Date
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What we discussed/How we spent our time
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Aug 24
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Syllabus. Text. HW. What is ``algebra''?
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Aug 26
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Reading: Review Chapter 7 of Dummit and Foote.
We discussed the process of creating
algebraic models.
The main topics and examples were
monoids and semigroups
small categories
Cayley representation theorem
rings and $k$-algebras
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Aug 26
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We continued working through the slides on
algebraization.
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Aug 31
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Rings are algebraic models for
the endomorphism structure of an abelian group.
($\textrm{End}_{\mathbb Z}(A)$).
$k$-algebras are algebraic models for
the endomorphism structure of a vextor space.
($\textrm{End}_{k}(V)$).
Commutative rings are interesting as generalizations of $\mathbb Z$
and also as auxiliary structures for studying spaces, $C(X)$.
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Sep 2
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Reading: Chapter 1 of A-M.
Discussion related to Chapter 1 of A-M.
The ring axioms are the identities satisfied by all
endomorphism structures $\textrm{End}_{\mathbb Z}(A)$.
Hom kernels correspond to ideals.
$\textrm{Idl}(R)$ is modular.
Operations on ideals: $I+J, I\cap J, IJ, (I:J)$.
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Sep 4
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We continued discussing Chapter 1 of A-M.
(Radical ideals.)
Solvability = nilpotence.
Nilradical.
Thm. If $R$ is a commutative ring and $S\subseteq R$
is a nonempty, multiplicatively closed subset not containing $0$,
then any ideal of $R$ maximal for being disjoint from $S$
is prime.
nilradical = intersection of all primes.
Stacks project link
for a meta-observation about prime ideals. |
Sep 7
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Labor Day!
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Sep 9
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We began discussing Spec$(R)$.
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Sep 11
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We continued discussing Spec$(R)$.
$\textrm{Idl}(R)/(\textrm{solvability})$ = a frame.
frames encode topological spaces.
points = characters. Can be identified with largest
element in $\chi^{-1}(\bot)$, a meet-irreducible frame element.
points can be identified back in $\textrm{Idl}(R)$
as meet-irreducible semiprime ideals ( = primes).
closed sets of topology = vanishing sets.
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Sep 14
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We began discussing spectral spaces.
Examples of spectra, including $\textrm{Spec}(\mathbb Z)$.
Nagata idealization. (Helps to produce example where
$R\not\cong S$, $\textrm{Spec}(R)\cong \textrm{Spec}(S)$.)
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Sep 16
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We finished spectral spaces.
Hochster's Theorem.
Sobriety.
Compact opens are finite unions of distinguished/principal opens.
($D(f)$ should be thought of as the support of $f$.)
Intersection of two compact opens is compact open.
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Sep 18
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We explained why Spec is a
contravariant functor.
Covariant and contravariant functors.
$\textrm{Spec}$ is the object part of a contravariant functors.
The morphism part is $F(\alpha)=\alpha^* = \alpha^{-1}$.
Contraction of a prime is prime.
Inverse image of principal open is principal open:
$(\alpha^*)^{-1}(D(f)) = D(\alpha(f))$.
Direct decompositions of $R$ correspond to
decompositions of $\textrm{Spec}(R)$ into complementary
clopen sets.
Stacks project link
for the spectrum of a ring.
Stacks project link
for open and closed subsets of spectra.
Stacks project link
for connected components of spectra.
Stacks project link
for irreducible components of spectra.
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Sep 21
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Chapter 1 Roundup!
(Especially the Jacobson radical.)
Wedderburn radical.
Jacobson radical of $R$ consists of elements that annihilate
simple $R$-modules = elements that act
nilpotently on $R$-modules of finite length.
Jacobson radical = intersection of maximal (left) ideals
= set of all $r\in R$ such that $(1+sr)$ is left invertible
for all $s\in R$.
Stacks project link
for the Jacobson radical of a ring.
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Sep 23
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Nakayama's Lemma.
(Still in these slides.)
Jacobson radical of a module.
Nontraditional proof of Nakayama's Lemma.
A Transfer Theorem from fields to commutative rings:
any positive universal sentence true in all fields
is true in all commutative rings. (E.g., Cayley-Hamilton Theorem.)
Traditional proof of Nakayama's Lemma,
using Cayley-Hamilton Theorem.
Proof that if $M$ is f.g. and $M=IM$, then
$(1+i)M=0$ for some $i\in I$.
Stacks project link
for the Cayley-Hamilton Theorem.
Stacks project link
for Nakayama's Lemma.
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Sep 25
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Reading: Chapter 2 of A-M.
Modules I.
Modules over rings and $k$-algebras.
Universal morphisms.
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Sep 28
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Modules II.
Free objects, products, coproducts in the categories
of $R$-modules, rings, and $k$-algebras.
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Sep 30
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Modules III.
Stacks project link
for products.
Stacks project link
for coproducts.
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Oct 2
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Tensor products.
A handout with more details.
Stacks project link
for tensor products.
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Oct 5
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Presentations.
Presentations are convenient.
Presentations are inconvenient.
Coproducts via presentations.
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Oct 7
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We continued with presentations.
Universal property for $\otimes$.
Not every element of $M\otimes N$ is a simple tensor.
Tensor products of vector spaces.
Testing whether two elements are equal
in $M\otimes N$.
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Oct 9
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We continued with presentations.
