Almost split sequences over \({\mathfrak p}\)-adic rings
Let R be a complete discrete rank one valuation ring with quotient field K, and let \(\Lambda\) be an R-order in a finite-dimensional semisimple K- algebra. \textit{M. Auslander} [in Ring Theory II, Proc. 2nd Okla. Conf. 1975, 1-44 (1977; Zbl 0361.16008)] and independently \textit{K. W. Roggenkamp} and \textit{J. W. Schmidt} [Commun. Algebra 4, 893-917 (1976; Zbl 0361.16007)] proved that almost split sequences exist in the category \(mod^ 0\Lambda\) of \(\Lambda\)-lattices. More precisely, if \(M\in mod^ 0\Lambda\) is indecomposable and not projective, these authors show that there is an almost split sequence \({\mathfrak S}: 0\to M^ +\to E\to M\to 0\) in \(mod^ 0\Lambda\), obtainable as follows. Let \(P\to ^{p_ 0}M\to 0\) be a projective cover of M in \(mod^ 0\Lambda\) and let Ṉ: mod\({}^ 0\Lambda \to mod^ 0\Lambda\) be the `Nakayama functor' (ṈX\(=Hom_ R(Hom_{\Lambda}(X,\Lambda),R)\), for \(X\in mod^ 0\Lambda)\). Since Ṉ is covariant and right exact there is an exact sequence \({\mathfrak R}: 0\to M^ +\to \underline NP\to \underline NM\to 0\) in \(mod^ 0\Lambda\), where \(M^ +:=Ker \underline Np_ 0\). The crucial fact is that \(Ext^ 1_{\Lambda}(M,M^ +)\), regarded as \((End_{\Lambda}(M),End_{\Lambda}(M))\)-bimodule, has simple socle: we then take \({\mathfrak S}\) to be the short exact sequence corresponding to (any) nonzero element of this socle. This paper begins with a short proof, based on ideas of Roggenkamp, of Auslander's natural isomorphism \(Ext^ 1_{\Lambda}(X,M^ +)\cong Hom_ R((M,X)/P(M,X),K/R)\) (here \((M,X):=Hom_{\Lambda}(M,X)\), and P(M,X) is the set of all \(f\in (M,X)\) which can be factored through a projective object of \(mod^ 0\Lambda)\). This isomorphism shows that \(Ext^ 1_{\Lambda}(M,M^ +)\) has simple socle; it is implicit in Auslander [loc. cit.], but there does not seem to be an explicit direct proof in the literature. In the present paper it is used to give a criterion for the short exact sequence \({\mathfrak R}\theta\) obtained from \({\mathfrak R}\) (see above) by pullback along a given element \(\theta \in Hom_{\Lambda}(M,\underline NM)\), to be almost split. When \(\Lambda\) is a symmetric Gorenstein order (and in particular when \(\Lambda =RG\), G a finite group) this criterion takes a very simple form. In this case there is a natural isomorphism ṈM\(\cong M\), and \({\mathfrak R}\) is isomorphic to the minimal projective presentation \({\mathfrak P}: 0\to \Omega M\to P\to M\to 0\). Taking \(\Lambda =RG\), the criterion now reads: Let \(\phi \in End_{RG}(M)\). Then \({\mathfrak P}\phi\) is almost split if and only if (i) \(Tr_ M(v\phi)\not\in | G| \cdot R\) for some \(v\in End_{RG}(M)\), and (ii) \(Tr_ M(v\phi)\in | G| \cdot R\) for all \(v\in J(End_{RG}(M))\). Here \(Tr_ M: End_{RG}(M)\to R\) is the usual trace map, and \(J(End_{RG}(M))\) is the radical of \(End_{RG}(M)\).
R-order in a finite-dimensional semisimple K-algebra
scientific article; zbMATH DE number 4023500
scientific article