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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.
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
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
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
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
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
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
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