Intereting Posts

List of Interesting Math Blogs
If a polynomial has only rational roots does that automatically mean it is solvable?
Linear homogeneous recurrence relations with repeated roots; motivation behind looking for solutions of the form $nx^n$?
Expected size of subset forming convex polygon.
How to sum 2 vectors in spherical coordinate system
Anti symmetrical relation
Is the countable union of measure-zero sets zero?
Irrationality of $ \frac{1}{\pi} \arccos{\frac{1}{\sqrt{n}}}$
Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?
Nesbitt inequality symmetric proof
An integral for $2\pi+e-9$
Is the function $F(x,y)=1−e^{−xy}$ $0 ≤ x$, $y < ∞$, the joint cumulative distribution function of some pair of random variables?
Question about Paul Erdős’ proof on the infinitude of primes
Transvection matrices generate $SL_n(\mathbb{R})$
Some co-finite subsets of rational numbers

This question came from a proof in Algebraic Geometry by Hartshorne (Chapt3, Corollary 9.6)

To be precise, Let $f:X \to Y$ be a flat morphism of schemes of finite type over a field $k$. Then is it true that the image of closed point of $X$ is also a closed point of $Y$?

Of course, one can restrict to affine schemes, say $X=\rm{Spec}A, Y=\rm{Spec}B$, and $\phi :B \to A$ is flat. Is there any lying over (or going up) property of flat map as it is in the case of integral extension? (If it has such property, one can prove the claim without difficulty).

- Why does surjectivity of the induced map show that a morphism of affine varieties has closed image?
- Best Algebraic Geometry text book? (other than Hartshorne)
- Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?
- Extension of regular function
- Open properties of quasi-compact schemes
- Degree of a Cartier Divisor under pullback

- About fibers of an elliptic fibration.
- Proof of $M$ Noetherian if and only if all submodules are finitely generated
- Help understanding Algebraic Geometry
- Surjective endomorphisms of finitely generated modules are isomorphisms
- Atiyah Macdonald Exercise 5.22
- Principal ideals having embedded components
- Why are K3 surfaces minimal?
- Neat way to find the kernel of a ring homomorphism
- Is an ideal generated by multilinear polynomials of different degrees always radical?
- Structure of Finite Commutative Rings

This is just a consequence of Nullstellensatz (the flatness is useless). If $x\in X$ is a closed point, then $k(x)$ is a finite extension of $k$ and so is $k(f(x))$ (being a subextension of $k(x)$). This is enough to show that $f(x)$ is a closed point.

If $X, Y$ are not finite type over a field, this is false even if $f$ is finite type (and faithfully flat if you like): consider a DVR $R$ with uniformizing element $\pi$, and let $f : X=\mathbb A^1_R \to Y=\mathrm{Spec}(R)$ be the canonical morphism. It is finite type and faithfully flat. The polynomial $1-\pi T\in R[T]$ defines a closed point of $X$ whose image by $f$ is the generic point of $Y$.

- An example of a bounded pseudo Cauchy sequence that diverges?
- Proving $\sum_{k=1}^n k k!=(n+1)!-1$
- Find $ ? = \sqrt {1 + \sqrt {1 + 2 \sqrt {1 + 3 \sqrt \cdots}}} $
- Applications of complex numbers to solve non-complex problems
- Prove $f(S \cup T) = f(S) \cup f(T)$
- To determine whether the integral $\int_0^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$ converges for $p>0$
- Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$
- Improper Riemann integral of bounded function is proper integral
- Let $f:\to\mathbb{R}$ be a continuous function. Calculate $\lim\limits_{c\to 0^+} \int_{ca}^{cb}\frac{f(x)}{x}\,dx$
- Elementary approach to proving that a group of order 9 is Abelian
- Uncountable injective submodule of quotient module of product of free modules
- Example of a finite non-commutative ring without a unity
- Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
- $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.
- antipodal map of complex projective space