2 tau is automorphism of I tensor I

This is an interesting question. I.e., I have no idea how to answer it ...

Here's a little bit of context to put this in. So $C$ is a symmetric monoidal category, with unit object $R$. Let $\mathrm{Pic}(C)$ be the collection of isomorphism classes of invertible objects in $C$; it's an abelian group using $\otimes$.

There's a group homomorphism $$\eta : \mathrm{Pic}(C) \to \text{(2-torsion subgroup of \mathrm{Aut}(R)).}$$ This sends an invertible object $I$ to its Euler characteristic..

Here's a different construction of $\eta$ which I find easier to understand. If $I$ is an invertible object, there is a canonical isomorphism $\mathrm{Aut}(I)\approx \mathrm{Aut}(R)$; to construct it, choose an isomorphism $I\otimes J\approx R$, so that an automorphism $f:I\to I$ gets sent to an automorphism $R\approx I\otimes J \xrightarrow{f\otimes 1} I\otimes J\approx R$. Now $\eta (I)$ is just the image of the braid map $(\tau: I\otimes I \to I\otimes I)\in \mathrm{Aut}(I)\approx mathrm{Aut}(I\otimes I)\approx \mathrm{Aut}(R)$. This definition makes it clear that $(\eta (I))^2=1$.

So you're asking why $\eta$ has trivial image when $C$ is a category of modules over a commutative ring. As you point out, in graded modules the image is ${\pm1}$. I got nothing here ...

1

This is an interesting question. I.e., I have no idea how to answer it ...

Here's a little bit of context to put this in. So $C$ is a symmetric monoidal category, with unit object $R$. Let $\mathrm{Pic}(C)$ be the collection of isomorphism classes of invertible objects in $C$; it's an abelian group using $\otimes$.

There's a group homomorphism $$\eta : \mathrm{Pic}(C) \to \text{(2-torsion subgroup of \mathrm{Aut}(R)).}$$ This sends an invertible object $I$ to its Euler characteristic..

Here's a different construction of $\eta$ which I find easier to understand. If $I$ is an invertible object, there is a canonical isomorphism $\mathrm{Aut}(I)\approx \mathrm{Aut}(R)$; to construct it, choose an isomorphism $I\otimes J\approx R$, so that an automorphism $f:I\to I$ gets sent to an automorphism $R\approx I\otimes J \xrightarrow{f\otimes 1} I\otimes J\approx R$. Now $\eta (I)$ is just the image of the braid map $(\tau: I\otimes I \to I\otimes I)\in \mathrm{Aut}(I)\approx \mathrm{Aut}(R)$. This definition makes it clear that $(\eta (I))^2=1$.

So you're asking why $\eta$ has trivial image when $C$ is a category of modules over a commutative ring. As you point out, in graded modules the image is ${\pm1}$. I got nothing here ...