Sub-lambda-rings of tensor products

[torstein-vik]

Conjecture: if $R, S$ are $\lambda$-rings and $X, Y$ respectively sub-$\lambda$-rings, then $X \otimes Y$ is a sub-$\lambda$-ring of $R \otimes S$.

Conjecture: if $R, S$ are $\lambda$-rings and $A$ is a sub-$\lambda$-ring of $R \otimes S$ then there exist $X, Y$ resp. sub-$\lambda$-rings of $R$ and $S$ such that $A \cong X \otimes Y$. Furthermore, $X$ is the pullback of $R \to R \otimes S$ and $U \to R \otimes S$ and $Y$ is the pullback of $S \to R \otimes S$ and $U \to R \otimes S$.

Final Conjecture: the category of sub-$\lambda$-rings of $R \otimes S$ is equivalent to the prodct category of sub-$\lambda$-rings of $R$ and sub-$\lambda$-rings of $S$.

[torstein-vik]

I believe that the first conjecture in the augmented case can be proven using a version of the monic four-lemma.

This is because we have a morphism $R \otimes S \to S$ given by $id \otimes \epsilon$ where $\epsilon$ is the augmentation. From this we can construct a composition $R \to R \otimes S \to S$ which maps $x$ to $\epsilon(x)$, which is kinda close to being exact... not really but at least if we have $0$ on the right we get that there must be something on the left, which is the most important part of the monic four-lemma.

[torstein-vik]

For the second conjecture (again $R$ and $S$ augmented), we can provide an injective homomorphism $P_1 \otimes P_2 \to A$ but I still don't know how to prove that this map is surjective (it certainly feels that way though)

The morphism is given by $P_1$ and $P_2$ both having morphisms to $A$, and then using that tensor product is the coproduct

To prove the morphism is injective, we employ a similar technique as in the previous comment, forming exact sequences: $$\mathbb{Z} \hookrightarrow R \hookrightarrow R \otimes S \to S \to \mathbb{Z}$$ $$\mathbb{Z} \hookrightarrow P_1 \hookrightarrow P_1 \otimes P_2 \to P_2 \to \mathbb{Z}$$ (where in both cases the latter two are surjective)

and then adding pullback-morphisms $P_1 \hookrightarrow R$ and $P_2 \hookrightarrow S$, as well as the universal morphism $\mu : P_1 \otimes P_2 \to A$, the given embedding $f : A \hookrightarrow R \otimes S$ and finally the pullback-morphism $P_1 \to A$. This will allow us to use a simple diagram-chase and end up with $\mu$ injective.