Maximal modifications and Auslander-Reiten duality for non-isolated singularities.

*(English)*Zbl 1308.14007The representation theory of commutative algebras is based on Auslander-Reiten (AR) duality for isolated singularities. This article establishes a version of AR duality for singularities with one-dimensional singular loci. Then the stable categories of Cohen Macaulay (CM) modules over Gorenstein singularities with one dimensional singular loci have a generalized Calabi-Yau property, meaning that it is possible to apply the methods of cluster tilting (CT) in representation theory to study singularities. Of particular interest is the AR theory of simple surface singularities which have only finitely many indecomposable CM modules, and the AR algebras \(\text{End}(\oplus_{M\text{ ind. CM}}M)\) are polite. This is not the case for singularities of dimension greater than two, then the representations are not finite, and the AR-algebras are not nice. To obtain the correct category on which higher AR theory is good, O. Iyama [Adv. Math. 210, No. 1, 22–50 (2007; Zbl 1115.16005)] introduced a notion of a maximal \(n\)-orthogonal subcategory and maximal \(n\)-orthogonal module for the category mod \(\Lambda\), called CT subcategories and CT modules. In the case of a higher dimensional CM singularity \(R\), the definition of a maximal \(n\)-othogonal subcategory and modules is applied to CM \(R\) and it is hoped that this handles higher-dimensional geometric problems. Strong evidence is given when \(R\) is a three dimensional normal isolated Gorenstein singularity: Then it is known that such objects have a relationship with noncommutative crepant resolutions (NCCR) given by M. Van den Bergh [Duke Math. J. 122, No. 3, 423–455 (2004; Zbl 1074.14013)].

The study if maximal \(n\)-orthogonal modules in CM \(R\) is not suited for studying non-isolated singularities because the Ext vanishing condition is too strong. So one needs a subcategory of CM \(R\) that can take the place of the maximal \(n\)-orthogonal subcategories. This is the main goal this article, but it gives more: The authors develop a theory which can deal with singularities in the crepant partial resolutions. As the endomorphism rings of maximal orthogonal modules have finite global dimension, these cannot work for the purpose. The notion of maximal modifying modules are introduced as corresponding to shadows of maximal crepant partial resolutions. This level always exist geometrically, but it can happen to be smooth. The level of generality is thus to view the case when the geometry is smooth as a coincidence: everything that can be done with NCCRs should be possible to do with maximal modifying modules.

When curves are flopped between varieties with canonical singularities which are not terminal, this doesn’t take place on the maximal level. Anyway, it should be homologically understandable. That is the motivation for defining modifying modules. This also leads to the development of the theory of mutation for modifying modules.

The authors remark that some parts of the theory of (maximal) modifying modules are analogues of cluster tilting theory, particularly for rings of Krull dimension three. Some of the main properties of CT theory are still true in the present setting, but new features also appear, e.g. a mutation sometimes does not change the given modifying module, which is a necessary feature since it reflects the geometry of partial crepant resolutions.

The present theory depends on a more general noncommutative setting. The authors use the language of singular Calabi-Yau algebras: Let \(\Lambda\) be an \(R\)-algebra, finitely generated over \(R\) (module finite). Then for \(d\in\mathbb Z\), \(\Lambda\) is called \(d\)-Calabi-Yau (=d-CY) if there is an isomorphism

\(\text{Hom}_{D(\text{Mod}\Lambda)}(X,Y[d])\simeq D_0\text{Hom}_{D(\text{Mod}\Lambda)}(Y,X)\) for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in D^b(\text{mod}\Lambda)\) where \(D_0\) is the Matlis-dual. \(\Lambda\) is called singular Calabi-Yau (d-sCY) if the isomorphism holds for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in K^b(\text{proj}\Lambda)\)

When \(\Lambda=R\), it is known that \(R\) is \(d\)-sCY iff \(R\) is Gorenstein and equi-codimensional with dim \(R=d\). Notice that purely commutative statements are proved in a commutative setting.

Now, let \(R\) be an equi-codimensional CM ring of dimension \(d\) with canonical module \(\omega_R\). CM \(R\) is the category of CM \(R\)-modules, \(\underline{\text{CM}} R\) is the stable category, and \(\overline{CM}R\) is the costable category. The AR stranslation is \(\tau=\text{Hom}_R(\Omega^d\text{Tr}(-),\omega_R):\underline{\text{CM}} R\rightarrow\overline{CM}R\).

