Résume | The notion of a ``matroid'' axiomatizes the linear algebra of a list of vectors. Matroid theory has proved to be a versatile language to deal with many problems on the interfaces of combinatorics, geometry and algebra. Many variants of this structure has been introduced: for example, a ``quasi-arithmetic matroid'', axiomatizing the linear algebra and the arithmetics of a list of elements of a finitely generated abelian group. After presenting these objects in a very elementary and self-contained way, we will introduce the notion of a ``matroid over a ring'', which is far more general and algebraic. Finally we will define and compute the Tutte-Grothendieck group of matroids over a Dedekind ring, and show several applications and examples. (Joint work with Alex Fink). |