ScalingStacks

Remark 8.1.21. There is a sequence of four adjoint functors between $A$-modules and $\hat{A}$-modules:

$$\bigl(A\otimes_{\hat{A}}-,\hat{A}(1-e)\otimes_{A}-,(1-e)\hat{A}\otimes_{\hat{A}}-,\operatorname{Hom}\nolimits_{A}((1-e)\hat{A},-)\bigr).$$

The first and fourth functors are not exact in general. Here,

• •

  $\hat{A}$ acts on the right on $A$ by right multiplication preceded by the composition

  $$\hat{A}\xrightarrow{{\mathrm{can}}}\hat{A}/\hat{A}e\hat{A}\xrightarrow[\sim]{g^{-1}}(1-e)\hat{A}(1-e)\xrightarrow[\sim]{h^{-1}}A$$

• •

  $\bigl(\hat{A}(1-e)\otimes_{A}-\bigr)=\bigl((1-e)\hat{A}(1-e)\otimes_{A}-\bigr)\xrightarrow[\sim]{h^{-1}}\bigl(A\otimes_{A}-\bigr)=\operatorname{Hom}\nolimits_{A}(A,-)$

• •

  $\bigl((1-e)\hat{A}\otimes_{\hat{A}}-\bigr)\xrightarrow[\sim]{{\mathrm{can}}}\bigl(\operatorname{Hom}\nolimits_{\hat{A}}(\hat{A}(1-e),\hat{A})\otimes_{\hat{A}}-\bigr)\xrightarrow[\sim]{{\mathrm{can}}}\operatorname{Hom}\nolimits_{\hat{A}}(\hat{A}(1-e),-)$.