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tensor products of algebras over a commutative ring

LEMMA 6.3   Let $ A$ be a commutative ring. Let $ B,C$ be $ A$ -algebras. Then the followings are true.
  1. The module

    $\displaystyle B\otimes_A C
$

    carries a natural structure of $ A$ -algebra.
  2. There exits $ A$ -algebra homomorphisms

    $\displaystyle \iota_B: B \ni b \mapsto b\otimes 1 \in B\otimes_A C,\qquad
\iota_C: C \ni c \mapsto 1\otimes c \in B\otimes_A C.
$

  3. The triple $ (B\otimes_A C, \iota_B,\iota C)$ has the following universal property: For any $ A$ -algebra $ D$ and for any $ A$ -algebra homomorphisms $ f:B\to D$ and $ g:C\to D$ , there exists a unique $ A$ -algebra homomorphism

    $\displaystyle h:B\otimes_A C \to D
$

    such that $ f=h\circ \iota_B $ and $ g=h \circ \iota_C$ .

PROOF.. An easy exercise.

$ \qedsymbol$

Note that in the situation of the above Lemma, if $ A$ is non commutative, then $ B\otimes_A C$ may not have a natural structure of ring. Sooner or later one needs to face this fact.


next up previous
Next: fiber products of schemes Up: fiber product Previous: definition of a fiber
2007-12-11