0P27
Theorem 5.1 . Given N , N ′ ∈ Ho b ( P − exact ) N,N^{\prime}\in\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-exact}\nolimits) K i ( N ⊗ P N ′ ) ≃ K i ( N ′ ⊗ P N ) K^{i}(N\otimes_{P}N^{\prime})\simeq K^{i}(N^{\prime}\otimes_{P}N) Ho ( P − modgr ) \operatorname{Ho}\nolimits(P\operatorname{\!-modgr}\nolimits)
Let s ∈ S s\in S z s z_{s} ( V ∗ ) S ∖ s (V^{*})^{S\setminus s} M ∈ Ho b ( P S ∖ s − exact ) M\in\operatorname{Ho}\nolimits^{b}(P_{S\setminus s}\operatorname{\!-exact}\nolimits)
•
K S i ( γ S ∖ s ( M ) ⊗ P F s ) ≃ ρ S ∖ s ∗ K S ∖ s i + 1 ( M ) [ − 1 ] in D ( P − modgr ) K^{i}_{S}\bigl(\gamma_{S\setminus s}(M)\otimes_{P}F_{s}\bigr)\simeq\rho_{S\setminus s}^{*}K^{i+1}_{S\setminus s}(M)[-1]\text{ in }D(P\operatorname{\!-modgr}\nolimits)
•
K S i ( γ S ∖ s ( M ) ⊗ P F s − 1 ) ≃ ρ S ∖ s ∗ K S ∖ s i ( M ) in D ( P − modgr ) K^{i}_{S}\bigl(\gamma_{S\setminus s}(M)\otimes_{P}F_{s}^{-1}\bigr)\simeq\rho_{S\setminus s}^{*}K^{i}_{S\setminus s}(M)\text{ in }D(P\operatorname{\!-modgr}\nolimits)
•
K S i ( γ S ∖ s ( M ) ) ≃ P ⊗ P S ∖ s K S ∖ s i + 1 ( M ) ⟨ − 1 ⟩ ⊕ P ⊗ P S ∖ s K S ∖ s i ( M ) in Ho ( P − modgr ) K^{i}_{S}\bigl(\gamma_{S\setminus s}(M))\simeq P\otimes_{P_{S\setminus s}}K^{i+1}_{S\setminus s}(M)\langle-1\rangle\oplus P\otimes_{P_{S\setminus s}}K^{i}_{S\setminus s}(M)\text{ in }\operatorname{Ho}\nolimits(P\operatorname{\!-modgr}\nolimits) .
0P28
Proof. Thanks to (2
C P en ( Tot ( N ⊗ P N ′ ) ) ≃ Tot 12 → 1 , 3 → 2 ( C P en ( N ) ⊗ P N ′ ) C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N\otimes_{P}N^{\prime})\bigr)\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}(C_{P^{\mathrm{en}}}(N)\otimes_{P}N^{\prime})
hence
P ⊗ P en C P en ( Tot ( N ⊗ P N ′ ) ) \displaystyle P\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N\otimes_{P}N^{\prime})\bigr) ≃ Tot 12 → 1 , 3 → 2 ( C P en ( N ) ⊗ P en N ′ ) \displaystyle\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}\bigl(C_{P^{\mathrm{en}}}(N)\otimes_{P^{\mathrm{en}}}N^{\prime}\bigr)
≃ Tot 12 → 1 , 3 → 2 ( N ′ ⊗ P en C P en ( N ) ) \displaystyle\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}\bigl(N^{\prime}\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}(N)\bigr)
≃ P ⊗ P en C P en ( Tot ( N ′ ⊗ P N ) ) \displaystyle\simeq P\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N^{\prime}\otimes_{P}N)\bigr)
and the first assertion follows.
We have
C P en ( M ⊗ k [ z s ] ) ≃ Tot 1 → 1 , 23 → 2 ( C P S ∖ s en ( M ) ⊗ X ) C_{P^{\mathrm{en}}}(M\otimes k[z_{s}])\simeq\operatorname{Tot}\nolimits^{1\to 1,23\to 2}\bigl(C_{P_{S\setminus s}^{\mathrm{en}}}(M)\otimes X\bigr)
where
X = 0 → k [ z s ] en ⟨ − 1 ⟩ → z s ⊗ 1 − 1 ⊗ z s k [ z s ] en → 0 X=0\to k[z_{s}]^{\mathrm{en}}\langle-1\rangle\xrightarrow{z_{s}\otimes 1-1\otimes z_{s}}k[z_{s}]^{\mathrm{en}}\to 0 − 1 -1 0 0
L = P ⊗ P en ( C P en ( Tot ( ( M ⊗ k [ z s ] ) ⊗ P F s ) ) ) . L=P\otimes_{P^{\mathrm{en}}}\biggl(C_{P^{\mathrm{en}}}\Bigl(\operatorname{Tot}\nolimits\bigl((M\otimes k[z_{s}])\otimes_{P}F_{s}\bigr)\Bigr)\biggr).
