Tame algebras and complete intersections
Abstract: For a bound quiver algebra A = K Q/I of a finite connected bound quiver (Q,I) over an algebraically closed field K and a nonnegative vector \bd in the Grothendieck group K_0(A) of A, we denote by \mod_A (\bd) the affine varitety of all A-modules (representations of (Q,I)) of dimension-vector \bd, and consider the natural action of the corresponding product G(\bd) of general linear groups on \mod_A (\bd). We shall discuss connections between the geometry of the module varieties \mod_A(\bd) and the representaion theory of A, and especially the tameness of A.
If I = 0, then it is well-known that the path algebra A = K Q is tame if and only if Q is a Dynkin quiver or an Euclidean quiver. Denote by \SI(Q,\bd) the algebra of semi-invariants on \mod_A(\bd) under the action of G(\bd).
Theorem [A. Skowronski--J. Weyman] The algebra A = K Q is tame if and only if the algebras \SI(Q,\bd), for all nonnegative \bd \in K_0(A), are complete intersections.
We shall show how, for A = K Q tame, the algebra structure of \SI(Q,\bd) can be described in terms of the representation theory of A.
An important role in recent investigations of finite dimensional algebras is played by quasi-tilted algebras, that is the algebras of the form A = \End_H(T), where T is a tilting object in a hereditary abelian K-category H. It has been proved by the speaker that a quasi-tilted algebra is tame if and only if the associated Euler integral quadratic form q_A is weakly nonnegative (q_A(\bd) \geq 0 for all \bd \geq 0 in K_0(A)). Moreover, if it is the case, then the dimension-vector of any indecomposable finite dimensional A-module is either a root or a radical vector of q_A.
Theorem [G. Bobinski--A. Skowronski] Let A be a tame quasi-tilted algebra and \bd the dimension vector of an indecomposable finite dimensional A-module. Then \mod_A(\bd) is a complete intersection of dimension \dim G(\bd) - q_A(\bd) and has at most two irreducible components. Moreover, \mod_A(\bd) is irreducible if and only \mod_A (\bd) is normal.
Similarly, we have
Theorem [G. Bobinski--A. Skowronski] Let \bd be the dimension-vector of a direct sum of indecomposable modules from a separating \bP_1(K)-family of stable tubes in the Auslander--Reiten quiver a tame algebra A.
Then \mod_A(\bd) is irreducible, normal, a complete intersection and \dim \mod_A(\bd) = \dim G(\bd) - q_A(\bd).