Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m …

Do disjoint unions and fiber products commute? In other words, is the following statement true? Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Le …

I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology. …

This is a question about a proof in Hartshorne, but let me try to formulate it without reference to Hartshorne. Let $X$ be a noetherian scheme (which is also integral, separated a …