Commutative Algebra
Professor A.J. de Jong,
Columbia university,
Department of Mathematics.
This Fall semester (2019) I am teaching our graduate course
on commutative algebra and algebraic geometry.
Tuesday and Thursday, 10:10 -- 11:25 AM in room 507 math.
The TA is Carl Lian.
Grades will be based on the weekly problem sets and a final exam.
Final Exam Thursday Dec 19 from 9AM - 12PM in Room 507.
The questions will be about commutative algebra only (and not on
the material about sheaves etc from the last 3 lectures).
Review of lectures and suggested readings
- Lecture I: In this lecture we discussed Chevalley's theorem
for constructible sets in C^{n}. We used the following
definitions
- An algebraic set is a subset of C^{n} which is
the common zero set of a collection of polynomials.
- A constructible set is a subset of C^{n} which
is a finite union of sets of the form Z ∩ {f ≠ 0} where Z is
an algebraic set.
With this terminology Chevalley's theorem says that the image of a
constructible set under a polynomial map
C^{n} → C^{r} is constructible.
I sketched how to prove this theorem, but we'll redo the proof in
the setting of rings and spectra of rings later.
- Lecture II: We discussed some notational conventions.
We discussed the spectrum of a ring, its topology, and its functoriality.
We gave a couple of examples. We discussed localization of a ring at a
multiplicative subset. We discussed the equalities
Spec(A/I) = V(I) and Spec(A_f) = D(f) as topological spaces.
- Lecture III: We discussed Noetherian rings.
We poved A[[x]] is Noetherian if A is Noetherian. We showed how Spec(A)
is a Noetherian topological space if A is a Noetherian ring. We discussed
decomposition into irreducible components. We discussed how Spec(A)
is a sober topological space in the sense that any irreducible closed subset
has a unique generic point. We proved a Noetherian topological space
has a finite number of irreducible components.
- Lecture IV: We defined a constructible subset of a Noetherian
topological space. We stated and proved Chevalley's theorem in
the setting of finite type homomorphisms of Noetherian rings.
We stated some facts in field theory. Finally, we
proved Hilbert Nullstellensatz (first version).
- Lecture V: We first talked about applications of the
Hilbert Nullstellensatz: closed points map to closed points
for maps Spec(B) → Spec(A) if A → B
is a map of finite type k-algebras and we discussed that
closed points are dense in constructible subsets of Spec(A).
This then implied the radical of an ideal I of A is the intersection
of all maximal ideals of A containing I. We discussed
trascendence degree of field extensions. We discussed
Krull dimension of topological spaces. We gave some examples
of weird behaviour of dimension. We discussed dimension functions.
We stated our goal theorem: the transcendence degree over k
of the residue field of a prime of A gives a dimension function on Spec(A).
- Lecture VI: We restated the goal. We first proved some lemmas on
Noetherian topological spaces and
dimension functions
(see [1] below):
- If X is sober Noetherian, then every point specializes to a closed
point.
- If X is sober Noetherian and has a dimension function δ,
then for every specialization x ⇝ y of X there is a sequence
x = x_0 ⇝ x_1 ⇝ x_2 ⇝ ... ⇝ x_l = y
of immediate specializations with l = δ(x) - δ(y).
- If X is sober Noetherian with a dimension function δ
and x ⇝ y is a specialization in X, then every maximal chain
of specializations between x and y has the same length equal to
δ(x) - δ(y) and consists of immediate specializations.
(This follows immediately from the previous one.)
- If X is sober Noetherian and has a dimension function δ
such that δ(x) = 0 for all x in X closed, then dim(X)
is equal to the maximal value of δ which is attained at a
generic point of an irreducible component.
Let A be a finite type k-algebra where k is a field. Denote
δ : Spec(A) → Z the function sending a prime ideal p
to the transcendence degree of the residue field k(p) over k.
We proved that for any closed point p of Spec(A) we have
δ(p) = 0. Thus, if we can prove our goal, then the dimension
of Spec(A) is going to equal the maximal value of the transcendence
degree of a residue field (at a generic point of an irreducible
component). We proved that δ(p) ≥ δ(q)
if the prime p contains the prime q. We explained what we
still have to prove to attain our goal. Finally,
we stated Noether normalization lemma and we explained
a very special case.
