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Assingments:PDF Dateien: 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 .Literature:There will be no official lecture notes to this lecture. However, you can download here the notes Simon Hampe took during the course a couple of years ago:Als weitere Literatur empfehle ich:
Content:A natural generalisation of the notion of a vector space over a field is that of a module over a (commutative) ring (e.g. every abelian group is a module over the ring of integers). Here, we merely dispense with the fact that the scalars have multiplicative inverses - but the effects are "devastating": a module generally no longer has a basis and we lose the notion of "dimension". Linear algebra was taught in the first semesters as a theory of finite-dimensional vector spaces, and much depended on the vector spaces considered having "finite dimension". We will learn in the lecture some "finiteness conditions" that generalise and replace the notion of finite dimension (finitely generated, noetherian, artinian, finite length). However, dispensing with multiplicative inverses also leads to a richer structure in the rings themselves. If one considers a field as a vector space over itself, i.e. one considers the elements as vectors of length one, then it has only two subspaces. If, on the other hand, a ring is considered as a module over itself, it usually has many "submodules", which are usually called ideals. Certain (classes) of these ideals have a special significance. The lecture will pay special attention to maximal ideals, prime ideals and primary ideals (primary decomposition, Nil radical, Jacobsen radical). These can be identified with the points of geometric objects, leading to a fascinating interrelation between geometry and algebra, which is the subject of algebraic geometry. Whenever one considers a structure (e.g. groups, vector spaces, topological spaces), one also considers structure-preserving mappings (e.g. group homorphisms, linear mappings, continuous mappings). In algebra, these are usually called "homomorphisms". Field homomorphisms are very restrictive. As soon as they do not map everything to zero, they are already injective. This is no longer the case with ring homorphisms. Again, rings allow a greater variety, which we will look at in excerpts in the lecture (integral ring extensions, Noether normalisation, going-up, going-down). Finally, there is the notion of localisation, which is simply the concept of fractions. Just as rational numbers are introduced as fractions of integers in school (keeping in mind the reduction rules) to remedy the lack of multiplicative inverses in the ring of integers (even though no teacher would say so), one can (under good conditions) allow fractions in other rings and obtain interesting new structures (quotient rings, local rings, Nakayama Lemma). The possibility of being able to decompose an integer into a product of primes makes the integer incredibly sympathetic ... and useful. Being able to usefully generalise this property to other rings therefore seems very desirable. Possible generalisations include factorial rings (such as the polynomial ring), Dedekind rings (which are of great interest in number theory) or, more generally, the theory of primary decomposition in Noetherian rings. The latter has an interesting geometric equivalent, namely the decomposition of a geometric space into its irreducible components (such as the splitting of the coordinate system defined by the equation x*y=0 into two straight lines). Keywords: Rings and ideals, modules, Nakayama lemma, localization, Noetherian and Artinian rings, primary decomposition, Noether normalization and applications (finite and integral extensions, integral closure, dimension, Hilbert's Nullstellensatz), Krull's Principle Ideal Theorem, Dedekind domains. |
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