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.

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.