exact sequence of k0 and k1

Proposition

Let 20241017-exact_sequence_of_k0_and_k1_89073d11f6b947f7741a40bd20dbd20aadcac38e.svg be a ring, an ideal
Then there exists an exact sequence

20241017-exact_sequence_of_k0_and_k1_17917c12141ed1e7f452a96dcb858fe1262bfdda.svg

natural in ideal pairs of rings

where

is the relative K1-Group
is K1 of a ring,

is the boundary map from K1 to K0 of rings

Proof

exact sequence with

We want to show that

20241017-exact_sequence_of_k0_and_k1_90b963cddef7748acd89ed94b7b045ab45cd8e8f.svg

is exact.

exact at , exact at

follows from relative infinite general linear group and projection

exact at

a)

Suppose there exist a preimage 20241017-exact_sequence_of_k0_and_k1_40077a4da9a98380d4c0b436bb4bff70f0c4041f.svg .
Then

20241017-exact_sequence_of_k0_and_k1_c6a7664288f4c35849b0398801a24728d2f64314.svg

defines an isomorphism.

Thus 20241017-exact_sequence_of_k0_and_k1_375124876ca78ba85917b0ff1e9edbd987315b58.svg or

20241017-exact_sequence_of_k0_and_k1_b1bc6260c62d93104407fd0ce2bbf852930b20a2.svg

b)

Let 20241017-exact_sequence_of_k0_and_k1_d8908d3a0ba4b71e88378ce382dd925d4ee3b58f.svg .
Then by equality in a grothendieck group

20241017-exact_sequence_of_k0_and_k1_f2e8e7faedabc1098d3a1c4e7474afa274859dd1.svg

or after replacing 20241017-exact_sequence_of_k0_and_k1_b3e96ca552de3062f7a1f387a7fc086410561073.svg with we get an isomorphism

20241017-exact_sequence_of_k0_and_k1_bcab28532d655e9d0655ef7cad692f2e97a2f85e.svg

then by some computation we may find a preimage of 20241017-exact_sequence_of_k0_and_k1_b3e96ca552de3062f7a1f387a7fc086410561073.svg (omitted)

applying snake lemma

We have exact rows

20241017-exact_sequence_of_k0_and_k1_a449b581db75951312c22b48db12e171b653413c.svg

Note that 20241017-exact_sequence_of_k0_and_k1_35108dfda2297845d74bc2e9556d5e5e33086574.svg is a normal subgroup
thus we may apply the snake lemma for groups and normal subgroups for

20241017-exact_sequence_of_k0_and_k1_d7257097a2812ca467e2c25ccd482481dd6d7822.svg

and get an exact sequence

20241017-exact_sequence_of_k0_and_k1_1780de26ca4c27ae87132d68c2087ff730dc850e.svg

and an induced morphism

20241017-exact_sequence_of_k0_and_k1_cb5233228f1d744317c6ab2b537be1a45539039f.svg

Now consider the exact rows

20241017-exact_sequence_of_k0_and_k1_d7d2d060e5152d4d5d1b5876174ce000148a04ae.svg

where we may apply the cokernel to

20241017-exact_sequence_of_k0_and_k1_45ebf7bc2a07352fd7752844403fac0814c92cdc.svg

first vertically and get

20241017-exact_sequence_of_k0_and_k1_a999242b479c61c86e013d9fe9ce8e7ac8f49929.svg

and then horizontally to 20241017-exact_sequence_of_k0_and_k1_14289b4f0d9684c92b61c7633e555547dc47a71c.svg

or first horizontally

20241017-exact_sequence_of_k0_and_k1_660a38fdbf1f2b59910636b211f04bc1d7d7a911.svg

and then vertically to 20241017-exact_sequence_of_k0_and_k1_edc55356cd16852be5a70e3d10f43450b5827e0a.svg

This gives us the isomorphism since colimits commute with colimits

20241017-exact_sequence_of_k0_and_k1_1ba4366825f4bf99b71bf372b29b215135302abb.svg

naturality

follows from functorality of 20241017-exact_sequence_of_k0_and_k1_cd0ee5c2c8707837ebed2ed12e3a8b772e9e5bfa.svg & some natural transformation (omitted)