Let C be a chain complex over a commutative ring R. Let M be a subchain of C such that M is chain equivalent to the zero chain complex. Must C/M and C be chain equivalent ?

Kolby Castillo

Kolby Castillo

Answered question

2022-09-26

Let C be a chain complex over a commutative ring R. Let M be a subchain of C such that M is chain equivalent to the zero chain complex. Must C/M and C be chain equivalent ?

Answer & Explanation

Willie Sharp

Willie Sharp

Beginner2022-09-27Added 8 answers

Step 1
Take complexes of abelian groups:
C := 0 Z × 2 Z 0 ,
M := 0 Z × 2 2 Z 0 ,
C / M := 0 0 Z / 2 Z 0 .
Step 2
Then M is isomorphic to the complex
0 Z id Z 0 ,
which is contractible (homotopy equivalent to the zero complex), but there are no nonzero maps C / M C, so C and C/M are not homotopy equivalent.

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