Upgrade to Pro — share decks privately, control downloads, hide ads and more …

導来代数幾何入門

 導来代数幾何入門

第三回関東すうがく徒のつどいでの発表スライドです。

Naoya Umezaki

March 29, 2019
Tweet

More Decks by Naoya Umezaki

Other Decks in Science

Transcript

  1. ಋདྷ୅਺زԿೖ໳
    ക࡚௚໵@unaoya
    2019/3/29 ୈ 3 ճؔ౦͢͏͕͘ెͷͭͲ͍
    1

    View full-size slide

  2. ͸͡Ίʹ
    ͜ͷߨԋͷ಺༰͸ɺ͍͔ͭ͘ͷղઆͱ࿦จͷΠϯτϩͷඇৗʹૈ
    ͍Ґ૬ͰͷషΓ߹ΘͤͰ͢ɻࢀߟʹͨ͠΋ͷ͸࠷ޙʹࢀߟจݙͱ
    ͯ͠Ұཡʹͯ͋͠Γ·͢ɻ
    ߨԋऀ͸ূ໌΍ਖ਼֬ͳఆٛΛϑΥϩʔ͍ͯ͠·ͤΜɻ
    ʢ࣍ճ࡞ʹظ଴ʣ
    εϥΠυ͸ެ։͍ͯ͠·͢ɻ
    ʢtwitter @unaoyaʣ
    2

    View full-size slide

  3. ໨࣍
    ୅਺زԿͱϗϞτϐʔ
    derived stack
    QC(X) ͱੵ෼ม׵
    දݱ࿦ͱ TFT ΁ͷԠ༻
    3

    View full-size slide

  4. ࠓ೔ͷ಺༰
    ୅਺زԿʹ͓͚Δۭؒʹରͯ͠ɺద੾ͳۃݶ΍༨ۃݶͷૢ࡞Λ༩
    ͑Δ࿮૊Έ͕ཉ͍͠ɻ௨ৗͷεΩʔϜ͸ Algk → Set ͱ͍͏ؔख
    Ͱద੾ͳ৚݅Λຬͨ͢΋ͷͰ͋Δɻ͜ΕΛ֦ுͯ͠ɺಋདྷεΩʔ
    ϜΛؔख dAlgk → sSet Ͱ͋ͬͯద੾ͳ৚݅Λ࣋ͭ΋ͷͱͯ͠ఆ
    ΊΔɻ
    ͜ΕΒ͸ϗϞτϐʔΛऔΓೖΕͨߏ଄Λ࣋ͭݍͰ͋ΓɺϗϞτ
    ϐʔΛߟྀͨ͠ద੾ͳۃݶૢ࡞Λఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ
    Ԡ༻্ͷಈػͱͯ͠ɺྫ͑͹ up to equivalence Ͱ෺ࣄΛ෼ྨ͢Δ
    Α͏ͳ໰୊Λߟ͍͑ͨɻྫ͑͹ಋདྷݍͷର৅Ͱద੾ͳ৚݅Λ࣋ͭ
    ΋ͷΛ෼ྨ͢Δɻ
    4

    View full-size slide

  5. ୅਺زԿͱϗϞτϐʔ
    5

    View full-size slide

  6. ͜ͷઅͷ໨ඪ
    ୅਺زԿʹ͓͍ͯϗϞτϐʔΛߟ͑ͨ͘ͳΔঢ়گɻ
    1. ؔख͕׬શͰͳ͍ͱ͖ɺಋདྷؔखΛߟ͑ΔɻෳମͷϗϞτ
    ϐʔΛ༻͍ͯఆٛ͢Δɻ
    2. εΩʔϜͷަ఺Λߟ͑Δͱɺॏෳ౓͕ݱΕΔɻ
    ʮॏෳ౓෇͖
    ͷۭؒʯΛ௚઀ѻ͍͍ͨɻ
    3. up to equivalence ͳϞδϡϥΠཧ࿦Λߟ͑Δɻྫ͑͹ಋདྷݍ
    ͷର৅Λ෼ྨ͢ΔͳͲɻ
    4. ϧʔϓۭؒΛߟ͑ΔɻS1 ͔Β X ΁ͷࣹΛ෼ྨ͢ΔۭؒΛߏ
    ੒͍ͨ͠ɻ
    6

    View full-size slide

  7. homotopy limit and homotopy colimit
    ௨ৗͷۃݶ΍༨ۃݶ͸ϗϞτϐʔͱ૬ੑ͕ѱ͍ɻ
    ∗ ←



    − ∗




    ∗ ←



    − ∗ ⨿ ∗
    S1 ←



    − [0, 1]




