Elementary commutative algebra – Lecture 11-12
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Lecture plan
Lecture 11 – February 28, Mads: 4.6. The polynomial ring is factorial.
Lecture 16 – April 6, Allan: 6.3. Cayley-Hamilton’s theorem. Emil: 6.4.
Nakayama’s lemma.
Lecture 18 – April 13, Michael: 7.2. The length.
Lecture 21 – April 25, Anja: 8.1. Modules and submodules.
Lecture comments
4. Fraction constructions
4.5. Homomorphism modules of fractions
4.5.1. The identification of fraction modules with change of rings gives a homomorphism between the fraction modules of homomorphisms. Beware, this is in
general neither injective nor surjective. When we later restrict attention to modules with good finiteness conditions, then this homomorphism becomes an isomorphism. The conclusion is: tensor product modules are preserved by fractions, 4.4.4,
and homomorphism modules are preserved for “good” finite modules, 6.5.8.
4.5.2. This is an interpretation of the missing lemma in 4.2 on universality of
homomorphisms out of a fraction module.
4.5.3. The construction of induced module to the ring of fractions is quite complicated.
4.5.4. The examples showing defects in 4.5.1.
4.6. The polynomial ring is factorial
Although factorization goes back to Euclid 300 b.c. it seems first to be rigorous
with Gauss 1800. The theorem in question 4.6.3 is proved by Hensel in 1927.
4.6.2. Gauss’ lemma is a handy way of bookeeping the divisibility of the coefficients in a polynomial.
4.6.3. The proposition also determines the irreducible elements in the small ring.
This is not made explicit here.
4.6.4. Is here made a theorem in honor of the role it plays in algebraic geometry. It
means that any variety of codimension 1 in an affine space is a hypersurface given
as solutions to only one polynomial equation.
5. Localization
Now we turn to the study of prime ideals and local rings. The local-global principle
makes many arguments easier and more clear.
5.1. Prime ideals
Elementary commutative algebra – Lecture 11-12
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5.1.1-2. There are enough maximal and prime ideals in any ring. It is told, that the
theorem is equivalent with Zorn’s lemma.
5.1.3-4. This is used again and again to spot focus on one prime ideal. Do the
details in the proof and get a reward later on.
5.1.5-6. The retraction and extension of prime ideals between a ring and a ring of
fractions give a bijective correspondence of prime ideals in the ring of fractions
and the subset of retractions of prime ideals. I think the statements in the notes are
clearer than this.
5.1.7-8. The nilpotent elements are determined by the prime ideals. There is a one
to one correspondence between prime ideals in a ring and prime ideals in the factor
ring dividing out by the nilradical.
5.1.9-11. Normally there is no minimal elements in a set of ideals in a ring. But
for prime ideals there are minimal prime ideals in any prime ideal. You may also
think of this as: there some domains as maximal factor rings.
5.2. Localization of rings
Some of the ideas about local rings come from the analog principle in number
theory. Later ideas from analytic and algebraic geometry were brought in.
5.2.1. Local rings and their homomorphism are defined.
5.2.2. The statement is wrong. Correct it assuming a proper ideal.
5.2.3 Local homomorphisms give residue field extensions.
5.2.4-6. The very important construction of a local ring at a prime ideal.
5.2.7. There are three groups of local rings, which in modern commutative algebra is seen to behave quite differently. (1) This is “equal characteristic zero”. The
ring contains the field of rational numbers and the residue field is a field extension
of the rational numbers. (2) This is “unequal characteristic”. The ring has characteristic zero, but the residue field has positive characteristic. (3) This is “equal
positive characteristic”. The ring has positive characteristic and the residue field
has positive characteristic.
5.2.11-12. The local ring at a prime ideal spots focus on the prime ideals contained
in the original ideal. The local ring at a smaller prime ideal may be constructed
from the local ring of the big prime ideal.