Bourbaki

Corollary 2

Tag 05VA, permanent. Cite it as [Bourbaki, Tag 05VA].

Algebra, chapter II, § 8. Restriction of the field ofscalars in vector spaces

Let f:V1→V2f : V_1 \to V_2 be a K\mathbf{K}-linear mapping rational over K′\mathbf{K}'. For every vector sub-K\mathbf{K}-space W1W_1 of V1V_1 (resp. W2W_2 of V2V_2) rational over K′\mathbf{K}', f(W1)f(W_1) (resp. f−1(W2)f^{-1}(W_2)) is a vector sub-K\mathbf{K}-space of V2V_2 (resp. V1V_1) rational over K′\mathbf{K}'.

In the notation of Proposition 3, for every vector sub-K′\mathbf{K}'-space W1′W'_1 of V1′V'_1, f(K)′(W1′⊗K′K)=f′(W1′)⊗K′Kf'_{(\mathbf{K})}(W'_1 \otimes_{\mathbf{K}'} \mathbf{K}) = f'(W'_1) \otimes_{\mathbf{K}'} \mathbf{K}; whence the assertion relating to W1W_1 (no. 2, Proposition 2). On the other hand, let W2′W'_2 be a vector sub-K′\mathbf{K}'-space of V2′V'_2 and let g′g' be the canonical K′\mathbf{K}'-linear mapping V2′→V2′/W2′V'_2 \to V'_2/W'_2; then f′(W2′)=Ker⁡(g′∘f′)f'(W'_2) = \operatorname{Ker}(g' \circ f'); hence, by Corollary 1,

f(K)′−1(W2′⊗K′K)=f′−1(W2′)⊗K′K, {f'_{(\mathbf{K})}}^{-1}(W'_2 \otimes_{\mathbf{K}'} \mathbf{K}) = {f'}^{-1}(W'_2) \otimes_{\mathbf{K}'} \mathbf{K},
whence the assertion relating to W2W_2.

Let V1,V2V_1, V_2 be two right vector K\mathbf{K}-spaces with K′\mathbf{K}'-structures V1′,V2′V'_1, V'_2 respectively. It is immediate that V1′×V2′V'_1 \times V'_2 is a K′\mathbf{K}'-structure on V1×V2V_1 \times V_2, called the product of the K′\mathbf{K}'-structures V1′V'_1 and V2′V'_2.

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

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Cites

Transcribed from the printed edition, 1998, Springer. Not translated.