    [0, 1] ←



    − ∗ ⨿ ∗
    ∗ −



    → ∗




    ∗ −



    → S1
    Z −



    → ∗




    R −



    → S1
    ͜ΕΛࠀ෰͢ΔͨΊʹϗϞτϐʔۃݶͱϗϞτϐʔ༨ۃݶͱ͍͏
    ֓೦Λఆٛ͢Δɻ
    7

    View full-size slide

  8. ୅਺زԿʹ͓͚ΔϑΝΠόʔੵ
    k ΛՄ׵؀ͱ͠ Algk
    ΛՄ׵ͳ k ୅਺ͷݍͱ͢Δɻ
    k ୅਺ A ʹରۭͯؒ͠ SpecA → Speck Λߏ੒͢Δɻ͜Ε͸
    B → Homk(A, B) ʹΑΓؔख Algk → Sets Λ༩͑Δɻ͜ΕͷషΓ
    ߹Θ͕ͤҰൠͷεΩʔϜ X Ͱ͋Δɻ͜Ε͸૬ରతͳٞ࿦ X → S
    Λѻ͏ͨΊͷ࿮૊ΈΛ༩͑Δɻ
    Xp −



    → X




    SpecFp −



    → SpecZ
    X0 −



    → X




    Speck −



    → Speck[t]/t2
    εΩʔϜͷϑΝΠόʔੵʹରԠ͢Δૢ࡞͕୅਺ͷςϯιϧੵɻ
    Spec(A1 ⊗B A2) ≃ SpecA1 ×SpecB SpecA2
    8

    View full-size slide

  9. ަ఺ཧ࿦
    Ճ܈ͷෳମͷϗϞτϐʔͱͦͷۃݶɺ༨ۃݶ΋ಉ༷ͷ໰୊ɻ
    k ⊗k[x]
    k ≃ k ͕ͩ k ⊗k[x]
    (k[x][−1] ⊕ k[x]) ≃ k[−1] ⊕ k ͱͳΔɻ
    0 −



    → k[x] 1→x




    → k[x] −



    → 0




    0 −



    → 0 −



    → k −



    → 0
    Tor1
    k[x]
    (k, k) = k Ͱ͋Γɺk ⊗L
    k[x]
    k ≃ k[ϵ−1] = k ⊕ k[−1] ͱ up to
    homotopy Ͱఆ·Δɻ
    SpecB ⊗A C −



    → SpecB




    SpecC −



    → SpecA
    Ͱ͸ͳ͘ɺ௚઀ SpecA ⊗L
    B
    C ΛزԿతͳର৅ͱ͍ͨ͠ɻͨͩ͠
    A ⊗L
    B
    C ͸ up to homotopy Ͱ͔ܾ͠·Βͳ͍͜ͱʹ஫ҙɻ
    9

    View full-size slide

  10. มܗཧ࿦
    A ͷඍ෼Ճ܈͸ Ω1
    A/k
    = I/I1, I = ker(A ⊗k A → A) Ͱ͋ΓɺX ͷ
    ઀ۭؒ͸ Homk(Speck[t]/t2, X) Ͱ͋ͬͨɻϗϞτϐʔۃݶΛ༻͍
    ͯɺX ʹର͠ɺ
    LX −



    → X





    X ∆




    → X ⊗ X
    ͱఆٛ͢Δͱ LX = SpecSymOX
    (LX [1]) ͱͳΔɻ
    ͜ͷ LX
    ͸ cotangent complex ͱݺ͹ΕΔ΋ͷͰɺX ͷมܗΛίϯ
    τϩʔϧ͢Δෳମɻ
    A ͕ smooth k-algebra ͳΒ LA ≃ Ω1
    A/k
    Ͱ͋Γɺπ0(LA) = Ω1
    π0A
    ͱ
    ͳΔɻ
    10

    View full-size slide

  11. stack
    εΩʔϜ X ͸ू߹ʹ஋Λ࣋ͭ૚ Algk → Set ΛఆΊΔɻྫ͑͹ A
    ʹର͠Մٯݩશମͷू߹ A× ΛରԠͤ͞Δ΋ͷ͸
    Gm = Speck[x, x−1] Ͱදݱ͞ΕΔɻ
    ͜ͷͱ͖૚ F ͷషΓ߹Θͤ৚݅͸ɺS ͷඃ෴ U• → S ʹରͯ࣍͠
    ͕׬શͰ͋Δ͜ͱɻ
    F(S) F(U) F(U ×S U)
    ྫ͑͹ G-torsor શମΛ෼ྨ͢Δۭؒ BG Λߟ͑Δɻͭ·Γؔख
    BG(S) = {S্ͷG-torsor શମ } Λߟ͑ΔɻಉܕྨΛద੾ʹॲཧ͢
    ΔͨΊʹɺSet Ͱ͸ͳ͘ Grpdʢશͯͷࣹ͕ಉܕͰ͋Δݍͷͳ͢ݍʣ
    ʹ஋Λ࣋ͭ૚Λߟ͑Δɻ͜ͷΑ͏ͳ΋ͷΛ stack ͱ͍͏ɻ
    stack F ͷషΓ߹Θͤ৚݅͸ίαΠΫϧ৚݅Λߟ͑ͯ
    F(S) F(U) F(U ×S U) F(U ×S U ×S U)
    Set ͸ Grpd ʹ཭ࢄతͳ΋ͷͱͯ͠ຒΊࠐΊΔɻ
    11