By (2
L ≃ P ⊗ P en Tot 13 → 1 , 2 → 2 ( C P en ( M ⊗ k [ z s ] ) ⊗ P F s ) ≃ Tot 13 → 1 , 24 → 2 ( ( F s ⊗ k [ z s ] en X ) ⊗ P S ∖ s en C P S ∖ s en ( M ) ) L\simeq P\otimes_{P^{\mathrm{en}}}\operatorname{Tot}\nolimits^{13\to 1,2\to 2}\bigl(C_{P^{\mathrm{en}}}(M\otimes k[z_{s}])\otimes_{P}F_{s}\bigr)\simeq\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl((F_{s}\otimes_{k[z_{s}]^{\mathrm{en}}}X)\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)
We have
F s ⊗ k [ z s ] en X ≃ θ s P θ s ⟨ 1 ⟩ P ⟨ 1 ⟩ m z s ⊗ 1 − 1 ⊗ z s 0 m ≃ θ s P θ s ⟨ 1 ⟩ P ⟨ 1 ⟩ m α s ⊗ 1 − 1 ⊗ α s 0 m F_{s}\otimes_{k[z_{s}]^{\mathrm{en}}}X\simeq\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 45.08356pt\hbox{{\hbox{\kern-7.48438pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.72223pt\hbox{$\textstyle{\theta_{s}}$}}}}}{\hbox{\kern 50.0007pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.41666pt\hbox{$\textstyle{P}$}}}}}{\hbox{\kern-13.87329pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\theta_{s}\langle 1\rangle}$}}}}}{\hbox{\kern 43.61179pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{P\langle 1\rangle}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 22.19449pt\raise 27.26903pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{m}$}}}\kern 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Indeed, there is c ∈ k ∗ c\in k^{*} s ( z s ) − z s = − 2 c α s s(z_{s})-z_{s}=-2c\alpha_{s} z s − c α s ∈ ( V ∗ ) s z_{s}-c\alpha_{s}\in(V^{*})^{s} z s ⊗ 1 − 1 ⊗ z s z_{s}\otimes 1-1\otimes z_{s} c ( α s ⊗ 1 − 1 ⊗ α s ) c(\alpha_{s}\otimes 1-1\otimes\alpha_{s}) θ s \theta_{s}
Lemma 5.2 s ∉ Z ( W ) s{\not\in}Z(W) P en P^{\mathrm{en}}
0 → P → α s ⊗ 1 + 1 ⊗ α s θ s → a ⊗ b ↦ a s ( b ) P s → 0 0\to P\xrightarrow{\alpha_{s}\otimes 1+1\otimes\alpha_{s}}\theta_{s}\xrightarrow{a\otimes b\mapsto as(b)}Ps\to 0
splits by restriction to P S ∖ s en P_{S\setminus s}^{\mathrm{en}} P s = P Ps=P P P a ∈ P a\in P s ( a ) s(a) s ∈ Z ( W ) s\in Z(W) P s = P S ∖ s ⊗ k [ α s 2 ] P^{s}=P_{S\setminus s}\otimes k[\alpha_{s}^{2}]
We deduce that there is an isomorphism of complexes of graded
P S ∖ s en P_{S\setminus s}^{\mathrm{en}}
( 0 → θ s → α s ⊗ 1 − 1 ⊗ α s θ s ⟨ 1 ⟩ → 0 ) ≃ P [ 1 ] ⟨ − 1 ⟩ ⊕ ( 0 → P s → a ↦ a α s ⊗ 1 − a ⊗ α s θ s → 0 ) ⟨ 1 ⟩ . \bigl(0\to\theta_{s}\xrightarrow{\alpha_{s}\otimes 1-1\otimes\alpha_{s}}\theta_{s}\langle 1\rangle\to 0\bigr)\simeq P[1]\langle-1\rangle\oplus\bigl(0\to Ps\xrightarrow{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}}\theta_{s}\to 0\bigr)\langle 1\rangle.
