Tagged Questions
3
votes
1answer
78 views
vector bundles on the affine line over a PID
Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial?
For $R=k[X]$ this is true by the Theorem of ...
4
votes
3answers
169 views
(geometric/intuitive) interpretation of ext
In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext.
But I don't have a feeling about ...
1
vote
0answers
67 views
Characterization of Hilbert schemes of points
Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a ...
3
votes
1answer
94 views
Flat sheaves over a non-flat base $X \rightarrow Y$
I have a relatively naive question. Suppose that $f: X \rightarrow Y$ is a map of schemes. Then, we get a map of local rings $\mathcal{O}_{Y,f(x)} \rightarrow \mathcal{O}_{X,x}$ and thus for any sheaf ...
0
votes
1answer
47 views
The variety associated to a polynomial ring with a particular grading
We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety.
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7
votes
1answer
264 views
is the pushforward of a flat sheaf flat?
Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$?
What's wrong with this argument? [EDIT: as Parsa points out, the (underived) ...
15
votes
4answers
662 views
Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module
Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module?
The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
