Dear George, First, a small correction: A@- should be considered as a functor to A-Alg, not k-Alg, in order for what I said to be correct (I thank Steve Lack for pointing that out). The square-zero extension is used to show that preservation of monomorphisms in k-Alg by A@- -- a priori a weaker condition than flatness -- actually implies preservation of monomorphisms in k-Mod, i.e flatness. After that it's the classical story you recalled. As for why - fair enough: I'm interested to know whether this property of commutative algebras is shared by other types of algebras (e.g algebras over k-linear operads, or more general algebraic theories like analytic or C-infinity rings). The fact that the coproduct coincides with the tensor product of underlying modules is a very special property of commutative algebras, so the classical proof fails already for associative algebras. So I wonder what general exactness results are available. For instance, I find Michael Barr's theorem mentioned by Richard very useful. Best, Dmitry On Fri, Oct 28, 2011 at 11:36 PM, George Janelidze <janelg@telkomsa.net> wrote:

Dear Dmitry,

Let me write then F(X) = A@X, denoting the tensor product over k by @. Hence, briefly, @ is + (coproduct) in k-Alg. The fact that, when A is flat as a k-module, A+(-) : k-Alg --> k-Alg preserves finite limits is absolutely classical, as well as its proof via modules. However, you do not need the fully faithful functor from k-Mod to k-Alg given by the square-zero extension for that: simply use the fact the forgetful functor k-Alg --> k-Mod not only preserves limits, but also reflects isomorphisms (Some people prefer to say/use "creates isomorphisms").

The preservation of finite limits by A+(-) : k-Alg --> k-Alg (already for modules) implies, for example, Grothendieck's Descent Theorem saying that if A is flat, then k --> A is an effective descent morphism. However, much stronger result is known now: k --> A is an effective descent morphism if and only if it is pure as a monomorphism of k-modules. This stronger result, commonly known as an unpublished theorem of A. Joyal and M. Tierney, was claimed several times by various authors, and I can tell you more if you are interested.

Anyway, since the category k-Alg has very bad exactness properties in a sense, modules were used by category-theorists themselves. And, in spite of nice results mentioned by Richard Gardner and Steve Lack, I still suggest not to go through a generalization of the abelian case. Well, such a suggestion surely cannot be "universal", and if you need a better suggestion, you should tell us why are you doing this - if I may say so.

Best regards,

George Janelidze

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