    View full-size slide

  12. higher stack
    Grpd ͷ nerve ΛͱΔ͜ͱͰ sSetʢ୯ମతू߹ͷͳ͢ݍʣ͕ఆ·
    Δɻ୯ମతू߹͸ɺ͓͓Αͦ఺΍ઢ෼ɺࡾ֯ܗɺ࢛໘ମͳͲΛద
    ੾ʹషΓ߹Θͤͨ΋ͷͱͯ͠Πϝʔδ͓ͯ͘͠ɻ
    ͜Εʹ޿͛Δ͜ͱͰΑΓ޿͍ moduli ໰୊Λߟ͑Δ͜ͱ͕Ͱ͖Δɻ
    ಛʹ up to euivalence Ͱ෼ྨ͍ͨ͠৔߹͕͋Δɻྫ͑͹ S ্ͷద
    ੾ͳ৚݅Λຬͨ͢૚ͷෳମΛ෼ྨ͍ͨ͠৔߹ͳͲɻ
    ͜ͷΑ͏ͳ໰୊Λߟ͑ΔͨΊʹ higher stack Λ sSet ʹ஋Λ࣋ͭ૚
    ͱͯ͠ఆΊΔɻషΓ߹Θͤ৚݅͸ߴ࣍ͷίαΠΫϧ৚݅Λߟ͑ͳ
    ͚Ε͹͍͚ͳ͍ɻ
    F(S) F(U) F(U ×S U) F(U ×S U ×S U) · ·
    12

    View full-size slide

  13. ϧʔϓۭؒ
    ۭؒ X ʹରͯ͠ S1 ͔Β X ΁ͷ࿈ଓࣸ૾શମΛద੾ʹҐ૬ۭؒͱ
    ࢥͬͨ΋ͷ͕ϧʔϓۭؒ LX Ͱ͋Δɻ
    ϗϞτϐʔ࿦ʹ͓͚Δϧʔϓۭؒ Map(S1, X) = Map(BZ, X)
    S1 ≃ BZ ≃ ∗ ⨿h
    ∗⨿h∗
    ∗ ͱͰ͖Δɻ
    ୅਺زԿʹ͓͍ͯϧʔϓۭؒΛ࡞ΔɻBZ ͸ελοΫͱͯ͠͸ఆ
    ٛͰ͖Δ͕ mapping stack ͸ࣗ໌ͳ΋ͷʹͳͬͯ͠·͏ɻ
    derived mapping stack Λߟ͑Δɻͭ·Γ T → Map(T × M, X) Ͱ
    ͸ͳ͘ T → Map(T ×h M, X) Λߟ͑Δɻ
    ϧʔϓۭؒ͸ X ×h
    X×X
    X ͱͯ͠ఆΊΔ͜ͱ͕Ͱ͖Δɻ
    13

    View full-size slide

  14. ϧʔϓۭؒͱ Chern ࢦඪ
    LBG = Map(S1, BG) = G/G ͱͳΔɻ͜Ε͸ S1 ্ͷ G-torsor Λ
    ߟ͑ΔͱɺషΓ߹Θ͕ͤ e ΛͲ͜ʹ͸Γ߹ΘͤΔ͔Ͱܾ·Δ͜ͱɺ
    G-torsor ͷಉܕ͕ G ಉมͰ͋Δ͜ͱͱ torsor ͷ࡞༻Λߟ͑Δͱɺ
    ಉܕΛ༩͑Δͷ͕ e → h ͱͨ͠ͱ͖ɺhg′ = gh ͱͳΔɻ
    V /X ͱ γ : S1 → X ʹର͠ɺ
    γ∗V −



    → V




    S1 γ




    → X
    ͷϞϊυϩϛʔͷ trace ΛରԠͤ͞Δ͜ͱͰɺCh(V ) ∈ O(LX)S1
    ͕ఆ·Δɻ͜Ε͕ Ch : K(X) → O(LX)S1
    = Hev
    DR
    (X) Λ༩͑Δɻ
    ಛʹ X = BG ͱ͢ΔͱɺLX = LBG = [G/G] Ͱ͋ΓɺV ͸ G ͷද
    ݱɺO(LX)S1
    = C(G/G) ͸ྨؔ਺ͰɺCh ͸දݱͷ trace Ͱ͋Δɻ
    14

