# 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.

# [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.

# 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.

# Generic Freeness and Chevalley’s Theorem II (Applications)

This post together with the last one is my work on the exercises 7.4 A-7.4.O of section 7.4 in Vakil’s note. This post is to be continued.

Following the last post, we now discuss some consequences of Chevalley’s Theorems.

View the full article as pdf here: Generic Freeness and Chevalley’s Theorem.

# Categorical Construction of Fibre Product of Schemes

Following our discussion of glueing schemes: Categorical descriptions for glueing sheaves and schemes. We now discuss the construction of the fibre product of schemes by glueing.

Given arbitrary schemes $X,Y,S$, let $q:X\to S$ and $r: Y\to S$ be the given morphisms. Let $\{S_i\}$ be an open affine cover of $S$. Let $X_i=q^{-1}(S_i)$, $Y_i=r^{-1}(S_i)$, choose an affine open cover $X_{ij}$ for $X_i$ and an affine open cover $Y_{jk}$ for $Y_k$. The fibre product is constructed by glueing various $X_i\times_{S_i} Y_i$  together.

We rewrite the glueing construction of fibre product in a more categorical way as follows. Note that the colimit here is glueing construction and the consequences of the two pullback squares should be clear thinking in terms of schemes.

View as pdf: Construction of fibre product

Theorem[Thm 3.3, [1]/ Thm 9.1.1, [2]] For any two schemes $X$ and $Y$ over a scheme $S$, the fibre product $X\times _S Y$ exists and is unique up to unique isomorphism.

# Variations of Yoneda Lemma; Monos, Epis and Isomorphisms of (Pre)sheaves

The first part is my work on a variation of Yoneda Lemma. The second part is my work on Exercises 2.4A, 2.4.C-2.4.D of section 2.4 in Vakil’s notes.

### 1.  Variations of Yoneda Lemmas (Monos, Epis and Isomorphisms of Presheaves)

Here are a few variations of Yoneda Lemma I played around a few years ago, which bear similar ideas of Yoneda Lemma. Recently I have been dealing with sheaves again, so I just reviewed some old stuff here. I used these variations to show that a morphism of presheaves is monic resp. epic if and only if it’s injective resp. surjective on the level of sections (For another proof see here https://stacks.math.columbia.edu/tag/00V5).

Here are the variations and proofs for presheaves: (pdf version: Variations of Yoneda lemma)

### 2.  Monos, Epis and Isomorphisms of Sheaves

Here we give a detailed discussion for sheaves, following exercises in Section 2.4 of Vakil’s notes: (pdf version: Monos, epis and isomorphisms of sheaves)

# Categorical descriptions for glueing sheaves and schemes

In this post, we give a categorical proof of Lemma 33.2 (Tag 00AK) by showing that a glued sheaf is defined as an equaliser. This equaliser provides a tool for calculating global sections of glued schemes. We later present some examples for calculating global sections and glueing constructions using (co)limits descriptions. (Sometimes people say they may lose insights about the details and the real maths behind abstraction. But it really depends on one’s approach and ways of thinking. ) The goal of this post is to give a structured summary of glueing constructions of schemes after meditating on explicit constructions, using categorical language. It is not meant to replace explicit arguments for schemes, but to give some ideas on how general a construction is and whether it can be transferred to a different setting. For example, we will know what to do if we are working on a site with a different Grothendieck topology instead of the Zariski topology. Note that some consequences of the categorical facts we use are indeed straightforward for schemes, for example, the two pullback squares in Example 4. This post is open-ended and more examples of glueing will be added.

Remark. Note that for a sheaf $\mathcal {F}$ on a topological space and an open cover $U=\bigcup U_i,$

$\displaystyle \mathcal {F}(U) =\mathrm{lim}_J \mathcal{F}(U_i),$

where $J$ is the covering sieve generated by the covering $\{U_i\}$, namely, $J$ is the collection of all those $V\subset U$ with $V\subset U_i$ for some $i.$ This limit is equivalent to the following equaliser diagram in the usual sheaf definition:

where for $t\in FU$, $e(t)=\{t|_{U_i} \mid i\in I\}$ and for a family $t_i\in FU_i, p\{t_i\}=\{t_i| _{U_i\cap U_j}\}, q\{t_i\}=\{t_j| _{U_i\cap U_j}\}.$

For a scheme $X=\bigcup X_i$ with an open cover $\{X_i\}$ in the Zariski topology, $X$ is the colimit indexed over the covering sieve generated by the covering $\{X_i\}$. This colimit can also be simplified to be a coequaliser diagram.

First, we include the explicit constructions for glueing sheaves and some sources for details check of these constructions. The categorical proof we are going to show is a categorical rephrasing by meditating on these constructions, which gives a more structured presentation.

### Glueing Morphisms

Proposition A.1  [Tag 00AK] Let $X$ be a topological space. Let $X=\cup U_i$ be an open covering. Let $\mathcal{F},\mathcal{G}$ be sheaves of sets on $X$. Given a collection

$\phi_i:\mathcal{F}|_{U_i}\to \mathcal {G}|_{U_i}$

of maps of sheaves such that for all $i,j\in I$ the maps $\phi_i,\phi_j$ restrict to the same map $\mathcal {F}_{U_i\cap U_j}\to \mathcal{G}_{U_i\cap U_j}$, then there exists a unique map of sheaves

$\phi: \mathcal {F}\to \mathcal{G}$

whose restriction to  each $U_i$ agrees with $\phi_i$.

