Proposition 3.1.6 (0P4U) · ScalingStacks

Proposition 3.1.6. If $W$ is finite, then

t_{S,I}:H^{\mathrm{nil}}(W)\to H^{\mathrm{nil}}(W_{I})\langle N-N_{I}\rangle

is a morphism of differential graded ${\mathbf{F}}_{2}$ -modules and Corollary 3.1.2 provides an isomorphism of differential graded $(H^{\mathrm{nil}}(W_{I}),H^{\mathrm{nil}}(W))$ -bimodules

\hat{t}_{S,I}^{\pm}:L^{\mp}(I,S)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\pm}(S,I)^{\vee}\langle N-N_{I}\rangle.

0P4V

Proof. Let $v\in W$ . There is a unique decomposition $v=v^{\prime}v^{\prime\prime}$ where $\ell(v)=\ell(v^{\prime})+\ell(v^{\prime\prime})$ , $v^{\prime\prime}\in W_{I}$ and $v^{\prime}\in W^{I}$ .

We have $d(T_{v})=d(T_{v^{\prime}})T_{v^{\prime\prime}}+T_{v^{\prime}}d(T_{v^{\prime\prime}})$ . If $u\in W$ and $u<v^{\prime}$ , then $u{\not\in}w_{S}W_{I}$ . It follows that

t_{S,I}(d(T_{v}))=t_{S,I}(T_{v^{\prime}}d(T_{v^{\prime\prime}}))=\delta_{v^{\prime},w^{I}}d(T_{v^{\prime\prime}})=d(t_{S,I}(T_{v})).

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