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Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \t …

Here is a (stupid in some sense) counterexample. Suppose $A$ is not trivial and $\kappa$ is a cardinal greater than the cardinality of $\textrm{Hom}_A (X, M)$ for all f.g. $A$-modules $X$ and $M$. The …

This is not true even for affine schemes. Let $k = \mathbb{Z}$, let $X = \operatorname{Spec} \mathbb{Z}$, let $Y = \operatorname{Spec} \mathbb{F}_p$, and let $Z \cong \operatorname{Spec} \mathbb{Z} [ …

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then …

One of the fundamental results in commutative algebra is the following: Let $M$ be an $A$-module. Then $M = 0$ if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m} \trianglelef …

Yes. In fact, one such example comes from homotopical algebra: Proposition. Let $\mathcal{C}$ be a small homotopical category and let $\gamma : \mathcal{C} \to \operatorname{Ho} \mathcal{C}$ be th …