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Isolated singularities and existence of almost split sequences. Notes by Louise Unger. (English) Zbl 0633.13007

Representation theory II, Groups and orders, Proc. 4th Int. Conf., Ottawa/Can. 1984, Lect. Notes Math. 1178, 194-242 (1986); identical with: Representations of algebras, Proc. 4th Int. Conf., Ottawa/Can. 1984, Vol. 1, Carleton-Ottawa Math. Lect. Note Ser. 1, W5, 49 p. (1985).
[For the entire collection see Zbl 0577.00007; respectively Zbl 0562.00002.]
Let (R,m) be a regular local ring and B an R-algebra (even noncommutative) which is a finitely generated free R-module. B is called nonsingular in \(p\in Spec(R)\) if \(gl\dim (B_ p)=\dim R_ P\). Then B is an isolated singularity if B is singular \((in\quad m)\) and nonsingular in every \(p\in Spec(R)\), \(p\neq m.\)
Suppose that R is complete and let \(P_ R(B)\) be the category of all finitely generated B-modules which are free over R. Then B is an isolated singularity if B is singular and there exists an almost split sequence \(0\to A\to B\to C\to 0\) in \(P_ R(B)\) for each indecomposable A in \(P_ R(B)\) such that \(Ext^ 1_ B(Y,A)\neq 0\) for some Y in \(P_ R(B)\) and for each indecomposable C in \(P_ R(B)\) such that \(Ext^ 1_ B(C,X)\neq 0\) for some X in \(P_ R(B).\)
If \(P_ k(B)\) contains only finitely many isomorphism classes of indecomposable objects then B is an isolated singularity or nonsingular.
Reviewer: D.Popescu

MSC:

13H05 Regular local rings
13D05 Homological dimension and commutative rings
16Gxx Representation theory of associative rings and algebras
13C05 Structure, classification theorems for modules and ideals in commutative rings