Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg <starrgazerr@gmail.com>]

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 commonly known as an unpublished theorem of A. Joyal and M. Tierney, was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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