Pushouts of noetherian rings

Question

Does the category of noetherian commutative rings have pushouts?

Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is only defined on the category of locally noetherian $S$-schemes (as far as I know). It is a group functor and in some situations it is representable. We then get a group object in the category of locally noetherian $S$-schemes, and I ask myself if it has a multiplication morphism. [Edit: Boyarsky has mentioned in the comments how to deal with this.]

Observe that the tensor product of noetherian commutative rings does not have to be noetherian (isn't this ugly?). Even for fields there is a counterexample: Let $L/K$ be a purely transcendental field extension of infinite transcendence degree. Then $\Omega^1_{L/K}$ is infinite-dimensional, from which you can concluce that the kernel of $L \otimes_K L \to L, a \otimes b \mapsto ab$ is not finitely generated. Thus $L \otimes_K L$ is not noetherian.

Of course, this does not disprove that $L \leftarrow K \rightarrow L$ has a pushout in the category of noetherian commutative rings. How can this be done? The question has a similar spirit as this one.

Answer 1

Let $k$ be a field, and let $\ell = k(x_1,x_2,\ldots)$ be the fraction field of $k[x_1,x_2,\ldots]$. Then $\ell \otimes_k \ell$ is the localisation of $k[x_1,x_2,\ldots][y_1,y_2,\ldots]$ at the multiplicative set $$S = \left\{fg\ \bigg|\ \begin{array}{ll}f \in k[x_1,x_2,\ldots]\setminus\{0\},\\g \in k[y_1,y_2,\ldots] \setminus\{0\}\end{array}\right\}.$$

Claim. The pushout $\ell \coprod_k \ell$ in the category of Noetherian rings does not exist.

Proof. If it does, it has natural maps $\iota_i \colon \ell \to \ell \coprod_k \ell$, hence a natural map $\phi \colon \ell \otimes_k \ell \to \ell \coprod_k \ell$. Consider the ideal $I = (x_1-y_1,x_2-y_2,\ldots) \subseteq \ell \otimes_k \ell$. Since $\ell \coprod_k \ell$ is Noetherian, the ideal in $\ell \coprod_k \ell$ generated by $\phi(I)$ is finitely generated. But if $\phi(I) (\ell \otimes_k \ell)$ is generated by $n$ elements, then the same is true for $\psi(I)A$ for any morphism $\psi \colon \ell \otimes_k \ell \to A$ for a Noetherian ring $A$. On the other hand, consider the ring \begin{align*} A &= (\ell \otimes_k \ell)/(x_{n+2}-y_{n+2},x_{n+3}-y_{n+3},\ldots)\\ &\cong k(z_{n+2},\ldots)[x_1,\ldots,x_{n+1},y_1,\ldots,y_{n+1}][T^{-1}]\\ &\cong k(z_{n+2},\ldots)(x_1,\ldots,x_{n+1}) \underset{{k(z_{n+2},\ldots)}}{\otimes} k(z_{n+2},\ldots)(y_1,\ldots,y_{n+1}), \end{align*} where $$T = \left\{fg\ \bigg|\ \begin{array}{ll}f \in k(z_{n+2},\ldots)[x_1,\ldots,x_{n+1}]\setminus\{0\},\\g \in k(z_{n+2},\ldots)[y_1,\ldots,y_{n+1}] \setminus\{0\}\end{array}\right\}.$$ Then $\mathfrak p = \psi(I)A = (x_1-y_1,\ldots,x_{n+1}-y_{n+1})$ is a prime ideal with $\operatorname{ht}(\mathfrak p) \geq n+1$, since we have a chain of prime ideals $$0 \subsetneq (x_1-y_1) \subsetneq \ldots \subsetneq (x_1-y_1,\ldots,x_{n+1}-y_{n+1}).$$ But now Tag 0BBZ (1) (a form of Krull's Hauptidealsatz) shows that $\mathfrak p$ cannot be generated by $n$ elements. $\square$