンプレートの使い方）出典検索?: コルク – ニュース · 書籍 · スカラー · CiNii · J-STAGE · NDL · dlib.jp · ジャパンサーチ · TWL（2017年4月）コルクを打ち抜いて作った瓶の栓コルク（木栓、�...

Adriano Sofri

Adriano Sofri nel 2014. Adriano Sofri (Trieste, 1º agosto 1942) è uno scrittore, opinionista e attivista italiano, ex leader di Lotta Continua, condannato a ventidue anni di carcere – dopo un lungo iter giudiziario – quale mandante, assieme a Giorgio Pietrostefani, dell'omicidio del commissario di polizia Luigi Calabresi, avvenuto nel 1972. Come esecutori materiali furono condannati invece i militanti di Lotta Continua Leonardo Marino e Ovidio Bompressi. Arrestato brevemente nel 1988, e...

Converge

Questa voce o sezione sugli argomenti gruppi musicali statunitensi e punk rock non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti del progetto di riferimento. Questa voce sull'argomento gruppi musicali statunitensi è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto ...

Second Donev Government

Government of Bulgaria, Feb 2023–Jun 2023 Donev Government101st Cabinet of Bulgaria2023Portrait of the Prime Minister Galab DonevDate formed2 February 2023 (2023-02-02)Date dissolved6 June 2023 (2023-06-06)People and organisationsPresidentRumen RadevPrime MinisterGalab DonevDeputy Prime MinistersLazar Lazarov, Ind.Hristo Alexiev, Ind.Ivan Demerdzhiev, Ind.Atanas Pekanov, Ind.No. of ministers20 (17 men and 3 women)Status in legislatureCaretaker GovernmentHistory...

Jeff Kostoff

American swimmer (born 1965) Jeff KostoffPersonal informationFull nameJeffrey James KostoffNicknameJeffNational teamUnited StatesBorn (1965-08-19) August 19, 1965 (age 58)Upland, CaliforniaHeight6 ft 1 in (1.85 m)Weight170 lb (77 kg)SportSportSwimmingStrokesFreestyle, individual medleyCollege teamStanford University Medal record Men's swimming Representing the United States Pan American Games 1983 Caracas 1500 m freestyle 1983 Caracas 400 m medley Jeffr...

Antigonos II Gonatas

AntigonosKoin Antigonos GonatasRaja MakedoniaBerkuasa277 – 274 SM(pertama kali)PendahuluSosthenesPenerusPyrrhos dari EpirosBerkuasa272 – 239 SM(kedua kali)PendahuluPyrrhos dari EpirosPenerusDemetrius AetolicusInformasi pribadiKelahiranskt. 319 SMKematian239 SMAyahDemetrius PoliorcetesIbuPhilaPasanganPhilaAnakDemetrius Aetolicus Antigonus II Gonatas (Yunani: Ἀντίγονος B΄ Γονατᾶς) (skt. 319–239 SM) merupakan seorang penguasa yang kuat yang memperkokoh posisi Dinasti Ant...

Menahem Golan

Menahem GolanMenahem Golan, difoto pada 2007.Lahir(1929-05-31)31 Mei 1929Tiberias, Mandat Palestina (sekarang Israel)Meninggal8 Agustus 2014(2014-08-08) (umur 85)Jaffa, Tel Aviv, IsraelKebangsaanIsraelNama lainJoseph GoldmanPekerjaanSutradara, produserDikenal atasPendiri Golan-Globus Menahem Golan (Ibrani: מנחם גולן; 31 Mei 1929 – 8 Agustus 2014) adalkah seorang sutradara dan produser Israel. Ia memproduksi film-film untuk bintang-bintang seperti Sean Connery, Sylvest...

Mark Grew

English footballer (born 1958) Mark Grew Grew in 2010Personal informationFull name Mark Stuart Grew[1]Date of birth (1958-02-15) 15 February 1958 (age 66)Place of birth Bilston, EnglandHeight 5 ft 10 in (1.78 m)[2]Position(s) GoalkeeperYouth career West Bromwich AlbionSenior career*Years Team Apps (Gls)1976–1983 West Bromwich Albion 33 (0)1978 → Wigan Athletic (loan) 4 (0)1979 → Notts County (loan) 0 (0)1983–1984 Leicester City 5 (0)1983 → Oldham ...

Festivali i Këngës 62

62nd edition of Festivali i Këngës Festivali i Këngës 2023DatesSemi-final 119 December 2023Semi-final 220 December 2023Semi-final 321 December 2023 (Nostalgia night)Final22 December 2023HostVenuePalace of CongressesTirana, AlbaniaPresenter(s)Adriana MatoshiKledi KadiuXhuliano DuleKrisa ÇaushiMusical directorElton DedaArtistic directorBojken LakoDirected byPali KukeExecutive supervisorAlfred PezaHost broadcasterRadio Televizioni Shqiptar (RTSH)ParticipantsNumber of entries31Number of fin...

MPS Group Championships 2009

MPS Group Championships 2009Sport Tennis Data6 aprile – 12 aprile Edizione30a SuperficieTerra verde CampioniSingolare Caroline Wozniacki Doppio Chuang Chia-jung / Sania Mirza 2010 Il MPS Group Championships 2009 è stato un torneo di tennis giocato sulla terra verde. È stata la 30ª edizione del MPS Group Championships, la 1a con la nuova denominazione, che fa parte della categoria International nell'ambito del WTA Tour 2009. Si è giocato nel Sawgrass Country Club di Ponte Vedra Beach in ...

Prasasti Kuburajo

/profillengkap.com/index.php/article/North Palisade

/profillengkap.com/index.php/article/Emilio Isierte

/profillengkap.com/index.php/article/Template talk:Hashemites

/profillengkap.com/index.php/article/Vattenfall Nederland

Grey Suit

Jenny Wilhelms

دوري إستونيا الممتاز 2008

العلاقات الأردنية اليونانية

Hitumessing, Leihitu, Maluku Tengah

Pachuca

Captain Arturo Prat Base