Tagged Questions
2
votes
1answer
70 views
Does the direct image functor on sheaves reflect epimorphisms?
Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
3
votes
3answers
82 views
Does the inclusion from affine schemes into schemes preserve pushouts?
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
$$
...
3
votes
1answer
54 views
Property kept under base change and composition is preserved by products
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 ...
4
votes
1answer
164 views
Basic definition of Inverse Limit in sheaf theory / schemes
I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of ...
3
votes
1answer
85 views
Does the functor of points commutes with inverse limits?
A scheme $X$ defines a (covariant) functor from commutative rings to sets by
$A\mapsto X(A)=Mor(SpecA,X)$
Does this functor commutes with inverse limits? It does when $X$ is affine, but I can't see ...
3
votes
1answer
130 views
For which categories is the Hom functor left exact?
Sorry for this silly question but I can't find a reference.
Let $\mathcal C$ be a preadditive category and $A\in \mathcal C$ an object.
Under what conditions (on $\mathcal C$) can one say that the ...
3
votes
1answer
126 views
On limits, schemes and Spec functor
I have several related questions:
Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
