Finite Gorenstein representation type implies simple singularity. (English) Zbl 1148.14004
Over an algebraically closed field \(K\) of characteristic 0, simple singularities are classically known as hypersurfaces \(R=K[[X_1,\dots ,X_{d+1}]]/(f)\), where \(f\) is (up to contact equivalence) one of the equations ADE [cf. V. I. Arnol’d, S. M. Gusejn-Zade and A. N. Varchenko, Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Monographs in Mathematics, Vol. 82. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)]. Among many others, those are equivalent conditions for a (formal) hypersurface:
(i) \(R\) is simple.
(ii) \(R\) is of finite CM-representation type (i.e. the number of isomorphism classes of indecomposable maximal Cohen-Macaulay modules is finite).
(iii) \(R\) is of finite deformation type.
This classification was extended by G.-M. Greuel and H. Kröning [Math. Z. 203, No. 2, 339–354 (1990; Zbl 0715.14001)] to arbitrary characteristic, where the equations \(f\) look slightly different in some cases.
The article under review gives a new characterization of simple singularities. In this context, \(R\) is simple iff it is a hypersurface of finite CM-representation type.
The authors discuss the following condition: Let \(R\) be a local noetherian ring and \((\mathbf{mod}_R)\) the category of finitely generated \(R\)-modules. Let \({\mathcal G}(R)\) denote the full subcategory of modules \( M\in (\mathbf{mod}_R)\) such that there exists an exact sequence \[ \dots \to F_n\to \dots \to F_1 \to \dots \to F_0 \to M\to 0 \] with \(F_i\) f.g. free and such that the dual \[ \dots \to F_n^*\to \dots \to F_1^* \to F_0^* \] of the complex \((F_i)_{i\in\mathbb N}\) is exact (\(^*\) denotes the algebraic dual \(\text{Hom} (- ,R)\)). Such modules are said to be of Gorenstein dimension 0 (in the sense of Auslander-Bridger) or totally reflexive (in the sense of Avramov-Martsinkovsky).
The authors show: Theorem A. Let \(R\) be complete. If the set of isomorphism classes of indecomposable modules in \({\mathcal G} (R)\) different from the class of \(R\) is finite and not empty, then \(R\) is simple.
There is, in fact, a close relation to the notion of MCM-representations. Thus it is reasonable to speak of finite Gorenstein representation type in the above case.
A theorem of J. Herzog [Math. Ann. 233, 21–34 (1978; Zbl 0358.13009)] states: If \(R\) is Gorenstein and of finite MCM-representation type then \(R\) is an abstract hypersurface, i.e. \(\hat{R} \cong A/xA\) with \(A\) regular, \(x\in m_A\).
On the other hand, if \(R\) is Gorenstein, \({\mathcal G} (R)= \mathcal{MCM}(R)\). Thus theorem A is a corollary of the following which does not make any assumption on the local noetherian ring \(R\).
Theorem B. Let \(R\) be a local noetherian ring. If the set of isomorphism classes of indecomposable modules in \({\mathcal G}(R)\) is finite, then \(R\) is Gorenstein or every module in \({\mathcal G}(R)\) is free.
In this general context, the known theory of CM-approximation is not appropriate any more, but it gives an idea how to prove the above result for the Gorenstein case. The authors develop an approximation theory of modules in \((\mathbf{mod}_R)\) with respect to \({\mathcal G}(R)\). More generally, it turns out necessary to do this for any reflexive subcategory \(\mathcal B\) of \((\mathbf{mod}_R)\) i.e. for a subcategory \(\mathcal B\) with the following properties: Denote \[ {\mathcal B}^\perp := \{ L\in (\mathbf{mod}_R) \mid \text{Ext}_R^i(B,L)=0 \;\;\forall B\in {\mathcal B}, i>0 \} . \] Then
\(\bullet\) \(R\in {\mathcal B} \cap {\mathcal B}^\perp \)
\(\bullet\) \(\mathcal B\) is closed under direct sums and direct summands.
\(\bullet\) \(\mathcal B\) is closed under syzygies.
\(\bullet\) \(\mathcal B\) is closed under algebraic duality.
\({\mathcal G} (R)\) is known to be the largest reflexive subcategory of \((\mathbf{mod}_R)\), and for any reflexive subcategory \(\mathcal B\) there are inclusions \[ {\mathcal F} (R) \subseteq {\mathcal B} \subseteq {\mathcal G} (R), \] where \({\mathcal F} (R)\) denotes the full subcategory of free modules in \((\mathbf{mod}_R)\).
Using a result of Takahashi, conditions on the existence of covers are obtained: Assuming the Krull-Remak-Schmidt property for \((\mathbf{mod}_R)\), a module \(M\) has a \(\mathcal B\)-precover iff it has a \(\mathcal B\)-approximation.
There is the following relation to MCM-approximation:
\({\mathcal G}(R) \subseteq \mathcal{MCM}(R)\) iff \(R\) is Cohen-Macaulay.
Central part in the proof of Theorem B is Theorem C, which reduces Theorem B to showing that the residue field \(k\) of \(R\) has a reflexive hull.
Corollary. For a local ring \(R\) such that \({\mathcal G}(R)\) contains not only free modules, the following conditions are equivalent:
\(\bullet\) \(R\) is Gorenstein.
\(\bullet\) The residue field \(R/m\) has a \({\mathcal G}(R)\)-approximation.
