Doubt regarding the definition of slice filtration

Question

Voevodsky defined the slice filtration on the motivic stable homotopy category $SH(S)$ over a Noetherian scheme $S$. In the article Open Problems in the Motivic Stable Homotopy Theory, I, Section 2, he defined $SH^{eff}(S)$ to be the smallest triangulated subcategory in $SH(S)$ which is closed under direct sums and contains suspension spectra of spaces. It follows from a result of Neeman that the inclusion functor $i_n:\Sigma^n_TSH^{eff}(S)\to SH^{eff}(S)$ has a right adjoint $r_n$. Since $i_n$ is full, we have a natural isomorphism $\mathrm{id}\simeq r_n\circ i_n$. Denote $f_n=i_n\circ r_n$. Applying the counit $f_{n+1}\to\mathrm{id}$ to $f_n$, we get a natural transformation $f_{n+1}\circ f_n\to f_n$. Voevodsky claims that $f_{n+1}=f_{n+1}\circ f_n$ so that we get a natural transformation $f_{n+1}\to f_n$. The slice functor $s_n$ is defined to be the cofiber of this map. Denoting the inclusion functor $\Sigma^n_TSH^{eff}(S)\to \Sigma^{n-1}_TSH^{eff}(S)$ by $j_n$, (so that $i_{n+1}=i_n\circ j_{n+1}$), I calculated that $$f_{n+1}=i_{n+1}\circ r_{n+1}=i_n\circ j_{n+1}\circ r_{n+1}=i_n\circ\mathrm{id}\circ j_{n+1}\circ r_{n+1}$$ $$\simeq i_n\circ r_n\circ i_n\circ j_{n+1}\circ r_{n+1}=f_n\circ f_{n+1}$$ but don't see why $f_{n+1}=f_{n+1}\circ f_n$. Note that $f_{n+1}=f_n\circ f_{n+1}$ also induces a natural transformation $f_{n+1}\to f_n$.

Am I missing something here? Or is there a typo in that paper?

Answer 1

The key here is that $SH^{eff}(S)$ is closed under suspensions, so there's an inclusion $j_{n+1}:\Sigma^{n+1}_T SH^{eff}(S)\subseteq \Sigma^n_T SH^{eff}(S)$. Hence you can write $i_{n+1}=i_n \circ j_{n+1}$ and $r_{n+1} = l_{n+1}\circ r_n$ where $l_{n+1}$ is the right adjoint of $j_{n+1}$ (which exists because $j_{n+1}$ commutes with colimits). So

$f_{n+1}f_n = i_{n+1}r_{n+1}i_nr_n = i_n j_{n+1} l_{n+1} r_n i_n r_n = i_n j_{n+1}l_{n+1}r_n = f_{n+1}\,.$