# Calculate (co)limits as (co)equalisers (two examples)

There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.

The constructions look like these (see the links above for details):

But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.

### The connected components [Background]

For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)

# [Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

Exercise 19.5. Prove that if $H$ is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in $H$ is not compact.

Proof. Choose an orthonormal basis $\{x_i\}$, then $||x_i-x_j||^2=||x_i||^2+||x_j||^2=2$. This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.

[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.

# Lusin’s Theorem and Continuous Extension

Here we give proofs for two versions of Lusin’s Theorem, one from Exercise 44, Ch2 in Folland’s Real Analysis and the other from the textbook used for my first year undergraduate mathematical analysis course in Beijing.  The latter version is a stronger result which in addition discusses the condition for a real-valued function defined on a subset of $\mathbb{R}^n$ to be extended to the whole of $\mathbb{R}^n$. A more general result in topology is the Tietze Extension Theorem.

See the full post here: Lusin’s Theorem and Continuous Extension

Here we let $\mu$ denote the Lebesgue measure on $\mathbb{R}$.

Lusin’s Theorem (Version 1)[Exercise 2.44, Folland]. Suppose $E\subset \mathbb{R}^n$ is Lebesgue  measurable, $f: E\to \mathbb{R}$ is Lebesgue measurable and $\epsilon> 0$, there is a compact set $F\subset E$ such that $\mu(F^c)<\epsilon$ and $f|_F$ is continuous.

Lusin’s Theorem(Version 2)[Huan]. Suppose $E\subset \mathbb{R}^n$ is Lebesgue measurable and $f: E\to \bar{\mathbb{R}}$ is a Lebesgue measurable extended real valued function with $\mu(|f|=+ \infty)=0$, then  $\forall \epsilon >0$, $\exists g\in C(E)$ such that $\mu(f\neq g)<\epsilon$, where $C(E)$ denotes the space of continuous function on $E$

Continuous Extension Theorem[Huan]. Suppose $E\subset \mathbb{R}^n$, then $f$ can be extended to a continuous function on $\mathbb{R}^n$ if and only if $f$ can be extended to a continuous function on the closure $\bar{E}$ of $E$.

Tietze Extension Theorem. Let $X$ be normal and $F \subset X$ be closed and let $f: F \to R$ be continuous. Then there is a map $g: X \to R$ such that
$g(x) = f(x)$ for all $x\in F$. (Note that in topology, by a map we mean a continuous function. )

# Krull’s Principal Ideal Theorem in Dimension Theory and Regularity

This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here:  2018-2019 Ring Theory.  Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.

Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].

See here for the full post: Application of Krull’s Principal Ideal Theorem

Please also let me know if you find any errors or have suggestions on any of my posts.

# The “Dunce Cap” Space Is Contractible

Here is the exercise 6 on P. 50 in the book Topology and Geometry by Glen Bredon. I put it here because I found the drawing of this cap very lovely. Indeed I like that most of the pictures in this book are lovely sketches.

Question. The “dunce cap” space is the quotient of a triangle (and interior) obtained by identifying all three edges in an inconsistent manner. That is, if the vertices of the triangle are $p, q, r$ then we identify the line segment $(p, q)$ with $(q, r)$ and with $(p, r)$ in the
orientation indicated by the order given of the vertices. (See Figure 1-6.) Show that
the dunce cap is contractible.

Following the development in the book, I will use the following theorem that the homotopy type of a mapping cylinder or cone
depends only on the homotopy class of the map [Theorem 14.18, Topology and Geometry by Glen Bredon]. The idea is to identify the dunce cap as a mapping cone.

Theorem 14.18. If $f_0\simeq f_1:X\to Y \text{ then } M_{f_0 } \simeq M_{f_1}\text { rel } X+Y \text{ and }C_{f_0}\simeq C_{f_1}\text{ rel } Y+\mathrm{vertex}.$

Proof of the Qustion. Suppose $f: S^1\to S^1$ is a map from $S^1$ to itself. The cone $C_f=M_f/S^1 \times\{1\}$ for $f$ is obtained by pinching the top of the mapping cylinder to a point. As $M_f$  is the cylinder $S_1\times [0,1]$ with the bottom pasted to $S_1$ by the map $f$, $C_f$ is $D_2$ with $\partial D_2$ pasted to $S_1$ by the map $f$. So the dunce cap is just $C_f \text{ with } f: S^1\to S^1$ defined as

$f(e^{2\pi i t})= \begin{cases} e^{2\pi i (3t)}, 0\leq t\leq 2/3\\ e^{2\pi i(2- 3t)}, 2/3\leq t\leq 1. \end{cases}$

which is homotopic to the identity by a linear homotopy (note that we make the choice of $f$ for an easy definition of the homotopy)

$H(e^{2\pi i t},s)= \begin{cases} e^{2\pi i (3t(1-s)+st}, 0\leq t\leq 2/3\\ e^{2\pi i[(2- 3t)(1-s)+st]}, 2/3\leq t\leq 1. \end{cases}$.