- Lecture VII: We discussed and proved Noether normalization
modulo two lemmas. Using Noether normalization we reformulated the
goal once more. Modulo a third lemma we proved if A ⊂ B is a finite
extension of domains, then any nonzero prime of B has a
nonzero intersection with A. We proved that if p is a nonzero prime ideal
in a UFD and there is no prime strictly between p and (0), then
p is generated by a prime element. We proved that if p is
a prime ideal in k[x_1, ..., x_n] which is nonzero and there
is no prime strictly between p and (0), then trdeg_k kappa(p) = n - 1.
To finish the proof of our goal we just need to prove going down
for integral over normal. The three lemmas
- A ring map R → S such that S is generated by a finite
number of elements integral over R is finite.
- A composition of finite ring maps is finite.
- If R → S is a finite ring map, then every element of S
is integral over R.
- Lecture VIII We proved "going down for integral over normal".
We gave an example of a finite extension of domains where going down
fails. Then we started talking about tensor products.
- Lecture IX We talked more about tensor products.
Then we talked about flat modules, faithfully flat modules,
flat ring maps, faithfully flat ring maps. A usful tool was the lemma:
given a ring map A → B, a prime q of B lying over the prime
p of A, an A-module M, and a B-module N, then we have
(M ⊗_{A} N)_{q} =
M_{p} ⊗_{Ap} N_{q}.
We used this, via
Tag 00HN
to show that you can check flatness of N
by checking flatness of N_q over A_p for all q and p
(and actually it suffices to check for q maximal).
Then we finally used this to show
- A flat ring map is faithfully flat if and only if
it is surjective on spectra.
- A flat local homomorphism of local rings is
faithfully flat.
- Going down for flat ring maps.
Next time we will deduce that flat finite type ring maps between
Noetherian rings are open.
- Lecture X
We proved that flat finite type ring maps between
Noetherian rings are open. We defined the completion of
a module with respect to an ideal and we defined when
a module is I-adically complete. We proved that the
completion is complete if the ideal is finitely generated.
We discussed (failure of) exactness of completion and we proved
that if we have Artin-Rees, then we have exactness of
completion on the category of finite modules.
- Lecture XI We proved Artin-Rees. We proved some
lemmas about complete rings. We proved the completion of
a Noetherian ring is complete. We proved flatness of the
completion of a Noetherian ring. We discussed the relationship
between a Noetherian local ring and its completion. We stated
(but did not prove) the Cohen structure theorem. We discussed
its implications.
- Lecture XII In this lecture we stated the
Hilbert Syzygy Theorem. As a corollary we obtained that
the Hilbert function of a finitely generated graded module
over the polynomial ring is eventually a polynomial function.
We introduced graded rings and graded modules. We gave an
example of a module which does not have a bounded resolution
by finite free module. We defined the Koszul complex in terms
of a differential graded algebra. We proved a lemma stating
that the Koszul complex on r elements f_1, ..., f_r
is the cone of multiplication by f_r on the Koszul complex
on f_1, ..., f_{r - 1}.
- Lecture XIII We deduced from the lemma that
the Koszul complex on x_1, ..., x_n is exact in positive degrees
over the polynomial ring k[x_1, ..., x_n]. We proved that a
finite module over a connected (A_0 = k) Noetherian graded k-algebra A
has a minimal resolution by finite graded free modules.
We proved that a finite module over a Noetherian local ring
has a minimal resolution by finite free modules.
We stated some properties of the Tor functors (existence/well definedness
and symmetry) and we
deduced the Hilbert Syzygy Theorem from these properties.
- Lecture XIV We did some homological algebra.
In particular we defined the left derived functors of a
right exact functor F in the presence of a suitable collection
P of objects such that
- every object is a quotient of an element of P,
- for any short exact sequence 0 → P → Q → R → 0
with Q, R in P we have P in P and
0 → F(P) → F(Q) → F(R) → 0 is exact.
Then we applied this to the functor - ⊗_{R} N for a
fixed R-module N. Finally, we proved that
Tor is symmetric. Thus we finished the proof of
the Hilbert Syzygy Theorem.
- Lecture XV We discussed dimension theory of
Noetherian local rings and in particular the proof of
00KQ
- Lecture XVI We defined the length of a module
over a Noetherian local ring. We revisited the dimension theory
of Noetherian local rings. We introduced regular local rings.
- Lecture XVII We proved some lemmas on regular local
rings. We proved regular local rings have finite global dimension.
