# $Ext$ preserves products

Let $\mathcal{A}$ be an $Ab4$ category. Define $\mathrm{Ext}^{n}(-,-):\mathcal{A}^{op}\times\mathcal{A}\rightarrow Ab$ bifunctor using n-extensions. Consider $A$ an object of $\mathcal{A}$. Does $\mathrm{Ext}^{n}(-,A)$ preserve products?

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This looks a lot like homework, and anyways is not a research-level math question. – Will Sawin Jun 4 '12 at 18:23
This is not a homework. – anonymous Jun 4 '12 at 18:28
It seems to me that this interesting question has been misunderstood so far. Let me rephrase it as follows: Assume we are in an abelian category $\mathcal{A}$ satisfying (AB4). Define $\mathrm{Ext}^n(-,-) : \mathcal{A}^{op} \times \mathcal{A} \to \mathrm{Ab}$ not as a derived functor (which is not possible since we didn't assume that $\mathcal{A}$ has enough injectives or projectives), but rather via n-extensions (see e.g. Wikipedia en.wikipedia.org/wiki/Ext_functor for a summary of this construction). Do we have $\mathrm{Ext}^n(\oplus_i A_i,A) \cong \prod_i \mathrm{Ext}^n(A_i,A)$? – Martin Brandenburg Jun 4 '12 at 20:33
Yes, it does. I trust that this is not homework; still, it is a good excercise in Yoneda Ext, and a not very difficult one. Just construct maps in both directions between the Ext from a direct sum and the product of the Exts; then check straightforwardly from the definition that both compositions are the idenitity maps (of sets). Let me know if any further directions are needed. – Leonid Positselski Jun 4 '12 at 21:21
You mention AB4, which is about infinite products/coproducts. Are you interested in finite products or infinite products? – David Speyer Jun 5 '12 at 13:39