Lemma 46.3.7. Let $A$ be a ring. Let $F$ be a module-valued functor on $\textit{Alg}_ A$. Assume that for $B \to B'$ flat the map $F(B) \otimes _ B B' \to F(B')$ is an isomorphism. Let $B$ be a graded $A$-algebra. Then
$F(B) = \bigoplus _{k \in \mathbf{Z}} F(B)^{(k)}$, and
the map $B \to B_0 \to B$ induces map $F(B) \to F(B)$ whose image is contained in $F(B)^{(0)}$.
Proof. Let $x \in F(B)$. The map $p : B \to B[t, t^{-1}]$ is free hence we know that
as indicated we drop the $F(p)$ in the rest of the proof. Write $F(a)(x) = \sum t^ k x_ k$ for some $x_ k \in F(B)$. Denote $\epsilon : B[t, t^{-1}] \to B$ the $B$-algebra map $t \mapsto 1$. Note that the compositions $\epsilon \circ p, \epsilon \circ a : B \to B[t, t^{-1}] \to B$ are the identity. Hence we see that
On the other hand, we claim that $x_ k \in F(B)^{(k)}$. Namely, consider the commutative diagram
where $a'(b) = s^{\deg (b)}b$, $f(b) = b$, $f(t) = st$ and $g(b) = t^{\deg (b)}b$ and $g(s) = s$. Then
and going the other way we see
Since $B \to B[s, t, s^{-1}, t^{-1}]$ is free we see that $F(B[t, s, t^{-1}, s^{-1}]) = \bigoplus _{k, l \in \mathbf{Z}} F(B) \cdot t^ ks^ l$ and comparing coefficients in the expressions above we find $F(a)(x_ k) = t^ k x_ k$ as desired.
Finally, the image of $F(B_0) \to F(B)$ is contained in $F(B)^{(0)}$ because $B_0 \to B \xrightarrow {a} B[t, t^{-1}]$ is equal to $B_0 \to B \xrightarrow {p} B[t, t^{-1}]$. $\square$
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