Download Lecture plan Lecture comments 4. Fraction constructions

Elementary commutative algebra – Lecture 9-10
1
Lecture plan
Lecture 9 – February 21, Anne: 4.1 Rings of fractions.
Lecture 10 – February 23, Emil: 4.3. Exactness of fractions.
Lecture 11 – February 28, Mads: 4.6. The polynomial ring is factorial.
Lecture 18 – April 13, Michael: 7.2. The length.
Lecture 21 – April 25, Anja: 8.1. Modules and submodules.
Lecture comments
4. Fraction constructions
4.1. Rings of fractions
The introduction of fractions seems to be as early as 1926 for domains by Grell
and around 1944 in general by Chevalley.
4.1.1-2. Definition of rings of fractions. There are two points of view built into
the definition. First two fractions are equal if the usual cross multiplication rule
for nominators and denominators is satisfied. Second you are allowed to cancel
a common factor in nominator and denominator. In a domain, the second rule is
automatically satisfied. When you have verified that the usual rules for fractions
are well defined, then you verify the conditions for a ring in the same way you once
convinced yourself that the field of rational numbers is a ring.
4.1.3. The ring of fractions is the “smallest” ring in which the given elements are
invertible. This is viewed as a universal property. This constructs homomorphisms
out of a ring of fractions uniquely.
4.1.4. We allow zero divisors to be inverted, so the ring of fractions is in general
not a ring extension. That is, the ring cannot be identified as a subring of the ring
of fractions.
4.1.5-7. For a domain we get a field of fractions. A construction analog to the
construction of the rational numbers starting with the ring of integers.
4.1.8-9. When we invert all nonzero divisors we get the total ring of fractions. This
is the “largest” ring of fractions in which the ring is identified as a subring. It works
as a substitute for the field of fractions of a domain.
4.1.10. If you invert only one element (and powers automatically), then the ring of
fractions is of finite type, 1.6.11. This is a quite important observation for many
applications. Note that this is seldom the case in general. E. g. the rational numbers
is not of finite type over the integers.
Elementary commutative algebra – Lecture 9-10
2
4.2. Modules of fractions
There is a functorial construction from modules of a ring to modules over the ring
of fractions.
4.2.1-2. Definition of modules of fractions. The construction of the module of
fractions is analog to the ring of fractions, 4.1.1-2.
4.2.3-4. The construction is functorial. So finite direct sums and products are
preserved.
4.2.5. The canonical homomorphism of a module to the module of fractions is a
natural map from the identity functor to the fraction functor. Notice this as an easy
way to remember many commutative diagrams.
4.2.6. In general the canonical homomorphism is not injective. Remark that this is
also a problem for modules over domains, in contrast to the situation for the ring
itself, 4.1.5.
4.2.7. The fraction functor preserves direct sums of any family. This is not true for
direct products. A lemma reflecting the later proposition 4.5.2 on universality of
homomorphisms out of a module of fractions is comfortable for the proof
4.3. Exactness of fractions
4.3.1-2. The fraction functor preserves exactness.
4.3.3-4. Any exact functor respects the constructions of kernel, cokernel and image. It follows that formation of sum, factor and intersection also is preserved by
the fraction functor.
4.3.5. The interpretations for ideals of exactness shall be widely used from now
and later on. Be sure to master this.
4.3.6. Playing back and forth between ideals in the ring and extended ideals in the
ring of fractions will be a major tool . Note that ideals in the ring of fractions are
determined by ideals in the ring.
4.3.7-8. When we restrict attention to prime ideals, this proposition allows us to
reconstruct ideals in the ring from ideals in rings fraction. This is the method of
localization in the next chapter.
4.3.9. Principal ideal domains are preserved by fractions.
4.3.10. Unique factorization domains are preserved by fractions.
4.4. Tensor modules of fractions
4.4.1-2. The module of fractions is shown to be exactly the same as change of
rings to the ring of fractions. So all the properties of change of rings are at our
disposable for the fraction functor.
4.4.3. The exactness 4.3.1 gives us that change of rings to the ring of fractions is a
flat change of rings.
4.4.4. Since change of rings respect tensor product, so does the fraction functor.
4.4.5. Just one of several identities that may easily be derived by the general formalism now.