Lemma 46.3.7 (06UZ)—The Stacks project

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

$F(B[t, t^{-1}]) = \bigoplus \nolimits _{k \in \mathbf{Z}} F(p)(F(B)) \cdot t^ k = \bigoplus \nolimits _{k \in \mathbf{Z}} F(B) \cdot t^ k$

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

$x = F(\epsilon )(F(a)(x)) = F(\epsilon )(\sum t^ k x_ k) = \sum x_ k.$

On the other hand, we claim that $x_ k \in F(B)^{(k)}$ . Namely, consider the commutative diagram

$\xymatrix{ B \ar[r]_ a \ar[d]_{a'} & B[t, t^{-1}] \ar[d]^ f \\ B[s, s^{-1}] \ar[r]^-g & B[t, s, t^{-1}, s^{-1}] }$

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

$F(g)(F(a'))(x) = F(g)(\sum s^ k x_ k) = \sum s^ k F(a)(x_ k)$

and going the other way we see

$F(f)(F(a))(x) = F(f)(\sum t^ k x_ k) = \sum (st)^ k x_ k.$

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 $B_0 \to B \xrightarrow {p} B[t, t^{-1}]$ . $\square$

The Stacks project

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