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Regular field factors can have a nonregular tensor product

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice), used below for the regular-local-domain theorem.

False claim: if $R$ and $S$ are regular Noetherian $k$ -algebras, then so is $R \otimes_{k} S$ . Let $k$ be a field of characteristic $p > 0$ and let $L / k$ be a nontrivial finite purely inseparable extension (Purely inseparable algebraic extensions). Then $L$ is a regular Noetherian ring (it is a field), while $L \otimes_{k} L$ has a nonzero nilpotent and is therefore not reduced and not regular.

Facts & Assumptions

Given: A field $k$ of characteristic $p > 0$ , a finite purely inseparable extension $L / k$ with $L \neq k$ , and the Axiom of Choice (The Axiom of Choice).

[F1]

Purely inseparable algebraic extensions: a finite extension $L / k$ in characteristic $p > 0$ is purely inseparable when for every $α \in L$ there is $n \in N$ with $α^{p^{n}} \in k$ ; the exponent $n = 0$ is allowed, so elements of $k$ are covered.

[F2]

The binomial theorem over an arbitrary commutative ring, A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : in characteristic $p$ the binomial coefficients $(\binom{p}{i})$ , $0 < i < p$ , are divisible by $p$ , so $(X + Y)^{p} = X^{p} + Y^{p}$ in every commutative ring of characteristic $p$ , and iterating gives the same identity for the exponent $p^{n}$ .

[F3]

regular noetherian ring, embedding dimension and regular local ring: a commutative Noetherian ring is regular when each of its prime localisations is a regular local ring; a nonzero Noetherian local ring $(R, m, κ)$ satisfies $edim R = \dim_{κ} (m / m^{2})$ and is regular exactly when $edim R = \dim R$ .

[F4]

regular local domain induction: under the Axiom of Choice, every regular local ring is an integral domain.

[F5]

Universal mapping property of the tensor product of commutative algebras: $L \otimes_{k} L$ is generated as a ring by the two copies of $L$ , with $b \otimes c \mapsto b c$ defining the multiplication, so that $a \otimes 1 = 1 \otimes a$ for $a \in k$ .

Counterexample

1.1

The element $α$ and the nilpotent $u$ . Since $L \neq k$ , choose $α \in L ∖ k$ ; by [F1] there is $n \geq 1$ with $a : = α^{p^{n}} \in k$ , and we fix such an $n$ (the minimal one). Put $u : = α \otimes 1 - 1 \otimes α \in L \otimes_{k} L$ .

F1givenchoose

2.1

$u \neq 0$ . Since $L$ is finite-dimensional over $k$ and $α \notin k$ , the elements $1, α$ are $k$ -linearly independent, so the linear functional on the plane $k \cdot 1 + k \cdot α$ with $λ (1) = 1$ , $λ (α) = 0$ extends to a $k$ -linear map $λ ⁣ : L \to k$ (finite-dimensional linear algebra). If $u = 0$ , then applying $λ \otimes id$ to the identity $α \otimes 1 = 1 \otimes α$ gives $λ (α) \cdot 1 = λ (1) \cdot α$ , that is $0 = α$ , a contradiction; hence $u \neq 0$ .

step 1.1algebra

3.1

$u$ is nilpotent. By [F2], in the commutative ring $L \otimes_{k} L$ of characteristic $p$ one has $u^{p^{n}} = (α \otimes 1)^{p^{n}} - (1 \otimes α)^{p^{n}} = α^{p^{n}} \otimes 1 - 1 \otimes α^{p^{n}}$ , and this is $a \otimes 1 - 1 \otimes a = 0$ because $a \in k$ satisfies $a \otimes 1 = 1 \otimes a$ by [F5]. So $u \neq 0$ is a nilpotent element and $L \otimes_{k} L$ is not reduced.

F2F5step 1.1step 2.1algebra

4.1

$L$ is regular and $L \otimes_{k} L$ is not. The field $L$ is Noetherian and its only prime is $(0)$ , whose localisation is the field $L$ itself: a field is a regular local ring of dimension $0$ with zero maximal ideal, so $edim = \dim = 0$ by [F3], and $L$ is regular. Suppose $L \otimes_{k} L$ were regular. It is a nonzero finite-dimensional $k$ -algebra, so some maximal ideal $m$ contains the annihilator of $u$ and then $u$ has nonzero image in the localisation at $m$ ; that localisation would be a regular local ring, hence a domain by [F4], in which the nilpotent image of $u$ must vanish, a contradiction. Therefore $L \otimes_{k} L$ is not regular, so regularity of the two field factors $L$ and $L$ does not pass to their tensor product over $k$ .

F3F4step 3.1algebra∎

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Sources

Stacks Algebra 10.166.1–2 (tags 0381, 0382) (standard reference, not scraped)