Lemma 3

Tag 04AH, permanent. Cite it as [Bourbaki, Tag 04AH].

Commutative Algebra, chapter II, § 4. Spectra of rings and supports of modules

Let $B$ be a local ring and $E$ and $E'$ two finitely generated $B$ -modules. If $E \neq 0$ and $E' \neq 0$ , then $E \otimes_B E' \neq 0$ .

Let $k$ be the residue field of $B$ . By virtue of § 3, no. 2, Proposition 4, $k \otimes_B E \neq 0$ and $k \otimes_B E' \neq 0$ ; then we deduce that

(k \otimes_B E) \otimes_k (k \otimes_B E') \neq 0

(Algebra, Chapter II, § 3, no. 7). But, since the tensor product is associative (loc. cit., § 3, no. 8), this tensor product is isomorphic to

E \otimes_B ((k \otimes_k k) \otimes_B E') = E \otimes_B (k \otimes_B E')

and therefore to $k \otimes_B (E \otimes_B E')$ , whence the lemma.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

§ 3, no. 8
§ 3, no. 2, Proposition 4
Algebra, Chapter II, § 3, no. 7