# Shouldn't the coaddition on $\mathbb Z[x]$ be a morphism of rings?

$\operatorname{Spec} \mathbb Z[x]$ is a group scheme. Then there is a morphism of schemes: $\operatorname{Spec} \mathbb Z[x]\times\operatorname{Spec} \mathbb Z[x]\rightarrow\operatorname{Spec} \mathbb Z[x]$ induced by the coaddition $a:\mathbb Z[x]\rightarrow \mathbb Z[x] \otimes\mathbb Z[x]$ mapping $x$ to $1\otimes x+x\otimes 1$. Now, the coaddition is not a morphism of rings, since it does not respect multiplication: $a(x^2)=(1\otimes x^2+x^2\otimes 1)\neq(1\otimes x + x\otimes 1)^2=a(x)\cdot a(x)$.

But $a$ should be a morphism of rings, since the addition on $\operatorname{Spec} \mathbb Z[x]$ is a morphism of affine schemes.

-

## 1 Answer

I think you are being confused by the notation. There is a canonical isomorphism $\mathbb{Z}[t] \otimes_{\mathbb{Z}} \mathbb{Z}[t] \cong \mathbb{Z}[x, y]$ such that $t \otimes 1 \mapsto x$ and $1 \otimes t \mapsto y$. In that light, coaddition is the unique ring homomorphism $\alpha : \mathbb{Z}[t] \to \mathbb{Z}[x, y]$ such that $t \mapsto x + y$, since $\mathbb{Z}[t]$ is freely generated by $t$. In particular, $\alpha(t^2) = x^2 + 2 x y + y^2 = \alpha(t)^2$.

-

Thank you very much! – Felix Wellen Dec 1 '11 at 15:04
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9572391510009766, "perplexity_flag": "head"}