In an arbitrary abelian category, does chain complex homology commute with coproduct? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:28:28Z http://mathoverflow.net/feeds/question/75795 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75795/in-an-arbitrary-abelian-category-does-chain-complex-homology-commute-with-coprod In an arbitrary abelian category, does chain complex homology commute with coproduct? Daniel Barter 2011-09-18T22:53:22Z 2011-09-19T01:41:44Z <p>On page 55 of Weibel's Introduction to homological algebra the following passage appears:</p> <blockquote> <p>Here are two consequences that use the fact that homology commutes with arbitrary direct sums of chain complexes</p> </blockquote> <p>I understand why homology commutes with arbitrary direct sums when the direct sum of a collection of monics is a monic (i.e the direct sum functor is exact) but I was under the impression that there were abelian categories where the direct sum functor is not exact. After a bit of thought, I realised that I don't know an example of an abelian category in which the coproduct functor is not exact.</p> <p>Sheaves of abelian groups on a fixed topological space give an example of an abelian category in which the product functor is not exact.</p> <blockquote> <p>Question 1: Is the passage from Weibel's book correct? If so, then why? </p> <p>Question 2: Is there an example of an abelian category where the direct sum functor is not exact?</p> </blockquote> http://mathoverflow.net/questions/75795/in-an-arbitrary-abelian-category-does-chain-complex-homology-commute-with-coprod/75809#75809 Answer by Sam Gunningham for In an arbitrary abelian category, does chain complex homology commute with coproduct? Sam Gunningham 2011-09-19T01:41:44Z 2011-09-19T01:41:44Z <p>I couldn't think of a natural example of an abelian category in which direct sums are not exact (I think this is called axiom AB4). For example, sheaves of abelian groups and R-modules both have this property. However there are natural examples of abelian categories where direct products are not exact (i.e. not satisfying AB4*), for example, the category of abelian sheaves on a space.</p> <p>Taking the opposite category of such a category will then give an example of a category not satisfying AB4 (albeit, not a very nice one).</p> <p>Once you have such an example, homology of chain complexes in this category will not commute with direct sum:</p> <p>if $A_i \to B_i$ is a sequence of monos such that $\bigoplus (f_i :A_i \to B_i)$ is not a mono, then consider the sequence of two-term complexes</p> <p>$A_i \to B_i$.</p> <p>$H^0$ of each of these complexes is zero, but $H^0$ of the direct sum is the kernel of $\bigoplus f_i$.</p> <p>Here is one way to see that Sh(X) does not satisfy AB4* (probably not the easiest). Assume for simplicity X = [0,1]. Take a finite open cover, $\mathcal U_i$ of X by balls of radius $1/i$. Let $A_i$ be the sheaf</p> <p>$\prod _{U \in \mathcal U_i} j_{U!} \mathbb Z_U$.</p> <p>This has an epimorphism to $\mathbb Z_X$, but the direct product of all of them together is not epimorphic: taking sections over any open set $V$ will kill off any $A_i$ when no $1/i$-ball contains $V$.</p> <p>I hope this is correct!</p>