We proved that a localization of a ring of finite global dimension
has finite global dimension. We proved that a minimal resolution
of the residue field of a Noetherian local ring has length at
least the dimension of the cotangent space.
- Lecture XVIII We finished the proof that the localization
of a regular local ring at a prime is a regular local ring. We did this
by proving that if over a Noetherian local ring A we have a complex
0 → F_e → F_{e - 1} → ... → F_1 → F_0
consisting of finite free modules with maps which are zero modulo
the maximal ideal which is exact in degrees e, e - 1, ..., 1, and
with F_e not zero, then the dimension of A is at least e. This in turn
was done by induction showing that in this case the maximal ideal of A
has at least one nonzerodivisor and induction. The statement about
the existence of a nonzerodivisor was done using a consideration
of supports and associated primes.
- Lecture XIX In this lecture we discussed the
existence of enough injectives in the category of modules over
a ring. Then we used this to define Ext functors using Hom(-, -)
and injective resolutions in the second variable or projective
resolutions in the first variable. Then we defined depth of
a finite module over a Noetherian local ring in terms of
regular sequences. At the end of the lecture we proved
that depth(M) is the minimal i such that Ext^i(residue field, M)
is nonzero.
- Lecture XX In this lecture we first proved some
lemmas about depth culminating in the following amazing result:
the function d_M which to a prime ideal p associates
d_M(p) = depth(M_p) + dim(A/p) has the following property:
if p ⊃ q then d_M(q) ≥ d_M(p). We deduced that if for
the maximal ideal m of A we have d_M(m) = dim(Supp(M)), then we get
d_M(p) = dim(Supp(M)) for all p in the support of M.
The condition d_M(m) = dim(Supp(M)) is the same as asking
depth(M) = dim(Supp(M)) which is the same as asking M to be
Cohen-Macaulay (CM) by definition. Thus if M is CM conclude
- Ass(M) is the set of generic points of the closed set Supp(M)
- dim(A/p) = dim(Supp(M)) for all generic points p of Supp(M)
- the space Supp(M) is catenary
- M_p is CM for all primes p
- Lecture XXI In this lecture we discussed examples of
CM and non-CM rings and modules. First we gave a precise definition
of when a finite module M over a Noetherian ring R is CM and when a
Noetherian ring R is CM. Next, we defined embedded associated primes
of M and embedded primes of R. A CM module M does not have any
embedded associated primes. This give a straightforward way to make
non-CM things. A Noetherian domain of dimension 1 is CM. If k is a field then
k[x, y]/(y^2) is CM. A (global) complete intersection over k is a k-algebra
of the form R = k[x_1, ..., x_n]/(f_1, ..., f_c) satifying
the condition that dim(R) = n - c. A complete intersection over k
is CM. The quotient of a CM local ring by a regular sequence is CM.
Examples k[x, y]/(xy) and k[x, y, z]/(xy) are CM. The ring
k[x, y, u, v]/(xu, yu, xv, yv) is not CM. Interesting rings are
often CM. At the end of the lecture I stated Serre's criterion
for normality.
- Lecture XXII Serre's criterion for normality.
We first introduced the condition (S_k) for finite modules M
over Noetherian rings and condition (R_k) for Noetherian rings R.
Then we proved that M has (S_1) if and only if it has no
embedded associated primes. Then we proved R is reduced if
and only if it has (R_0) and (S_1). Next, given a Noetherian
local ring (A, m , k) the following are equivalent
- dim(A) is 1 and A is a normal domain
- A is regular of dimension 1
- A is a discrete valuation ring,
- A is a valuation ring not a field
- A is a domain and m is principal and not zero,
- A is a PID and not a field
- there is a t in m such that every nonzero element of A
can be uniquely written as (unit) t^{n}.
We defined a valuation ring. We proved the implication (1) implies (2)
using an argument of Kollar. We stated Serre's criterion: R is normal
if and only if R is (R_1) and (S_2). In the proof of Serre's criterion
we observed that Kollar's argument gives that a Noetherian
normal local ring of dimension ≥ 2 has depth ≥ 2.
We computed some examples of normalizations to show the
utility of Serre's criterion as a way to determine when
you are done.
