Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object.  In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've found something disturbing I'd like to resolve.

Following 7.2 of [[DG12]][1], we can identify $\text{Sh}(B\mathbf{G}_m)$ with the modules in $\text{Sh}(\text{pt})$ for the algebra $B=H^*(\mathbf{G}_m,k)^\vee\in\text{Sh}(\text{pt})$. Note $B=k[u]/u^2$ where $|u|=-1$. Cohomology is
$$H^*(B\mathbf{G}_m,-)\ :\ M \ \longrightarrow\ \text{Ext}_B(k,M).$$
For instance, $k\mapsto k[t]$ where $|t|=2$ and $B\mapsto k$.

Now let $V$ be a vector bundle over $B\mathbf{G}_m$ (=vector space with $\mathbf{G}_m$ action) of rank $r$. Write $i$ for the zero section and $j$ for the complementary open embedding. Applying $i^*$ to the distinguished triangle $i_!i^!k\to k\to j_*j^*k$ gives the *Gysin sequence*
$$i^!k \ \stackrel{e}{\longrightarrow} \ i^*k \ \longrightarrow \ i^*j_*j^*k$$
Note that $i^!k\simeq k[-2r]$ and $e$ is multiplication by the Euler class. In particular, if $V$ has nonzero $\mathbf{G}_m$ weights, then on cohomology $e=t^{2r}:k[t]\to k[t]$ so $H^*(i^*j_*j^*k)$ is finite dimensional. 

**Question**: however, what on earth is $i^*j_*j^*k$? Forgetting the $B$-module structure, 
$$i^*j_*j^*k\ =\ k\oplus k[-2r+1]$$
by pulling back the Gysin sequence to $\text{pt}$. In the rank $r=1$ case, this means that as a $B$ module it's $k\oplus k[-1]$ or $B$. Only the latter has finite dimensional cohomology it must be that:
$$i^*j_*j^*k\ = \ B \ =\ H^*(\mathbf{G}_m)^\vee.$$
*However*, when the rank is greater than one there seems to be no $B$-module structure on $k\oplus k[-2r+1]$ which gives finite dimensional cohomology!

 What on earth is going on here?




  [1]: https://arxiv.org/abs/1108.5351