# Tagged Questions

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### Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
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### Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$... 1answer 92 views ### A new(?) partial order on the set of continuous maps Let X,Y be topological spaces. Define a partial order on \hom(Y,X) as follows: f \leq g if f^{-1}(U) \subseteq g^{-1}(U) for all open subsets U \subseteq X. Equivalently, f(y) is a ... 1answer 111 views ### Is this square diagram cocartesian for every regular local ring? Let K be a field and R=\{f\in K[X]\mid f(0)=f(1)\} the K-algebra obtained by pulling back K[X]\to K\times K, X\mapsto (0,1) along the diagonal. Is the induced square \begin{eqnarray} ... 1answer 160 views ### What are the “correct” modules over locally ringed spaces?$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...