Subject: categories: Curious fact

From: Michael Barr <[email-masked]>
Date: Sat, 1 Apr 2000 14:44:04 -0500 (EST)

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   ...    )
          (   .     .     .   .      )
          (   .     .     .    .     )
          (   .     .     .     .    )
 then C is exact.  The proof is by using elements and although AB4 (at
least countable) is clearly a necessary condition for this (it is the
case that all the del's are 0), it sure doesn't look sufficient.

BTW, both are false if you change direct sum to product in the first or
product to sum in the second.

Michael