    View full-size slide

  15. ͜ͷઅͷ·ͱΊ
    Algk Sets
    Grpd
    sSets
    scheme
    stack
    higherstack
    ϧʔϓۭؒΛਖ਼͘͠ఆٛ͢ΔͨΊʹ͸ɺۭؒͷϗϞτϐʔۃݶ͕
    ඞཁͰ͋Δɻ͜ͷ΋ͱͰ
    LX = Map(S1, X) = X ×h
    X×X
    X
    ͱఆٛͰ͖Δ͸ͣɻ
    15

    View full-size slide

  16. derived stack
    16

    View full-size slide

  17. ͜ͷઅͷ໨ඪ
    Algk Sets
    Grpd
    dAlgk sSets
    Sch
    St
    hSt
    dSt
    ӈଆΛ sset ʹ͢Δͱ moduli problem ΛΑΓ޿͍΋ͷΛѻ͏͜ͱ͕
    Ͱ͖Δɻྫ͑͹ಋདྷݍͷର৅Λ෼ྨ͢ΔɺಋདྷݍΛ෼ྨ͢ΔͳͲ
    up to equiv Ͱ෼ྨ͍ͨ͠৔߹ͳͲʹඞཁɻ
    ࠨଆΛ derive ͢Δͱʮਖ਼͘͠ʯۃݶΛͱΔ͜ͱ͕Ͱ͖ɺ
    ʮਖ਼͍͠ʯ
    ۭؒΛఆٛͰ͖Δɻ
    ྆ଆʹϗϞτϐʔ͕ఆ·͍ͬͯͯɺͦΕʹ͍ͭͯ੔߹తͳؔखɻ
    17

    View full-size slide

  18. derived topology
    (higher) stack ͸ Algk → sSet Ͱ૚ʹͳΔ΋ͷͩͬͨɻ͜ΕΛ֦ு
    ͯ͠ dAlgk → sSet Ͱ૚ʹͳΔ΋ͷͱͯ͠ derived stack Λఆٛ
    ͢Δɻ
    dAlgk
    ͸ྫ͑͹Մ׵ dg k ୅਺ͷݍɻdg ୅਺ͱ͸ ⊕i Ai
    Ͱ࣍਺ −1
    ͷࣹ d Ͱ͋ͬͯ d2 = 0 ͳΔ΋ͷɻ
    ૚Λఆٛ͢ΔͨΊʹ͸Ґ૬͕ඞཁɻ
    ఆٛ
    dAlgop
    k
    ʹ derived ´
    etale topology ΛҎԼͰఆΊΔɻ{A → Bi }i
    ͕
    ´
    etale covering ͱ͸ɺ{π0(A) → π0(B)} ͕௨ৗͷՄ׵؀ͱͯ͠
    ´
    etale covering Ͱ͋ΓɺπnA ⊗π0A π0Bi → πnBi
    ͕ಉܕɻ
    ͜Ε͸ infinitesimal lifting Ͱಛ௃෇͚Δ͜ͱ΋Ͱ͖Δɻ
    18

    View full-size slide

  19. derived stack
    ૚͸લ૚Ͱ͋ͬͯɺషΓ߹Θͤ৚݅Λຬͨ͢΋ͷɻ
    ఆٛ
    derived stack ͱ͸ؔख F : dAlgk → sSet Ͱ͋ͬͯɺweak
    equivalence Λอͪɺ࣍ͷ descent ৚݅Λຬͨ͢ɻ
    ೚ҙͷ etale h-hypercovering B• → A ʹରͯ͠
    F(A) → holimF(B•) ͕ Ho(sSet) ʹ͓͚Δಉ஋
    F(A) holim(F(B) F(B ⊗L
    A
    B) · · · )
    19

    View full-size slide

  20. derived affine stack
    RSpec : dAlgk → dStk
    ͕ఆ·Γ஧࣮ॆຬɻ
    ʢderived Yonedaʣ͜Ε
    ͸ A → (B → Map(A, B)) ͰఆΊΔɻ
    RSpecB ×h
    RSpecA
    RSpecC ≃ RSpec(B ⊗L
    A
    B)
    Map(F, G) : H → Map(F ×h H, G)
    ͳͲͱͯ͠ɺinternal hom ΍ holim ͕ఆ·Δɻ
    Ұൠͷ derived stack ͸ affine derived stack ͷ colimit Ͱ͔͚Δɻ
    ఆٛҬΛ੍ݶ͢Δ͜ͱͰ t0 : dSt → St ͕ఆ·Γɺafiine ΛషΓ߹
    ΘͤΔ͜ͱͰ i : St → dSt ͕ఆ·Δɻt0(RSpecA) = Specπ0(A)
    ͱͳΔɻ·ͨ it0X → X ͸ดຒΊࠐΈͰɺX ͱ t0(X) ͷ small
    etale site ͸Ұக͢Δɻ͔͠͠ i ͸ holim ΍ Map Λอͨͣɺderived
    tangent ΍ derived fibered product ͸ਅʹ derived ͳ৘ใΛؚΉɻ
    20

