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I try to compute a pushout in the category of commutative $A$-algebras, where $A$ is a commutative ring with unity. My question is if there is some abstract nonsense which gives me a simple description of such a pushout, for instance in terms of some tensor products, or some colimit in the category of commutative rings.

In fact, in my case, I have some special situation which I think should make the problem trivial : the target of one of the morphisms is $A$ itself, and the other morphism factors through $A$. Geometric intuition suggests me that the pushout is just trivial, i.e. isomorphic to $A$ itself. But I cannot write down the proof.

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The category of commutative $A$-algebras is isomorphic to the (co)slice category ${}^{A /} \mathbf{CRing}$. As such, the forgetful functor ${}^{A /} \mathbf{CRing} \to \mathbf{CRing}$ creates all connected colimits. In particular, it creates pushouts; in other words, pushouts in ${}^{A /} \mathbf{CRing}$ are the same as in $\mathbf{CRing}$, where they are computed by tensor product.

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  • $\begingroup$ Thanks, I did not know this ! $\endgroup$ Mar 31, 2015 at 19:40

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