Asymmetry of Ext-groups
For an algebra \(R\) over a fixed base field \(k\) and finitely generated right \(R\)-modules \(M\) and \(N\), the intersection multiplicity of \(M\) and \(N\) has been defined as \(M\cdot N :=(-1)^{GK\dim R-GK\dim M}\xi(M,N),\) where \(GK\dim M\) is the Gelfand-Kirilov dimension of \(M\) and \(\xi (M,N) = \sum _{i \in {\mathbb{Z^+}}} (-1)^{i} \dim_{k} \mathrm{Ext}_{R}^{i}(M,N)\) (where \({\mathbb{Z^+}}\) denotes the positive integers) is the Euler form of \(M\) and \(N\) [\textit{I. Mori} and \textit{S. P. Smith}, J. Pure Appl. Algebra 157, 279--299 (2001; Zbl 0976.16033)]. For a good intersection theory, the property \(M \cdot N = N \cdot M\) is desirable and is often true if both \(M \cdot N\) and \(N \cdot M\) are well-defined. The author is interested in the question when \(M \cdot N\) being well defined implies that \(N \cdot M\) is well defined and in this paper constructs counterexamples. This is done by looking at two types of symmetric behaviour of the \(\mathrm{Ext}\)-group, which together imply that \(N \cdot M\) will be well-defined if \(M \cdot N\) is: \(\bullet\) {\textbf{(EE)}}: for all finitely generated right R-modules \(M\) and \(N,\) \(\dim_{k} \mathrm{Ext}^{i}_{R} (M,N) < \infty\) for all \(i \in {\mathbb {N}}\) if and only if \(\dim_{k} \mathrm{Ext}^{i}_{R} (N,M)< \infty\) for all \(i \in {\mathbb{N}}\) \(\bullet\) \textbf{ {(ee)}}: for all finitely generated right R-modules \(M\) and \(N \) \(\mathrm{Ext}^{i}_{R} (M,N) =0\) for all \(i \gg 0\) if and only if \(\mathrm{Ext}^{i}_{R} (N,M)=0\) for all \(i \gg 0\) Techniques of noncommutative projective geometry are used to construct examples of algebras not satisfying \textbf{ {(EE)}} and \textbf{ {(ee)}}. It is shown that, in contrast to the commutative case, a simple example exists of a noncommutative noetherian Gorenstein (Frobenius)local algebra satisfying the uniform Auslander condition \textbf{ {(uac)}}: there is an integer \(d_{R} \in {\mathbb{N}}\) such that, for all finitely generated right R-modules \(M\) and \(N\) if \(\mathrm{Ext}^{i}_{R} (M,N) =0\) for all \(i \gg 0\), then \(\mathrm{Ext}_{R}^{i} (M,N) = 0\) for all \(i > d_{R}\), but not satisfying \textbf{ {(ee)}}.
scientific article; zbMATH DE number 5614711
scientific article