    View full-size slide

  21. derived mapping stack
    MapdStk
    (F, G) : H → HomdStk
    (F ×h H, G) ͱఆΊΔɻ͜Ε͕ dStk
    ʹ͓͚Δ internal hom Ͱ͋Δɻ
    Σ ͕Ґ૬ۭؒ΍୯ମతू߹ͷ࣌ɺinternal hom XΣ = Map(Σ, X)
    ͕ derived stack ͱͯ͠ఆ·Δɻ͜͜Ͱ Σ ͸ constant stackɻ
    ͜ͷͱ͖ i : Stk → dStk
    ͸ Map ͱަ׵͠ͳ͍ɺͭ·Γ
    iMap(F, G) ≃ RMap(iF, iG) ͱͳΔͱ͸ݶΒͳ͍ɻ
    ҰํͰ t0 : dStk → Stk
    ͱ͸ަ׵͢Δɻͭ·Γ
    t0
    RMap(F, G) ≃ Map(t0F, T0G) ͱͳΔɻಛʹ F, G ͕ St(k) ͔Β
    དྷΔͱ͖ɺt0
    RMap(iF, iG) ≃ Map(F, G) Ͱ͋Δɻ
    ʢt0iF ≃ F Ͱ͋
    Δ͜ͱʹ஫ҙʣͭ·Γ derived mapping stack ͸ mapping stack Λ
    ଠΒͤͨ΋ͷɻ
    mapping space ͕ͣΕΔྫͱͯ͠ɺ࣍ͷ loop stack ͷྫΛݟΔɻ
    21

    View full-size slide

  22. derived loop stack
    LX = XS1
    = Map(S1, X) ͸ internal hom ͰఆΊΔɻ
    LX ≃ X ×h
    X×X
    X Ͱ͋Δɻ
    X ͕Ґ૬ۭ͔ؒΒఆ·Δ constant stack ͷ৔߹ɺLX ͸௨ৗͷ
    loop space ͔Βఆ·Δ constant stack
    stack ͱͯ͠ͷ Map(BZ, X) ͸ X ͦͷ΋ͷʹͳΔ͕ɺderived
    stack ͱͯ͠ͷ Map(BZ, X) = X ×h
    X×hX
    X ͱͳΔɻ
    ∗ ×A1
    ∗ ≃ k[ϵ−1] ͷܭࢉ
    X = BG ͷͱ͖ LX = LBG = G/G
    X ͕ smooth scheme over char 0 field ͷ࣌͸ TX [−1]
    22

    View full-size slide

  23. cotangent complex
    scheme ͷ cotangent complex, Ω1 ͱͷؔ܎ɺมܗཧ࿦
    derived ring Di = RSpeck[ϵ] = RSpec(k ⊕ k[i]) ͱ͢Δɻdegree 0
    ͱ-i ʹ͋Δɻ
    ͜ͷͱ͖ Exti
    k
    (LX,x , k) ≃ RHom∗(Di , (X, x)) ͱͳΔɻExti ͸
    derived stack ʹ͓͍ͯ͸දݱՄೳɻ
    derived tangent stack Λ TX = Map(Speck[ϵ], X) ͱ͢Δɻ
    Y ͕ scheme ͳΒ TiY ≃ RSpecY
    (SymOY
    LY ) ͱͳΔɻ
    Vectn(X) ͸ඇࣗ໌ͳ derived extension Λ࣋ͭɻRVectn(X) ͱ
    ͢Δɻ
    23

    View full-size slide

  24. ͜ͷઅͷ·ͱΊ
    derived affine stack RSpecA ͱͦͷషΓ߹ΘͤͰ derived stack X
    ͕ಘΒΕΔɻ͜Ε͸ؔख X : dAlgk → sSet Ͱ͋ͬͯɺϗϞτϐʔ
    ΛอͪɺషΓ߹Θͤ৚݅Λຬͨ͢΋ͷɻ
    ͜ͷ࿮૊Έʹ͓͍ͯɺ
    1. ϧʔϓۭؒ LX
    2. cotangent complex LX
    3. ަ఺ੵ X×h
    X
    ͕ਖ਼͘͠ఆٛͰ͖Δɻ
    24

    View full-size slide

  25. QC(X) ͱੵ෼ม׵
    25

    View full-size slide

  26. ͜ͷઅͷ໨ඪ
    ·ͣ derived stack X ্ͷ૚ͷݍ QC(X) Λఆٛ͢Δɻ·ͨ͋Δछ
    ͷ༗ݶੑ৚݅Λຬͨ͢΋ͷͱͯ͠ perfect stack X Λఆٛ͢Δɻ
    ͜ͷԼͰੵ෼ม׵ͷݍ͕४࿈઀૚ͷݍͱಉ஋ʹͳΔ͜ͱΛΈΔɻ
    X → Y , X′ → Y ʹରͯ͠
    QC(X ×Y X′) ≃ FunY (QC(X), QC(X′))
    K → (F → (f∗(g∗F ⊗ K)))
    ੵ෼ม׵͸ X × Y ্ͷ֩ؔ਺ K(x, y) Λ༻͍ͯ Y ্ͷؔ਺ f (y)
    ͔Β x ্ͷؔ਺ΛఆΊΔɻ
    K(x, y) → (f (y) → (x →
    Y
    f (y)k(x, y)dy))
    26

