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

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

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

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

୅਺زԿͱϗϞτϐʔ 5

͜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

homotopy limit and homotopy colimit ௨ৗͷۃݶ΍༨ۃݶ͸ϗϞτϐʔͱ૬ੑ͕ѱ͍ɻ ∗ ← − − − − ∗ ⏐ ⏐ ⏐ ⏐ ∗ ← − − − − ∗ ⨿ ∗ S1 ← − − − − [0, 1] ⏐ ⏐ ⏐ ⏐ [0, 1] ← − − − − ∗ ⨿ ∗ ∗ − − − − → ∗ ⏐ ⏐ ⏐ ⏐ ∗ − − − − → S1 Z − − − − → ∗ ⏐ ⏐ ⏐ ⏐ R − − − − → S1 ͜ΕΛࠀ෰͢ΔͨΊʹϗϞτϐʔۃݶͱϗϞτϐʔ༨ۃݶͱ͍͏ ֓೦Λఆٛ͢Δɻ 7

୅਺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

ަ఺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

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

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

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

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

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

͜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

derived stack 16

͜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

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

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

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

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

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

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

͜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

QC(X) ͱੵ෼ม׵ 25

͜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

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

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

༗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

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

⊗ 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

⊗ 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

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

ੵ෼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

͜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

දݱ࿦ͱ TFT ΁ͷԠ༻ 36

Ԡ༻ 1. Hecke category 2. Ґ૬త৔ͷཧ࿦ 37

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

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

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

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