Abstract
Let be an artin algebra. In his seminal Philadelphia Notes published in 1978, Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these investigations (they also form part of the final chapter of the Auslander–Reiten–Smalo book and could and should be seen as its culmination). The theory presented by Auslander has to be considered as an exciting frame for working with the category of -modules, incorporating all what is known about irreducible maps (the usual Auslander–Reiten theory), but the frame is much wider and allows for example to take into account families of modules—an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of -modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of “morphisms being determined by modules”. Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander’s powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander’s ideas and to show the wealth of these ideas by exhibiting many examples.
Highlights
There are two basic mathematical structures: groups and lattices, or, more generally, semigroups and posets
A first glance at any category should focus the attention on these two structures: to symmetry groups, as well as to the posets given by suitable sets of morphisms, for example by looking at inclusion maps, or at the possible factorizations of morphisms
Its aim is to report on the work of Auslander in his seminal Philadelphia Notes published in 1978
Summary
There are two basic mathematical structures: groups and lattices, or, more generally, semigroups and posets. Since the concept of “right determination” looks (at least at first sight) technical and unattractive, let us first describe the set C [→ Y only in the important case when C is a generator: in this case, C [→ Y consists of the (right equivalence classes of the right minimal) maps f ending in Y with kernel in add τ C (we denote by τ = D Tr and τ − = Tr D the Auslander–Reiten translations). The quiver Grassmannians Ge (Hom(C, Y )) corresponds under the Auslander bijection ηCY to the set C [→ Y e of all right equivalence classes of right C-determined maps which end in Y and have type e. In Parts III, we discuss some special cases; these are the Sects. 16 to 18
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