    View full-size slide

  27. QC(X) ͷఆٛ
    ҰൠʹΞʔϕϧݍ A ͔Βͦͷෳମͷͳ͢ dg ݍ Ch(A) Λ࡞Γɺ͞
    Βʹ͔ͦ͜Β ∞ ݍ Ndg (Ch(A)) ΛఆΊΔ͜ͱ͕Ͱ͖Δɻ͜ΕΛ
    ModA
    ͱ͢ΔɻX = SpecA ͕ affine derived scheme ͷ࣌ɺ
    QC(X) = ModA
    ͱ͢Δɻ
    Ұൠͷ derived stack ʹ͍ͭͯ͸ɺX Λ affine derived stack ͷ
    colimit Ͱॻ͖ɺಉ͡ਤࣜͰ QC ͷ limit Λ ∞-cat of ∞-cats Ͱ
    ͱΔɻ
    X ͕ qc Ͱ affine diagonal ∆ : X → X × X Λ࣋ͯ͹ɺcosimplical
    diagram ͷ totalization Ͱ͔͚Δɻ
    27

    View full-size slide

  28. perfect stack
    ఆٛ
    1. A Λ derived commutative ring ͱ͢ΔɻA Ճ܈ M ͕ perfect ͱ
    ͸ɺModA
    ͷ smallest ∞ category Ͱ finite colimit ͱ retract Ͱ
    ͱͨ͡΋ͷʹଐ͢Δ͜ͱɻ
    2. derived stack X ʹର͠ɺPerf (X) ͸ QC(X) ͷ full
    ∞-subcategory Ͱ͋ͬͯɺ೚ҙͷ affine f : U → X ΁ͷ੍ݶ
    f ∗M ͕ perfect module Ͱ͋Δ΋ͷ͔ΒͳΔ΋ͷɻ
    3. derived stack X ͕ prefect stack ͱ͸ QC(X) ∼
    = IndPerf (X) Ͱ
    ͋Δ͜ͱɻ
    4. f : X → Y ͕ perfect ͱ͸ɺ೚ҙͷ affine U → Y ʹ͍ͭͯɺ
    X ×Y U ͕ perfect ͳ͜ͱɻ
    28

    View full-size slide

  29. ༗ݶੑ৚݅
    compact ͱ dualizable ͱ perfect ͷؔ܎ɻstable ∞-category C ͷ
    ର৅ M ʹ͍ͭͯ
    1. compact ͱ͸ HomC (M, −) ͕ coproduct ͱަ׵͢Δ͜ͱɻ
    2. dualizabule ͱ͸͋Δ M∨ ͱ u : 1 → M ⊗M∨, τ : M ⊗M∨ → 1
    ͕ଘࡏͯ͠ɺM → M ⊗ M∨ ⊗ M → M ͕ idM
    ͱͳΔ΋ͷɻ
    Vect/k ʹ͓͚Δ༗ݶ࣍ݩϕΫτϧۭؒɻV ∨ = Hom(V , k) ͱ͢
    Δɻ1 → V ⊗ V ∨ Λର֯ߦྻɺV ⊗ V ∨ → 1 Λ trace ͱ͢Δͱɺ্
    ͷ৚݅Λຬͨ͢ɻ
    ಛʹ X ͕ affine diagonal Λ࣋ͪ perfect ͳͱ͖ɺQC(X) ʹ͓͍ͯ
    dualizable ͱ compact ͱ perfect ͸ಉ஋ɻ
    29

    View full-size slide

  30. base change ͱ projection formula
    ໋୊ (BFN, proposition 3.10)
    f : X → Y Λ perfect ͱ͢Δɻ͜ͷ࣌
    1. f∗ : QC(X) → QC(Y ) ͸ small colimit ͱަ׵͠ɺprojection
    formula Λຬͨ͢
    2. ೚ҙͷ derived stack ͷࣹ g : Y ′ → Y ʹର͠ɺbase chage
    map g∗f∗ → f ′

