ScalingStacks

Proof. Thanks to (2), we have

$$C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N\otimes_{P}N^{\prime})\bigr)\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}(C_{P^{\mathrm{en}}}(N)\otimes_{P}N^{\prime})$$

hence

	$\displaystyle P\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N\otimes_{P}N^{\prime})\bigr)$	$\displaystyle\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}\bigl(C_{P^{\mathrm{en}}}(N)\otimes_{P^{\mathrm{en}}}N^{\prime}\bigr)$
		$\displaystyle\simeq\operatorname{Tot}\nolimits^{12\to 1,3\to 2}\bigl(N^{\prime}\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}(N)\bigr)$
		$\displaystyle\simeq P\otimes_{P^{\mathrm{en}}}C_{P^{\mathrm{en}}}\bigl(\operatorname{Tot}\nolimits(N^{\prime}\otimes_{P}N)\bigr)$

and the first assertion follows.

We have

$$C_{P^{\mathrm{en}}}(M\otimes k[z_{s}])\simeq\operatorname{Tot}\nolimits^{1\to 1,23\to 2}\bigl(C_{P_{S\setminus s}^{\mathrm{en}}}(M)\otimes X\bigr)$$

where $X=0\to k[z_{s}]^{\mathrm{en}}\langle-1\rangle\xrightarrow{z_{s}\otimes 1-1\otimes z_{s}}k[z_{s}]^{\mathrm{en}}\to 0$, the non-zero terms being in degrees $-1$ and $0$. Let

$$L=P\otimes_{P^{\mathrm{en}}}\biggl(C_{P^{\mathrm{en}}}\Bigl(\operatorname{Tot}\nolimits\bigl((M\otimes k[z_{s}])\otimes_{P}F_{s}\bigr)\Bigr)\biggr).$$

By (2), we have

$$L\simeq P\otimes_{P^{\mathrm{en}}}\operatorname{Tot}\nolimits^{13\to 1,2\to 2}\bigl(C_{P^{\mathrm{en}}}(M\otimes k[z_{s}])\otimes_{P}F_{s}\bigr)\simeq\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl((F_{s}\otimes_{k[z_{s}]^{\mathrm{en}}}X)\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)$$

We have

$$F_{s}\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 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\ignorespaces\ignorespaces{\hbox{\kern-45.08356pt\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{z_{s}\otimes 1-1\otimes z_{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\ignorespaces\ignorespaces{\hbox{\kern 56.90521pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern 3.0pt}}}}}}\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\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 3.0pt}}}}}}\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}}}}\simeq\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 47.26572pt\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 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\ignorespaces\ignorespaces{\hbox{\kern-47.26572pt\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{\alpha_{s}\otimes 1-1\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\ignorespaces\ignorespaces{\hbox{\kern 56.90521pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern 3.0pt}}}}}}\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\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 3.0pt}}}}}}\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}}}}$$

Indeed, there is $c\in k^{*}$ such that $s(z_{s})-z_{s}=-2c\alpha_{s}$. Then $z_{s}-c\alpha_{s}\in(V^{*})^{s}$, hence $z_{s}\otimes 1-1\otimes z_{s}$ and $c(\alpha_{s}\otimes 1-1\otimes\alpha_{s})$ are equal in $\theta_{s}$.

Lemma 5.2 below shows that, when $s{\not\in}Z(W)$, then the exact sequence of $P^{\mathrm{en}}$-modules

$$0\to P\xrightarrow{\alpha_{s}\otimes 1+1\otimes\alpha_{s}}\theta_{s}\xrightarrow{a\otimes b\mapsto as(b)}Ps\to 0$$

splits by restriction to $P_{S\setminus s}^{\mathrm{en}}$. Here, $Ps=P$ as a left $P$-module, and the right action of $a\in P$ is given by multiplication by $s(a)$. Note that when $s\in Z(W)$, then $P^{s}=P_{S\setminus s}\otimes k[\alpha_{s}^{2}]$, and the splitting of the sequence holds trivially.