\(\bullet\) All modules in \( (\mathbf{mod}_R)\) have a minimal \({\mathcal G} (R)\)-approximation.
(i) \(R\) is simple.
(ii) \(R\) is of finite CM-representation type (i.e. the number of isomorphism classes of indecomposable maximal Cohen-Macaulay modules is finite).
(iii) \(R\) is of finite deformation type.
This classification was extended by G.-M. Greuel and H. Kröning [Math. Z. 203, No. 2, 339–354 (1990; Zbl 0715.14001)] to arbitrary characteristic, where the equations \(f\) look slightly different in some cases.
The article under review gives a new characterization of simple singularities. In this context, \(R\) is simple iff it is a hypersurface of finite CM-representation type.
The authors discuss the following condition: Let \(R\) be a local noetherian ring and \((\mathbf{mod}_R)\) the category of finitely generated \(R\)-modules. Let \({\mathcal G}(R)\) denote the full subcategory of modules \( M\in (\mathbf{mod}_R)\) such that there exists an exact sequence \[ \dots \to F_n\to \dots \to F_1 \to \dots \to F_0 \to M\to 0 \] with \(F_i\) f.g. free and such that the dual \[ \dots \to F_n^*\to \dots \to F_1^* \to F_0^* \] of the complex \((F_i)_{i\in\mathbb N}\) is exact (\(^*\) denotes the algebraic dual \(\text{Hom} (- ,R)\)). Such modules are said to be of Gorenstein dimension 0 (in the sense of Auslander-Bridger) or totally reflexive (in the sense of Avramov-Martsinkovsky).
The authors show: Theorem A. Let \(R\) be complete. If the set of isomorphism classes of indecomposable modules in \({\mathcal G} (R)\) different from the class of \(R\) is finite and not empty, then \(R\) is simple.
There is, in fact, a close relation to the notion of MCM-representations. Thus it is reasonable to speak of finite Gorenstein representation type in the above case.
A theorem of J. Herzog [Math. Ann. 233, 21–34 (1978; Zbl 0358.13009)] states: If \(R\) is Gorenstein and of finite MCM-representation type then \(R\) is an abstract hypersurface, i.e. \(\hat{R} \cong A/xA\) with \(A\) regular, \(x\in m_A\).
On the other hand, if \(R\) is Gorenstein, \({\mathcal G} (R)= \mathcal{MCM}(R)\). Thus theorem A is a corollary of the following which does not make any assumption on the local noetherian ring \(R\).
Theorem B. Let \(R\) be a local noetherian ring. If the set of isomorphism classes of indecomposable modules in \({\mathcal G}(R)\) is finite, then \(R\) is Gorenstein or every module in \({\mathcal G}(R)\) is free.
In this general context, the known theory of CM-approximation is not appropriate any more, but it gives an idea how to prove the above result for the Gorenstein case. The authors develop an approximation theory of modules in \((\mathbf{mod}_R)\) with respect to \({\mathcal G}(R)\). More generally, it turns out necessary to do this for any reflexive subcategory \(\mathcal B\) of \((\mathbf{mod}_R)\) i.e. for a subcategory \(\mathcal B\) with the following properties: Denote \[ {\mathcal B}^\perp := \{ L\in (\mathbf{mod}_R) \mid \text{Ext}_R^i(B,L)=0 \;\;\forall B\in {\mathcal B}, i>0 \} . \] Then
\(\bullet\) \(R\in {\mathcal B} \cap {\mathcal B}^\perp \)
\(\bullet\) \(\mathcal B\) is closed under direct sums and direct summands.
\(\bullet\) \(\mathcal B\) is closed under syzygies.
\(\bullet\) \(\mathcal B\) is closed under algebraic duality.
\({\mathcal G} (R)\) is known to be the largest reflexive subcategory of \((\mathbf{mod}_R)\), and for any reflexive subcategory \(\mathcal B\) there are inclusions \[ {\mathcal F} (R) \subseteq {\mathcal B} \subseteq {\mathcal G} (R), \] where \({\mathcal F} (R)\) denotes the full subcategory of free modules in \((\mathbf{mod}_R)\).
Using a result of Takahashi, conditions on the existence of covers are obtained: Assuming the Krull-Remak-Schmidt property for \((\mathbf{mod}_R)\), a module \(M\) has a \(\mathcal B\)-precover iff it has a \(\mathcal B\)-approximation.
There is the following relation to MCM-approximation:
\({\mathcal G}(R) \subseteq \mathcal{MCM}(R)\) iff \(R\) is Cohen-Macaulay.
Central part in the proof of Theorem B is Theorem C, which reduces Theorem B to showing that the residue field \(k\) of \(R\) has a reflexive hull.
Corollary. For a local ring \(R\) such that \({\mathcal G}(R)\) contains not only free modules, the following conditions are equivalent:
\(\bullet\) \(R\) is Gorenstein.
\(\bullet\) The residue field \(R/m\) has a \({\mathcal G}(R)\)-approximation.
\(\bullet\) All modules in \( (\mathbf{mod}_R)\) have a minimal \({\mathcal G} (R)\)-approximation.
Reviewer: Marko Roczen (Berlin)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 13C14 | Cohen-Macaulay modules |
Keywords:
approximations; Cohen-Macaulay representation type; covers; Gorenstein dimension; precovers; simple singularity; totally reflexive modulesReferences:
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