    g′∗ ͸ಉ஋
    X′ g′




    → X
    f ′


    f


    Y ′ g




    → Y
    QC(X′) g′∗




    − QC(X)
    f ′



    f∗


    QC(Y ′) g∗




    − QC(Y )
    30

    View full-size slide

  31. ⊗ ͱ ×
    ໋୊ (BFN, Proposition 4.6)
    X1, X2 perfect, : QC(X1)c ⊗ QC(X2)c ∼
    = QC(X1 × X2)c
    1. ⊗ ͱ pullback ͸ dualizable ΛอͪɺX = X1 × X2
    ͕ perfect ͳ
    ͜ͱ͔Βɺ֎෦ੵ͕ compact Λอͭ
    2. QC(X1 × X2)c ͕֎෦ੵͰੜ੒
    3. projection formula
    ʹΑΓূ໌ɻ͞Βʹ
    1. Ind : st → PrL ͕ summetric monoidal
    2. IndQC(X)c ≃ QC(X)
    ͔Βɺ : QC(X1) ⊗ QC(X2) ≃ QC(X1 × X2) ͕੒ཱɻ
    31

    View full-size slide

  32. ⊗ ͱ ×
    ఆཧ (BFN ͷ Theorem 4.7)
    X1, X2, Y ͕ perfect ͷ࣌ɺ
    QC(X1 ×Y X2) = QC(X1) ⊗QC(Y )
    QC(X2)
    Y ͕Ұൠͷ࣌ͷূ໌ͷํ਑ʢͲ͜ʹ Y ͕ perfect Λ࢖͏ʁʣ
    X1 ×Y X2 → X1 × Y • × X2
    ͔Β
    QC(X1 ×Y X2) ← QC(X1 × Y •X2) Λ࡞Δɻ͢Ͱʹূ໌ͨ͜͠ͱ
    ͔Β QC(X1) ⊗ QC(Y )• ⊗ QC(X2) ͱͳΓɺ͜Εͷ geometric
    realization Ͱ QC(X1) ⊗QC(Y )
    QC(X2) ͕ܭࢉͰ͖Δɻ
    1. QC(X1 ×Y X2) = ModTgeom
    (QC(X1 × X2)) by Barr-Beck
    2. QC(X1) ⊗QC(Y )
    QC(X2) = ModTalg
    (QC(X1 × X2)) by
    Barr-Beck
    3. Talg = Tgeom by base change
    32

    View full-size slide

  33. self-duality
    ܥ (BFN, Corollary 4.8)
    π : X → Y map of perfect stacks ͱ͢ΔɻQC(X) ͸ self dual
    QC(Y )-mod Ͱ͋Δɻͭ·Γ
    FunQC(Y )
    (QC(X), QC(X′)) ≃ QC(X) ⊗QC(Y )
    QC(X′) ͱͳΔɻ
    33

    View full-size slide

  34. ੵ෼ม׵
    ఆཧ (BFN ͷ Theorem 4.14)
    X, Y dst with affine diagonalɺf : X → Y Λ perfectɺg : X′ → Y
    ͸೚ҙɻ͜ͷ࣌ QC(X ×Y X′) ≃ FunY (QC(X), QC(X′)) ͸ಉ஋ɻ
    1. ؔखͷߏ੒ M → ˜
    f∗(M ⊗ ˜
    g∗−) ͱ͢Δɻ˜
    f ͕ perfect ͳͷͰ
    colimit Λอͪ QC ʹҠΔɻ·ͨ projection formula ʹΑΓ
    QC(Y ) ઢܗʹͳΔɻ
    2. X′ ʹ͍ͭͯ local ͳͷͰʢ×, lim, colim, QC ͷަ׵ؔ܎ʣ
    ɺ
    affine ʹؼண͢Δɻ
    QC(X ×Y SpecA) ≃ FunY (QC(X), ModA) Λࣔ͢ɻ
    3. Y = SpecB ͷ࣌ɻલͷܥ 4.8 ͔Β QC(X) ͸ ModB
    ্ self
    dual Ͱɺલͷ໋୊ 4.13 ͔Β QC ͱ ⊗ ͷަ׵͕Θ͔ΔͷͰ
    FunB(QC(X), ModA) ≃ FunB(ModB, QC(X)∨ ⊗B ModA) ≃
    QC(X) ⊗B ModA QC(X ×B SpecA) ≃ QC(X) ⊗B ModA
    ͱ
    ܭࢉͰ͖Δɻ
    4. Y ͕Ұൠͷ࣌ɻ
    34

    View full-size slide

  35. ͜ͷઅͷ·ͱΊ
    1. derived affine scheme X = RSpecA ʹର͠ QC(X) = ModA
    Λ ∞ ݍͱͯ͠ఆٛͨ͠ɻ
    2. Ұൠͷ derived scheme X ʹର͠ QC(X) Λ
    X = colimi
    RSpecAi
    ͷͱ͖ ModAi
    ͷషΓ߹ΘͤͰఆٛͨ͠ɻ
    ͜Ε͸Ճ܈Ͱ͋Δɻ
    ʢstable symmetric monoidal categoryʣ
    3. perfect ͱ͍͏ΫϥεΛఆٛͨ͠ɻ༗ݶੑͷ৚݅
    4. ੵ෼ม׵ͷͳ͢ݍ͕ϑΝΠόʔੵͷ QC ͱಉ஋Ͱ͋Δ͜ͱΛ
    ࣔͨ͠ɻX ͕ Y ্ perfect ͳͱ͖
    QC(X ×Y X′) ≃ FunY (QC(X), QC(X′))
    K → (F → (f∗(g∗F ⊗ K)))
    35

