The Stacks project

Lemma 15.89.12. Assume $\varphi : R \to S$ is a flat ring map and let $I = (f_1, \ldots , f_ t) \subset R$ be an ideal. Then $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ is an abelian category, and the functor $\text{Can}$ is exact and commutes with arbitrary colimits.

Proof. Given a morphism $(\varphi ', \varphi _ i) : (M', M_ i, \alpha _ i, \alpha _{ij}) \to (N', N_ i, \beta _ i, \beta _{ij})$ of the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ we see that its kernel exists and is equal to the object $(\mathop{\mathrm{Ker}}(\varphi '), \mathop{\mathrm{Ker}}(\varphi _ i), \alpha _ i, \alpha _{ij})$ and its cokernel exists and is equal to the object $(\mathop{\mathrm{Coker}}(\varphi '), \mathop{\mathrm{Coker}}(\varphi _ i), \beta _ i, \beta _{ij})$. This works because $R \to S$ is flat, hence taking kernels/cokernels commutes with $- \otimes _ R S$. Details omitted. The exactness follows from the $R$-flatness of $R_{f_ i}$ and $S$, while commuting with colimits follows as tensor products commute with colimits. $\square$


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