A new proof of Gabriel's Lemma

Abstract

The technical result [2, Lemma 3.2] of Gabriel, called often Gabriel's Lemma (for the precise formulation see Section 2), played a crucial role in proofs of two famous theorems in representation theory of algebras: the theo- rem of Gabriel on openness of the set of ¯nite representation type algebras in a variety of all algebras with a ¯xed dimension (see [2]) and the Geiss Theorem saying that degeneration of a wild algebra is also wild (see [3]). The original proof of Gabriel's Lemma is rather involved and uses geometry of schemes. An alternative proof, proposed by H. Kraft for the case of characteristic 0, ap- plies essentially invariant theory and geometric quotients (see [5]). We present here a new, quite simple proof, which uses only basic projective geometry and adapts some arguments presented in [1].