Bourbaki

Exercise 12, § 11

Tag 00E3, permanent. Cite it as [Bourbaki, Tag 00E3].

Algebra, VIII, p. 210

Let f:A→Bf: A\rightarrow B be a surjective ring homomorphism, a\mathfrak{a} its kernel, and ii the canonical injection of a\mathfrak{a} into A. Construct a homomorphism ∂:K1(B)→K0(a)\partial : K_1(B)\rightarrow K_0(\mathfrak{a}) such that the sequence

∗∗∗***
K1(A)⟶fK1(B)⟶∂K0(a)⟶iK0(A)⟶fK0(B) K_1(A)\longrightarrow^fK_1(B)\longrightarrow^{\partial}K_0(\mathfrak{a})\longrightarrow^iK_0(A)\longrightarrow^fK_0(B)

is exact (argue as in Exercise 11, c)).

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Cites

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