    View full-size slide

  36. දݱ࿦ͱ TFT ΁ͷԠ༻
    36

    View full-size slide

  37. Ԡ༻
    1. Hecke category
    2. Ґ૬త৔ͷཧ࿦
    37

    View full-size slide

  38. affine Hecke category
    G ͸؆໿܈ɻHaff
    G
    Λ StG = ˜
    G ×G
    ˜
    G ্ͷ G ಉม४࿈઀૚ͷͳ͢
    ∞-category ͱ͢Δɻ͜͜Ͱ ˜
    G → G ͸ Grothendieck-Springer
    resolution Ͱ ˜
    G = {(g, B), g ∈ B, B ͸ Borel} ͱ͢Δɻ
    StG = ˜
    G ×G
    ˜
    G ͱ͢ΔɻZ(QC(X ×Y X)) ≃ QC(LY ) Λ
    X = ˜
    G/G → Y = G/G = LBG ʹద༻͢Δ͜ͱͰ
    Z(Haff
    G
    ) = Z(QC(StG )) ≃ Z(QC(X ×Y X)) ≃ QC(LY ) ≃
    QC(LLBG) ≃ QC(LocG (T2)) ͱͳΔɻ
    38

    View full-size slide

  39. finite Hecke algebra
    X → Y ʹରͯ͠ D(X ×Y X) Λߟ͑Δɻಛʹ BB → BG ʹରͯ͠
    X ×Y X = B\G/B ͱͳΔɻ
    Hecke category ͸ Hecke algebra ͷ categorification
    ·ͨ Loop space ͱͯ͠ͷղऍ͔Β
    D(B\G/B) ≃ Coh[(B×B)/G]
    (Stu/G)S1
    loc ͱͯ͠ affine Hecke
    catgoory ͱ finite Hecke category Λ݁ͼ͚ͭΔ͜ͱ͕Ͱ͖Δɻ
    coherent D-module ͷݍ D(B\G/B) ͷ Drinfeld center ͱ G ্ͷࢦ
    ඪ૚ͷݍͷಉҰࢹɻ͞Βʹࢦඪ૚ͷ Langlands ૒ର͕͋Δɻ
    39

    View full-size slide

  40. TFT
    extended TFT ͱ͸ (∞, 2)-cat ͷؒͷ symmetric monoidal functor
    Z : 2Cob → 2Alg ͷ͜ͱɻ2Cob ͸఺Λ 0 ର৅ɺ఺ͷؒͷ 1 ࣍ݩ
    bordism ͕ 1 ର৅ɺ1 ࣍ݩ bordism ͷؒͷ 2 ࣍ݩ bordism ͕ 2 ର৅ɻ
    2Alg ͸୅਺͕ 0 ର৅ɺbimodule ͕ 1 ର৅ɺͦͷؒͷࣹ͕ 2 ର৅ɻ
    ໋୊
    perfect stack X ʹର͠ extended 2d TFT ∃ZX
    ͕
    ZX (S1) = QC(LX), ZX (Σ) = Γ(XΣ, OXΣ
    ) ͱͯ͠ఆ·Δɻ
    ZX ((S1)⨿m) = QC((LX)×m) ≃ QC(LX)⊗m = ZX (S1)⊗m
    ͱͳΓɺsymmetric monoidal ʹͳΔɻ
    ಛʹ X = BG ͷ৔߹͕਺ཧ෺ཧతʹ΋ڵຯΛ࣋ͨΕΔɻ
    40

    View full-size slide

  41. ࢀߟจݙ
    • D. Ben-Zvi, J. Francis and D. Nadler, Integral transforms and
    Drinfeld centers in derived algebraic geometry.
    • D. Ben-Zvi and D. Nadler, Loop Spaces and Connections.
    • D. Ben-Zvi and D. Nadler, The character theory of a complex
    group.
    • D. Ben-Zvi and D. Nadler, Loop Spaces and Langlands
    Parameters.
    • D. Ben-Zvi and D. Nadler, Loop Spaces and Representations.
    • B. To¨
    en, Higher and Derived Stacks: a global overview.
    • B. To¨
    en and G. Vezzosi, A note on Chern character, loop
    spaces and derived algebraic geometry.
    • D. Gaitsgory and N. Rozenblyum, A study in derived algebraic
    geometry
    • J. Lurier, Higher Algebra.
    41

    View full-size slide