- Lectures XXIII, XXIV, XXV We talk a little bit about
sheaves on topological spaces, pushforward and pullback of sheaves,
stalks of sheaves, sheafification, sheaves on bases, ringed spaces,
morphisms of ringed spaces, locally ringed spaces, morphisms of ringed
spaces, Spec(A) as a locally ringed space, and finally we defined schemes.
- If you intend to continue with the next part of this course next
semester, it would be great if you could read about sheaves and maybe
do some exercises to familiarize yourself with that material.
Homework: (make sure you refresh the page)
- Due Thursday, Sep 12 in lecture: All exercises from
0FJ3
and the following exercises
02CZ
02D0
02D1
02D2
from the section on the spectrum of a ring.
- Due Thursday, Sep 19 in lecture:
02D4
02D8
02DD
02DF
0767
02DZ
0FKE
- Due Thursday, Sep 26 in lecture:
0CVP
02DR
0E9F
(affine variety means irreducible algebraic set)
0D5I
- Due Thursday, Oct 3 in lecture:
0D5J
0D5K
0EER and
carefully prove Lemma 2.7 of handout
- Due Thursday, Oct 10 in lecture:
02CR
02CU
02CW
02DW
- Due Thursday, Oct 17 in lecture:
02DM
078P
078R
078S
- Due Thursday, Oct 24 in lecture:
0CYH
0CRC
0D8Y
(hint: you may use that Ext functors are right derived
functors of Hom; please recall that sometimes exercises
are too hard and you can just say something to get partial credit)
- Due Thursday, Oct 31 in lecture:
02LU
07DH
0D5M
- Due Thursday, Nov 7 in lecture:
0CT2
0D1T
0D5L
0CS1
- Due Thursday, Nov 14 in lecture:
0CR8
0CRA
02CL
(the text says this should be easy, but the second part isn't).
- Due Thursday, Nov 21 in lecture:
0CT4
0D1S
02DT
- Due Thursday, Dec 5 in lecture: none.
Sometimes an exercise may be too hard. Then just skip it and
start with the next one.
Reading materials:
- The Stacks project.
Closely related material is listed here, but be aware that the material
in the Stacks project is often more general than what we discussed in
the lecture.
- Spectrum of a ring
00DY
- Localization
00CM
- Noetherian rings
00FM
- Noetherian topological spaces
0050
- Irreducible components
004U
- Chevalley's theorem
00FE
- Hilbert Nullstellensatz
00FS
- Transcendence degree of fields
030D
- Dimension of topological spaces
0054
- Dimension functions
02I8
- Finite ring maps
0562
- Integral ring maps
00GH
- Noether normalization
00OW
- Going down for integral over normal
037E
- Finite and integral ring maps
handout
- Yoneda lemma
001L
- Limits and colimits
002D
- Adjoint functors
0036
- Abelian categories
00ZX
- Tensor products
00CV
- Checking zero locally
00EN
- Flat ring maps
000H9
- Mittag-Leffler and short exact sequences
03CA
- Completion
00M9
- Completion of Noetherian rings
0BNH
- Koszul complex
0621
- Regular sequences
0AUH
- Ext groups (for stuff about resolutions)
00LO
- Tor groups and flatness
00LY
- Dimension theory
00KD
- Jordan-Holder
0FCK
- Length
00IU
- Noetherian local rings
00K4
- Regular local rings
00NN
- Rings of finite global dimension
00O2
- Length of resolution of residue field
00OA
- Depth
00LE
- Depth (advanced)
0AVY
- What makes a complex exact?
00MR
- Supports
080S
- Associated primes
00L9
- Serre's criterion
031O
- Valuation rings
00I8
- Local complete intersections
00S8
- Polynomial rings are regular
00OQ
- The book "Algebraic Geometry" by Hartshorne (mostly for what
I will say about schemes not so much about commutative algebra)
- The book "Commutative Algebra" by Matsumura (I have the second
edition published in 1980). People call this book the "old Matsumura"
to distinguish from the next item
- The book "Commutative ring theory" by Matsumura (translated
by Miles Reid.
- The book "Commutative algebra. With a view toward algebraic geometry"
by David Eisenbud. I never read this myself, but I think this is a good
choice to look at. It has a lot of stuff in it and it is a bit more wordy
if you like that.
- The book "Introduction to commutative algebra" by Atiyah and Macdonald.
Many of those whose first introduction to commutative algebra comes
from this book, swear by this book. I think this is a psychological
effect (bc I swear by the old Matsumura).