1 Apr 21:44 2000

### Curious fact

Michael Barr <barr <at> barrs.org>

2000-04-01 19:44:04 GMT

2000-04-01 19:44:04 GMT

Has anyone seen this before, or can anyone give a principled explanation. Begin with the following observation. Suppose del del ... -----> (C_n,d) -----> (C_{n-1},d) ---> ... ---> (C_0,d) --> 0 is a chain complex of differential abelian groups, except that we assume d.del = -del.d. Suppose (C_n,d) is exact for each n. Let C be the direct sum of the C_n with boundary given by the matrix ( d 0 0 ... ) ( del d 0 ... ) ( 0 del d ... ) ( . . . . ) ( . . . . ) ( . . . . ) The finite truncations F^n = C_0 +...+ C_n are readily shown to be exact: F^0 = C_0 is by hypothesis and there is an exact sequence 0 --> F^{n-1} --> F^n --> C_n --> 0. AB5 now shows that the direct limit, which is C, is exact. Of course, this is going to be false for a category, such as compact abelian groups, that is not AB5. Except it is true. Using duality, we can translate it to the following statement for abelian groups. Suppose del del ... <----- (C_n,d) <----- (C_{n-1},d) <--- ... <--- (C_0,d) <-- 0 is a cochain complex of differential abelian groups, except that we assume d.del = -del.d. Suppose (C_n,d) is exact for each n. Let C be the direct product of the C_n with boundary given by the matrix ( d del 0 ... ) ( 0 d del ... ) ( 0 0 d ... ) ( . . . . )(Continue reading)