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Commutative Algebra

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Commutative Algebra

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Please register for the exercise class through the online system URM by October 28th.

Content of the lecture

A natural generalization of the concept of a vector space over a field is that of a module over a (commutative) ring (for example, every abelian group is a module over the ring of integers). Passing from vector spaces to modules simply means to ignore the fact that the scalars have multiplicative inverse - but the effects are "devastating": a module generally has no basis and we lose the concept of "dimension". Linear algebra was taught in the first semesters as the theory of finite-dimensional vector spaces, and much depended on the fact that the vector spaces considered had a finite dimension. In this lecture, we will learn of some "finite conditions" which generalize and replace the concept of the finite dimension (finitely generated, Noetherian, Artinian, of finite length).


The absence of a multiplicative inverse also leads to a richer structure in the rings themselves. If we consider a field as a vector space over itself, that is, we consider the elements as vectors of length one, then it has only two subspaces. However, if one takes a ring as a module over itself, it has a lot of "submodules", which are usually called ideals. Certain classes of ideals are of particular interest. The lecture will focus on maximal ideals, prime ideals and primary ideals (primary decomposition, Nilradical, Jacobson radical). These can be identified with the points of geometric objects and thus lead to a fascinating relationship between geometry and algebra, which is the subject of algebraic geometry.


Whenever you study a particular structure (e.g. groups, vector spaces, topological spaces), you have to study also the structural maps (e.g. group homomorphisms, linear maps, continuous maps). In algebra these are usually called "homomorphisms". Field homomorphisms are very restrictive. If they do not map everything to zero, they are automatically injective. This is no longer the case with ring homomorphisms. Here, again, rings allow for a greater variety, which we will consider in sections of the lecture (integral ring expansions, Noether normalization, going-up, going-down).

And finally there is the concept of localization, which is simply the concept of fractions. Just as in the school the rational numbers are introduced as fractions of integer numbers (one has to consider the rules of cancellation), in order to correct the absence of multiplicative inverses in the ring of the integers (even if no teacher said so) also allows us to create (under good conditions) fractions in other rings and to obtain interesting new structures (total quotient rings, localized rings).

The ability to decompose an integer into a product of prime numbers makes the integers incredibly friendly and useful. Thus it seems very desirable to generalize this property to other rings. Possible generalizations are factorial rings (such as the polynomial ring), Dedekind rings (which are of great interest in Number Theory), or generally the theory of primary decomposition in Noetherian rings. The latter has an interesting geometrical correspondence, namely the decomposition of a geometric objects into their irreducible components (for example, the splitting of the coordinate axes defined by the equation x·y=0 into two lines).

Dates

  • Lecture dates: (Entry in KIS)

    • Wed, 11:45 - 13:15, 48-438, Beginn: 25.10.2017
    • Fri, 10:00 - 11:30, 48-438, Beginn: 27.10.2017

  • Exercises dates: (Entry in KIS)

    • Wed, 15:30 - 17:00, 46-268, Beginn: 08.11.2017

    Teaching assistant: Laura Tozzo

Exercise sheets

Please see the exact submission date of the exercise sheets. You can submit alone or in groups of up to three students.

Nr.DownloadNotes
1Sheet 1
2Sheet 2
3Sheet 3
4Sheet 4
5Sheet 5
6Sheet 6
7Sheet 7
8Sheet 8
9Sheet 9
10Sheet 10
11Sheet 11
12Sheet 12

Credits and Exam

A regular, active and successful participation in the exercise groups is a prerequisite for obtaining an exercise certificate. This includes, in particular, that you reach at least 40% of the exercise points and present your own solutions at the blackboard.

At the end of the semester there will be an oral examinations, for which you can receive credit points.

If you have further questions, please contact your academic advisor. If you are studying mathematics, you can find it here.

Literature

Images

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Lecture notes

For the lectures there are Lecture notes with additional comments by Felix Boos. For further work we recommend the Lecture notes by Thomas Markwig.