Let Y 1 = P ⟨ − 1 ⟩ P 0 0 2 α s and Y 2 = P s 0 θ s ⟨ 1 ⟩ P ⟨ 1 ⟩ a ↦ a α s ⊗ 1 − a ⊗ α s \text{Let }Y_{1}=\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 16.07117pt\hbox{{\hbox{\kern-16.07117pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{P\langle-1\rangle}$}}}}}{\hbox{\kern 50.0007pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.41666pt\hbox{$\textstyle{P}$}}}}}{\hbox{\kern-5.5pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.22223pt\hbox{$\textstyle{0}$}}}}}{\hbox{\kern 51.40521pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.22223pt\hbox{$\textstyle{0}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 23.51813pt\raise 28.51764pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75555pt\hbox{$\scriptstyle{2\alpha_{s}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 50.0007pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-16.53987pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 56.90521pt\raise-16.53987pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 51.40521pt\raise-22.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\text{ and }Y_{2}=\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 56.29254pt\hbox{{\hbox{\kern-9.24826pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.41666pt\hbox{$\textstyle{Ps}$}}}}}{\hbox{\kern 51.40521pt\raise 22.76208pt\hbox{\hbox{\kern 3.0pt\raise-3.22223pt\hbox{$\textstyle{0}$}}}}}{\hbox{\kern-13.87329pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\theta_{s}\langle 1\rangle}$}}}}}{\hbox{\kern 43.61179pt\raise-22.76208pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{P\langle 1\rangle}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 51.40521pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-56.29254pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75555pt\hbox{$\scriptstyle{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-14.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 56.90521pt\raise-14.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 43.61179pt\raise-22.76208pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}
We have an exact sequence of bicomplexes of graded
P en P^{\mathrm{en}}
0 → Y 1 → F s ⊗ k [ α s ] en X → Y 2 → 0 . 0\to Y_{1}\to F_{s}\otimes_{k[\alpha_{s}]^{\mathrm{en}}}X\to Y_{2}\to 0.
It splits after restricting to P S ∖ s en P_{S\setminus s}^{\mathrm{en}} ? ( i , ∗ ) ?^{(i,*)} P en P^{\mathrm{en}}
0 → H d 2 i ( Tot 13 → 1 , 24 → 2 ( Y 1 ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) → H d 2 i ( L ) → → H d 2 i ( Tot 13 → 1 , 24 → 2 ( Y 2 ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) → 0 . 0\to H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{1}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\to H^{i}_{d_{2}}(L)\to\\
\to H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\to 0.
We have an exact sequence of P en P^{\mathrm{en}}
0 → P s → a ↦ a α s ⊗ 1 − a ⊗ α s θ s → 𝑚 P → 0 . 0\to Ps\xrightarrow{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}}\theta_{s}\xrightarrow{m}P\to 0.
So,
the morphism of bicomplexes Y 2 → Y 2 ′ Y_{2}\to Y^{\prime}_{2}
P s \textstyle{Ps\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} a ↦ a α s ⊗ 1 − a ⊗ α s \scriptstyle{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}} 0 \textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 0 \textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 0 \textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces} θ s ⟨ 1 ⟩ \textstyle{\theta_{s}\langle 1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} m \scriptstyle{m} m \scriptstyle{m} P ⟨ 1 ⟩ \textstyle{P\langle 1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} id \scriptstyle{\operatorname{id}\nolimits} P ⟨ 1 ⟩ \textstyle{P\langle 1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces} id \scriptstyle{\operatorname{id}\nolimits} P ⟨ 1 ⟩ \textstyle{P\langle 1\rangle}
induces an isomorphism of complexes
H d 2 i ( Tot 13 → 1 , 24 → 2 ( Y 2 ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) → ∼ H d 2 i ( Tot 13 → 1 , 24 → 2 ( Y 2 ′ ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y^{\prime}_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)
and these complexes vanish in Ho b ( P en − modgr ) \operatorname{Ho}\nolimits^{b}(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)
We deduce that
H d 2 i ( L ) \displaystyle H^{i}_{d_{2}}(L) ≃ H d 2 i ( Tot 13 → 1 , 24 → 2 ( Y 1 ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) \displaystyle\simeq H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{1}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)
≃ ρ S ∖ s ∗ H d 2 i ( Tot 13 → 1 , 24 → 2 ( P S ∖ s [ ( − 1 , 1 ) ] ⊗ P S ∖ s en C P S ∖ s en ( M ) ) ) \displaystyle\simeq\rho_{S\setminus s}^{*}H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(P_{S\setminus s}[(-1,1)]\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)
≃ ρ S ∖ s ∗ H d 2 i + 1 ( P S ∖ s ⊗ P S ∖ s en C P S ∖ s en ( M ) ) [ − 1 ] \displaystyle\simeq\rho_{S\setminus s}^{*}H^{i+1}_{d_{2}}\bigl(P_{S\setminus s}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)[-1]
in D b ( P en − modgr ) D^{b}(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits) P en → P P^{\mathrm{en}}\to P D b ( P − modgr ) D^{b}(P\operatorname{\!-modgr}\nolimits) F s − 1 F_{s}^{-1}
We have
k [ z s ] ⊗ k [ z s ] en X ≃ k [ z s ] ⟨ − 1 ⟩ [ 1 ] ⊕ k [ z s ] k[z_{s}]\otimes_{k[z_{s}]^{\mathrm{en}}}X\simeq k[z_{s}]\langle-1\rangle[1]\oplus k[z_{s}]