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Apr
18
comment Atiyah-Macdonald, Exercise 4.6
Related
Mar
18
comment Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$?
Well, $\mathbb R = (-\infty, 0) \cup [0, \infty)$. These intervals are connected and disjoint. You need more in order to show that each subset is a distinct component.
Mar
17
comment Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$?
You also need to show that each of $f(\mathbb R^+)$ and $f(\mathbb R^-)$ is open (or closed).
Mar
14
comment How to find the elements of a finite field?
The statement in your last paragraph is true. It's usually a theorem proved soon after introducing polynomial rings over fields. What reference are you using?
Feb
28
comment Minimal prime ideals of $\mathcal O_{X,x}$ correspond to irreducible components of $X$ containing $x$
Let $\{U_j\}$ be the irreducible components of $U$ containing $x$. The map $U_j \mapsto \overline U_j^X$ gives a bijection from $\{U_j\}$ to $\{X_i\}$ whose inverse is $X_i \mapsto U \cap X_i$. This is an exercise in point-set topology.
Feb
25
comment Compactness of $Y$ implies compactness of $X$
I half-typed an answer but euouae posted one before me. What's wrong with that proof?
Feb
21
comment Compactness of $Y$ implies compactness of $X$
This may be easier to prove with the formulation of compactness that uses the finite intersection property. Are you interested in such a proof?
Feb
17
revised Subgroups containing kernel of group morphism to an abelian group are normal.
added 14 characters in body
Feb
16
comment Field theory extensions.Proof
$K(\alpha_1, \ldots, \alpha_k)$ is the smallest subfield of $L$ that contains both $K$ and $\{\alpha_1, \ldots, \alpha_k\}$.
Feb
16
reviewed Reject What's the point in being a “skeptical” learner
Feb
14
comment cofinite topology
You don't have to look at the real line specifically. If a space has the cofinite topology, then it's compact. If the space is infinite, then it's also connected. Try to prove this.
Feb
13
reviewed Approve How do I use homomorphism theorem to show the assertion?
Feb
12
comment Why is a discrete algebraic subset of $K^n$ finite?
@Drike I don't think so. $\operatorname{Spec} R$ is always quasi-compact, regardless of whether $R$ is Noetherian or not.
Feb
11
comment Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
Hint: A space $X$ is Hausdorff iff the diagonal is closed in $X \times X$.
Feb
10
comment Hartshorne Exercise II.2.18(d)
@Manos No. $\mathfrak p$ is a point in $\Spec A$. Think of $B$ as an $A$-algebra, and localize it at the prime ideal $\mathfrak p$ of $A$. This is a valid operation that gives us the homomorphism $\varphi_\mathfrak p$. Remember that $f^\#$ is a morphism of sheaves on $\Spec A$. This is why we consider points in $\Spec A$.
Feb
10
comment Hartshorne Exercise II.2.18(d)
@Manos If we consider $B$ as an $A$-algebra via $\varphi$, then the notation $B_\mathfrak p$ makes sense.
Feb
10
reviewed Approve Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?
Feb
10
answered Hartshorne Exercise II.2.18(d)
Feb
10
comment Hartshorne Exercise II.2.18(d)
(2) is indeed superfluous. I haven't checked your proof, but note that the result quickly follows from the fact that being surjective is a local property of morphisms of sheaves and homomorphisms of algebras.
Feb
9
comment Is torus w. disc removed homotopic to klein bottle w. disc removed?
Are you familiar with the fundamental polygons of those spaces? This should enable you to prove that both spaces are homeomorphic to a wedge sum of two circles.