So the dunce cap is homotopic to $C_{id}\simeq D^2$ which is contractible.

# [My Answer] On A Question about Quintic Del Pezzo 3-fold of Degree 5

A question from my classmate:

Question. The Quintic Del Pezzo 3-fold $V(5)$ of Degree $5$ is the intersection of $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$ with a codimension $3$ linear subspace. Show that for any point $p\in V(5)$, there is a line not passing through that point.

Proof. I posted my answer on MO here: https://mathoverflow.net/a/354809/144294. Sasha gave a short answer in the same post using the fact that the Hilbert scheme of lines on $V(5)$ has dimension $2$ (it’s isomorphic to $\mathbb{P}^2$). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of $\mathbb{P}^9$ characterising lines on $V(5)$ is $2$ by an incidence correspondence argument.

# [Soft Question] Why higher category theory?

Here is a MO post asking a question that I’ve had in mind for a while: Why higher category theory?

Having studied some elementary topos before, I have been interested in higher topos since I attended a summer school lecture last year. Besides formal generalisations, I expect to see applications which provide new results or meaningful insights. Though the meaning of “new” and “useful” very much diverse between different mathematical cultures.

Here are some important applications of higher categories in K-theory (added on Mar 05, 2020), suggested by my supervisor Schlichting.

Dustin-Mathew-Morrow, Algebraic K-theory and descent for blow-ups

Nikolaus-Scholze, On topological cyclic homology

Clausen-Mathew-Morrow, K-theory and topological cyclic homology of henselian pairs

Antieau-Gepner-Heller, K-theoretic obstructions to bounded t-structures

Blumberg-Gepner-Tabuada, A universal characterization of higher algebraic K-theory

Barwick, On exact infinity-categories and the Theorem of the Heart

# Characterisation of Quasicoherent Sheaves by Distinguished Inclusions

This is my work on the six exercises Exercise 13.3.D-13.3.I in Section 13.3.3 of Vakil’s notes. We look at a useful characterisation of quasicoherent sheaves in terms of distinguished inclusions and prove some properties in reasonable circumstances (quasicompact and quasiseparated).

Here is the pdf file: Characterisation of quasicoherent sheaves

# Generic Freeness and Chevalley’s Theorem I

This post and the next is my work on Exercises 7.4 A-7.4.O of section 7.4 in Vakil’s note. We discuss Chevalley’s Theorem and prove it using Grothendieck’s Generic Freeness Lemma.

We first discuss some properties of constructible sets and then we prove Grothendieck’s generic freeness lemma following a sequence of exercises in Vakil’s notes.  Then we use Generic Freeness to prove Chevalley’s Theorem. Though there are more direct ways to prove it, such as the proof we did in Thursday’s lecture (06/02/2020) in Applied Scheme Theory (proof of Theorem 2.2.9 in Algebraic Geometry II by Mumford and Oda). We only use Generic Freeness here as we will use it again in the future for generic flatness. Note that except proposition 1.2 the rests of the first section on the properties of constructible sets are not needed later.

We will discuss some applications of Chevalley’s Theorem including its implication of Hilbert’s Nullstellensatz in the next post: Generic Freeness and Chevalley’s Theorem II (Applications). I divided the post into two parts since the next part about applications is to be continued. The pdf file below contains the full article so far. Later more will be added into the next post.

Note that except proposition 1.2 the rests of the first section on the properties of constructible sets are not needed later.

View the full article as pdf here: Generic Freeness and Chevalley’s Theorem[16/02/2020]

As we remarked that the proof of Lemma 1.1 applied to the case that $M_{i+1}/M_{i}$ projective is the induction step of the proof of a generalised result. We include this result here. It’s not related to another part of the post so I add it as an appendix.

The reference for this proof is the lecture notes for the course Ring Theory at Warwick, 2018: Ring Theory Lecture notes 2018.