Proof. Take any $s\in \mathcal{F}(V)$, where $V\subset X$ is open, and let $V_i=U_i\cap V$. Then we have an element $\phi_i (s| _{V_i})\in \mathcal{F}(V_i)$ and $\phi_i (s| _{V_{ij}})=\phi_j (s| _{V_{ij}})$ by the glueing condition. Thus by the sheaf condition for $G$, the sections $\phi_i(s|_{V_i})\in G(V_i)$ patch together to give a section in $G(V)$, define this section to be $\phi(s)$. (We omitted the checking details. )                                                   $\square$

### Glueing Sheaves

Explicit construction of glueing sheaves is given in Lemma 6.33.2, Tag 00AK, but the details of checking have been omitted.  For some details of a reality check,  see this post.

### Glueing Schemes

To glue schemes, one needs to define the glued topological spaces which will be a quotient space of the disjoint union of the glued spaces and then verify the structure sheaves of the glued schemes satisfy the condition of Lemma 6.33.2 [see Tag 01JA].

Example 1. (The affine line with doubled origin is not affine).   Let $k$ be a field. Let $X= \text{Spec} (k[t])$, $Y= \text{Spec}(k[u])$. Let $U=D(t) =\text{Spec}k[t,1/t]\subset X \text{ and } V= D(u)=\text{Spec}k[u,1/u]\subset Y$.  Consider the ismorphism $U\cong V$ given by $t\leftrightarrow u$.  Let $Z$ be the glued scheme, from the equaliser diagram in the proof of Lemma 6.33.2,  we see that the structure sheaf $\mathcal{O}_Z$ is given by

$\mathcal{O}_{Z}(W) = \mathcal{O}_X(W \cap X) \times_{\mathcal{O}_{X(W \cap X \cap Y)} \cong \mathcal{O}_{Y}(W \cap X \cap Y)} \mathcal{O}_Y(W \cap Y)$.

Thus the global section $\mathcal{O}_Z(Z) \text{ is } k[t] \times_{k[t,t^{-1}] \cong k[u,u^{-1}]} k[u] \cong k[t]$. From this we see that $Z$ is not affine, since  $Z=\text{Spec}(k[t])$ which is not the case: the underlying topological space $Z$ has one more point- the doubled origin.

Example 2. (Quasiseparated scheme is glued from affine schemes). Note that every scheme is a colimit of affine schemes. This is true in general by the fact that every sheaf is a colimit of representables and that the Zariski topology is subcanonical (see here for more details).

For the case that a scheme $X$ is separated (for which the intersection
of any two affine open sets is affine), take an affine open cover $\bigcup U_i=X$ such that each intersection $U_{ij}$ is affine, then $X$ is just the coequaliser of the diagram $\coprod_{i,j} U_{ij} \rightrightarrows \coprod_{i} U_i$. This is the glueing construction as we described above.

If $X$ is not separated, one can still write $X$ as a colimit of affines but not with the same diagram as we used for glueing, see here for a description of the diagram.

Remark. One implication of viewing $X$ as a colimit is: let Sch and Rings be the categories of schemes and rings respectively, given that $\text{Hom}_{\textbf{Rings}}(A,B)\cong \text{Hom}_{\textbf{Sch}}(\text{spec}(B),\text{spec}(A))$, one can deduce that for any scheme $X$$\text{Hom}_{\textbf{Rings}}(R,\Gamma(X,\mathcal{O}_x))\cong \text{Hom}_{\textbf{Sch}}(X,\text{spec}(R))$ by the fact Hom-functor preserves (co)limits.

Example 3 (Proj construction). [Section 4.5.7, Vakil]

Let $S_\bullet=\oplus _{n\in \mathbb{Z}} S_n$ be a $\mathbb{Z}$-graded ring and $S_+=\oplus_{i>0} S_i$ be the irrevalant ideal. Suppose $f\in S_+$  is homogeneous, there is a bijection between the prime ideals of $((S_\bullet)_f)_0$ and the homogeneous prime ideal of $(S_\bullet)_f$. The projective distinguished open set $D(f)= \mathrm{Proj} S_\bullet \setminus V(f)$ is identified with $\mathrm{Spec}((S_\bullet)_{f})_0$. If $f,g\in S_{+}$ are homogeneous and nonzero, $D(f)\cap D(g)= \mathrm{Spec} ((S_\bullet)_{fg})_0)$ is isomorphic to the distinguished open subset $D(g^{\mathrm{deg} f}/f^{\mathrm{deg}g})$ of $\mathrm{Spec} ((S_\bullet)_f)_0$, similarly for $\mathrm{Spec} ((S_\bullet)_g)_0$. $\mathrm{Proj}S_\bullet$ is glued from various $\mathrm{Spec}((S_\bullet)_{f})_0$ along the pairwise intersections $\mathrm{Spec}((S_\bullet)_{fg})_0$.

Example 4. (Fibre product of schemes)[For detailes of a categorical proof of this construction, see the post: Construction of Fibre Product of Schemes or the pdf here: Construction of fibre product.]

Given arbitrary schemes $X,Y,S$, let $q:X\to S$ and $r: Y\to S$ be the given morphisms. Let $\{S_i\}$ be an open affine cover of $S$. Let $X_i=q^{-1}(S_i)$, $Y_i=r^{-1}(S_i)$, choose an affine open cover $X_{ij}$ for $X_i$ and an affine open cover $Y_{jk}$ for $Y_k$. The fibre product is constructed by glueing various $X_i\times_{S_i} Y_i$  together.

# Notes and Remarks on Unstable Motivic Homotopy Theory

I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.

Affine representability results:

For remarks on section 2.3 and Lemma 2.3.2 of [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent.  By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], we show that how this implies classifying space of a stack satisfies hyperdescent.

After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.

Exercises on the construction of motivic homotopy theory:

I also have some informal notes about the exercises in [1]. The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.

[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry95, 305-370.

[2] Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology22(2), 1181-1225.

[3] Jardine, J. F. (2015). Local homotopy theory. Springer.