Testing whether $M\otimes_R N = 0$.
Tensor product of objects versus tensor product of objects.
Tensor product of rings and algebras.
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Oct 12
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We finished presentations.
Restriction and extension of scalars.
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Oct 14
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We began discussing exact sequences,
especially how they arise in connection
with the extension problem.
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Oct 16
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More on exact sequences.
Definition of a complex. Homology modules.
Factoring a complex into short complexes.
Homology of a complex can be computed from
its short factors.
Exact sequences split on the right or left.
Section versus retraction.
A SES splits iff the middle term is the biproduct
of the outer terms.
Stacks project link
for complexes.
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Oct 19
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Continuation of exact sequences.
We proved that representable functors between module
categories are additive. We proved that additive functors
preserve zero morphisms, zero objects,
direct sums, and split exact sequences.
We proved that hom functors are left exact
but not necessarily exact.
We introduced projective and injective modules.
Projective = retract of free = direct sum of free.
Injective = no proper essential extension.
Stacks project link
for preadditive and additive categories.
Stacks project link
for additive functors.
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for projectives.
Stacks project link
for injectives.
Stacks project link
for flatness.
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Oct 21
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Continuation of exact sequences.
A surprise worksheet!
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Oct 23
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Ext and Tor.
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Oct 26
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Reading: Chapter 3 of A-M.
Key points about localization.
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Oct 28
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(Unfortunately, this lecture was not recorded.)
We discussed
this worksheet.
Then we completed the
last localization slides.
Finally, we started discussing
primary decomposition.
Stacks project link
for zero divisors and the total rings of fractions.
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Oct 30
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Reading: Chapter 4 of A-M.
We continued with the
primary decomposition
slides, ending with the proof of the Krull Intersection Theorem.
Stacks project link
for the Krull intersection theorem.
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Nov 2
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(Unfortunately, this lecture was not recorded.)
We discussed
associated primes.
Stacks project link
for associated primes.
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for weakly associated primes.
Stacks project link
for embedded primes.
Stacks project link
for support and dimension.
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Nov 4
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Reading: Chapter 5 of A-M.
We started on
integral dependence.
Stacks project link
for finite and integral ring extensions.
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Nov 6
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This Chapter has a lot of interesting exercises,
and Exercise 16 (Noether Normalization) is probably the most important one.
Exercise 18 (Zariski's Lemma) is also important. Please read Exercises
1, 3, 5 as well.
We discussed the Cohen-Seidenberg Theorems.
(Lying Over, Incomparability, Going Up, Going Down)
Stacks project link
for Noether normalization.
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Nov 9
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Reading: Chapters 6 and 7 of A-M.
We started discussing chain conditions.
Stacks project link
for Noetherian rings.
Stacks project link
for more on Noetherian rings.
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Nov 11
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Reading: Chapter 8 of A-M.
We started the lecture by explaining why a complemented
modular lattice satisfies ACC iff DCC iff all chains are finite.
For such a lattice there is a number $k$ such that
all maximal chains have length $k$.
Next, we discussed the structure of commutative
Artinian rings. In particular, we argued that
a commutative Artinian ring
has nilpotent radical.
is Noetherian. (Commutative version of Hopkins-Levitski.)
has finitely many maximal ideals.
has Krull dimension zero.
is a finite product of Artinian Local rings.
Stacks project link
for Artinian rings.
Stacks project link
for length.
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Nov 13
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Reading: Chapter 9 of A-M.
We started the lecture by proving that
a commutative ring is Artinian iff
it is Noetherian of Krull dimension zero.
(Half the proof was given on November 11.)
Next, we began discussing
Dedekind domains.
Stacks project link
for Dedekind domains,
starting with Definition 10.119.12.
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Nov 16
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We finished the first set of slides on
Dedekind domains,
where the main new material was the proof that
$\mathcal{O}_K$ is a Dedekind domain when
$K$ is a finite extension of the rationals.
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Nov 18
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We started to discuss different characterizations of
Dedekind domains.
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Nov 20
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We proved that if $D$ is an integral domain,
then every fractional ideal is invertible iff $D$
is Noetherian, integrally closed, and
dimension $1$. We then proved that such a
domain has unique factorization of nonzero ideals
into prime ideals. We started proving that if
$D$ has unique factorization into prime ideals,
then all fractional ideals are invertible.
(We reduced this to proving the following statement:
If $D$ has unique factorization of nonzero ideals
into prime ideals, then every invertible prime is maximal.)
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Nov 23
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We completed the proof that a Dedekind domain
has unique factorization of ideals into primes.
We proved that PID=UFD+Dedekind.
We started discussing the
ideal class group.
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Nov 25
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We discussed how the ideal class group
of a Dedekind domain classifies its f.g.
projective modules, following the slides
about the
ideal class group.
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Nov 27
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We completed the discussion of how the
ideal class group
classifies the f.g. projective modules over a Dedekind
domain.
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Nov 30
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We began discussing the local structure
of Dedekind domains.
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Dec 2
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We started with a brief discussion
of uniserial domains,
and ended by finishing the slides about the
local structure
of Dedekind domains.
Stacks project link
for valuation rings.
Stacks project link
for a valuation ring is discrete iff its value group
is isomorphic to $\mathbb Z$.
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Dec 4
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We began a discussion
of the Nullstellensatz.
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Dec 7
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We completed the
Nullstellensatz slides.
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