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Results tagged with schemes
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user 4790
The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.
12
votes
Extending vector bundles on a given open subscheme
This is false as stated; for example, if $X$ is obtained from a projective geometrically connected smooth surface over a field $k$ by gluing two points together and $U$ is the complement of the singul …
- 27k
11
votes
Accepted
Nonnegative additive functions on coherent sheaves
I suppose that "additive" means that "additive over short exact sequences". If so, this is does not seem too hard, at least if $X$ is separated.
By noetherian induction, you may assume that for all p …
- 27k
11
votes
Accepted
How does descent theory imply a sheaf is a scheme?
Anyway, the general question is: suppose that we have an fpqc covering of schemes $Y'\to Y$ and a scheme $X' \to Y'$ with descent data. When can I conclude that $X'$ descends to a scheme over $Y$? …
- 27k
8
votes
Accepted
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular...
Well, $f^{-1}Z$ could easily be non-reduced (for example, take the relative Frobenius morphism $\mathbb A^1_k \to \mathbb A^1_k$, defined by the embedding $k[y] = k[x^p] \subseteq k[x]$, where $k$ is …
- 27k
8
votes
Accepted
More on universal homeomorphisms
Let $f\colon X \to Y$ be a universal homeomorphism of locally noetherian schemes. Assume that $X$ and $Y$ are integral and $Y$ is normal, and the function field $k(Y)$ has characteristic 0. …
- 27k
7
votes
Accepted
Codimension of points in fibered products
Suppose that the codimension of $\pi(x)$ is at least two; then there exist a chain of closed irreducible subsets $V_0 \subset V_1 \subset V_2$ containing $\pi(x)$ (the inclusions are proper). The inve …
- 27k
6
votes
Accepted
Universal homeomorphisms and the étale topology
I think that the following might work. Let $X_0$ be a reduced scheme over a field $k$ of characteristic $p > 0$, and let $X$ be the product of $X_0$ with the ring of dual numbers $k[\epsilon]$. Then $ …
- 27k
5
votes
Accepted
Non-flat locus for smooth schemes
Suppose that $f \colon X \to Y$ is generically finite. Then the locus in $X$ where $f$ is not flat is the locus where the fiber is positive dimensional.
Now, let $Z$ be a projective threefold with a …
- 27k