November 2012) for having provided me with congenial venues for the expository
lectures that gave rise to these notes.
Further conventions. Throughout, Λ will be a basic finite dimensional algebra
over an algebraically closed field K, and J will denote the Jacobson radical of Λ.
We thus do not lose any generality in assuming that Λ = KQ/I for a quiver Q and
an admissible ideal I of the path algebra KQ. The vertex set Q0 = {e1,...,en}
of Q will be identified with a full set of primitive idempotents of Λ. Moreover, we
let Si = Λei/Jei be the corresponding representatives of the simple modules. The
absolute value of a dimension vector d = (d1,...,dn) is |d| =

We will systematically identify isomorphic semisimple modules. The top of a
(left) Λ-module M is top(M) = M/JM. The radical layering of M is the sequence
of semisimple modules S(M) =
, where L + 1 is the Loewy
length of Λ. In particular, the zero-th entry of S(M) equals the top of M.
For our present purpose, it suffices to consider classical quasi-projective vari-
eties. By a subvariety of a such a variety we will mean a locally closed subset.
2. Affine and projective parametrizations of the Λ-modules of
dimension vector d
Suppose that C is a class of objects in some algebro-geometric category, and let
be an equivalence relation on C.
Riemann’s classification philosophy in loose terms. (I) Identify dis-
crete invariants of the objects in C, in order to subdivide C into finitely many (or
countably many) subclasses Ci, the objects of which are sufficiently akin to each
other to allow for a normal form characterizing them up to the chosen equivalence.
(II) For each index i, find an algebraic variety Vi, together with a bijection
Vi ←→ {equivalence classes in Ci},
which yields a continuous parametrization of the equivalence classes of objects in
Ci. (The idea of “continuity” will be clarified in Section 4. Typically, such a
parametrization will a priori or a posteriori be a classification of normal forms.)
Once a parametrization that meets these ciriteria is available, explore potential
universal properties. Moreover, investigate the interplay between the geometry of
Vi on one hand and structural properties of the modules in Ci on the other.
We will focus on the situation where C is a class of representations of Λ. In
this situation, the most obvious equivalence relation is isomorphism, or graded iso-
morphism if applicable. Riemann’s philosophy then suggests the following as a first
step: Namely, to tentatively parametrize the isomorphism classes of modules with
fixed dimension vector in some plausible way by a variety. We will review two such
parametrizations, both highly redundant in the sense that large subvarieties map
to single isomorphism classes in general. In each case, the considered parametrizing
variety carries a morphic action by an algebraic group G whose orbits capture the
redundancy; in other words, the G-orbits are precisely the sets of points indexing
objects from the same isomorphism class of modules. Since each of these settings
will have advantages and downsides compared with the other, it will be desirable
to shift data back and forth between them. Such a transfer of information between
Scenarios A and B will turn out to be optimally smooth. We will defer a detailed
Previous Page Next Page