We deduce that there is an isomorphism of complexes of graded $P_{S\setminus s}^{\mathrm{en}}$-modules

$$\bigl(0\to\theta_{s}\xrightarrow{\alpha_{s}\otimes 1-1\otimes\alpha_{s}}\theta_{s}\langle 1\rangle\to 0\bigr)\simeq P[1]\langle-1\rangle\oplus\bigl(0\to Ps\xrightarrow{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}}\theta_{s}\to 0\bigr)\langle 1\rangle.$$

$$\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 $P^{\mathrm{en}}$-modules

$$0\to Y_{1}\to F_{s}\otimes_{k[\alpha_{s}]^{\mathrm{en}}}X\to Y_{2}\to 0.$$

It splits after restricting to $P_{S\setminus s}^{\mathrm{en}}$ and applying $?^{(i,*)}$. It follows that we have an exact sequence of complexes of graded $P^{\mathrm{en}}$-modules

$$0\to H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{1}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\to H^{i}_{d_{2}}(L)\to\\ \to H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\to 0.$$

We have an exact sequence of $P^{\mathrm{en}}$-modules

$$0\to Ps\xrightarrow{a\mapsto a\alpha_{s}\otimes 1-a\otimes\alpha_{s}}\theta_{s}\xrightarrow{m}P\to 0.$$

So, the morphism of bicomplexes $Y_{2}\to Y^{\prime}_{2}$:

induces an isomorphism of complexes

$$H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y^{\prime}_{2}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)$$

and these complexes vanish in $\operatorname{Ho}\nolimits^{b}(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$.

We deduce that

	$\displaystyle H^{i}_{d_{2}}(L)$	$\displaystyle\simeq H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(Y_{1}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)$
		$\displaystyle\simeq\rho_{S\setminus s}^{*}H^{i}_{d_{2}}\Bigl(\operatorname{Tot}\nolimits^{13\to 1,24\to 2}\bigl(P_{S\setminus s}[(-1,1)]\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)\Bigr)$
		$\displaystyle\simeq\rho_{S\setminus s}^{*}H^{i+1}_{d_{2}}\bigl(P_{S\setminus s}\otimes_{P_{S\setminus s}^{\mathrm{en}}}C_{P_{S\setminus s}^{\mathrm{en}}}(M)\bigr)[-1]$

in $D^{b}(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$. Note that the multiplication map $P^{\mathrm{en}}\to P$ is a split surjection of algebras. We deduce the first and last terms of the sequence of isomorphisms above are actually isomorphic in $D^{b}(P\operatorname{\!-modgr}\nolimits)$. This shows the second assertion. The proof of the assertion involving $F_{s}^{-1}$ is similar.

We have $k[z_{s}]\otimes_{k[z_{s}]^{\mathrm{en}}}X\simeq k[z_{s}]\langle-1\rangle[1]\oplus k[z_{s}]$ and the last assertion follows immediately. ∎

Proof. We will show that $\phi:P_{S\setminus s}\otimes_{S(L)}P_{S\setminus s}\to P,\ a\otimes b\mapsto as(b)$ is an isomorphism. It is a morphism of algebras, and a morphism of graded left $P_{S\setminus s}$-modules.

By assumption, there is $t\in S\setminus s$ such that $m_{st}\not=2$, so that $s(\alpha_{t})-\alpha_{t}$ is a non-zero multiple of $\alpha_{s}$. It follows that $\phi(\alpha_{t}\otimes 1-1\otimes\alpha_{t})\in k^{\times}\alpha_{s}$. Since $V^{*}=(V^{S\setminus s})^{\perp}\oplus k\alpha_{s}$, we have $P=P_{S\setminus s}\otimes k[\alpha_{s}]$ and we deduce that $\phi$ is surjective.

Since $\phi$ is a morphism of graded free left $P_{S\setminus s}$-modules with the same graded ranks $1+t+t^{2}+\cdots$, we deduce that $\phi$ is an isomorphism. ∎