When \(R\) is an isolated singularity, one of the fundamental properties of the category CM \(R\) is the existence of AR duality: \(\underline{\text{Hom}}(X,Y)\equiv D_0\text{Ext}^1_R(Y,\tau X)\) for all \(X,Y\in\) CM \(R\). Letting \(D_1:=\text{Ext}^{d-1}_R(-,\omega_R)\) be the duality on the category of CM-modules of dimension 1, the authors prove that AR duality generalizes to mildly non-isolated singularities, in a precise sense. This generalised AR-duality also implies a generalized \((d-1)\)-Calabi-Yau property of the triangulated category \(\underline{\text{CM}} R\), and the symmetry in the Hom groups gives the tool needed to move from the CT level to the MM level. It is analogues to the symmetry given in cluster theory.

Recall that if \(R\) is a CM ring and \(\Lambda\) a module finite \(R\)-algebra, then \(\Lambda\) is an \(R\)-order if \(\Lambda\in\text{CM}R\), an \(R\)-order \(\Lambda\) is non singular if gl.dim\(\Lambda_{\mathfrak p}=\text{dim}R_{\mathfrak p}\) for all primes \(\mathfrak p\subset R\), and an \(R\)-order \(\Lambda\) has isolated singularities if \(\Lambda\) is a non-singular \(R_{\mathfrak p}\)-order for all non-maximal primes \(\mathfrak p\subset R\). Then for \(R\) CM, a noncommutative crepant resolution (NCCR) of \(R\) is \(\Gamma:=\text{End}_R(M)\) where \(M\) is reflexive and non-zero such that \(\Gamma\) is a non-singular \(R\)-order.

The article gives the following classification of cluster tilting (CT) modules: For a field of characteristic zero, \(S=k[x_1,\dots,x_d]\), and \(G\) a finite linear group. Let \(R=S^G\), then \(S\) is a CT \(R\)-module. The main result for CT modules is that for \(R\) a normal \(3-s\)CY ring, (1) CT modules are those reflexive generators which give NCCRs, (2)\(R\) has a NCCR\(\Longleftrightarrow R\) has a NCCR given by a CM generator \(\Longleftrightarrow R\) has a CT module.

After giving precise definitions of modifying and maximal modifying models the authors prove that when a NCCR exists, then the Maximal modifying algebras are exactly the same as NCCRs.

The authors explain their results on derived equivalences. They show that any algebra derived equivalent to a modifying algebra also has the form \(\text{End}_R(M)\). They give a result detailing the relationship between modifying and maximal modifying modules on the level on categories, and a relationship between maximal modifying modules and tilting modules. A main issue is the construction of modifying modules by mutations, by a method using exchange sequences. Defining left and right mutations allows the mutation of any NCCR of any dimension, at any direct summand, and gives another NCCR together with a derived equivalence, and this process together with its main results are covered in the article.

This is a long and advanced article. However, it is rather explicit with nice examples, and this makes it a good introduction and survey of AR-duality and its generalization. The article rather self contained, and illustrates the development of the categorical representation theory.

The study if maximal \(n\)-orthogonal modules in CM \(R\) is not suited for studying non-isolated singularities because the Ext vanishing condition is too strong. So one needs a subcategory of CM \(R\) that can take the place of the maximal \(n\)-orthogonal subcategories. This is the main goal this article, but it gives more: The authors develop a theory which can deal with singularities in the crepant partial resolutions. As the endomorphism rings of maximal orthogonal modules have finite global dimension, these cannot work for the purpose. The notion of maximal modifying modules are introduced as corresponding to shadows of maximal crepant partial resolutions. This level always exist geometrically, but it can happen to be smooth. The level of generality is thus to view the case when the geometry is smooth as a coincidence: everything that can be done with NCCRs should be possible to do with maximal modifying modules.

When curves are flopped between varieties with canonical singularities which are not terminal, this doesn’t take place on the maximal level. Anyway, it should be homologically understandable. That is the motivation for defining modifying modules. This also leads to the development of the theory of mutation for modifying modules.

The authors remark that some parts of the theory of (maximal) modifying modules are analogues of cluster tilting theory, particularly for rings of Krull dimension three. Some of the main properties of CT theory are still true in the present setting, but new features also appear, e.g. a mutation sometimes does not change the given modifying module, which is a necessary feature since it reflects the geometry of partial crepant resolutions.

The present theory depends on a more general noncommutative setting. The authors use the language of singular Calabi-Yau algebras: Let \(\Lambda\) be an \(R\)-algebra, finitely generated over \(R\) (module finite). Then for \(d\in\mathbb Z\), \(\Lambda\) is called \(d\)-Calabi-Yau (=d-CY) if there is an isomorphism

\(\text{Hom}_{D(\text{Mod}\Lambda)}(X,Y[d])\simeq D_0\text{Hom}_{D(\text{Mod}\Lambda)}(Y,X)\) for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in D^b(\text{mod}\Lambda)\) where \(D_0\) is the Matlis-dual. \(\Lambda\) is called singular Calabi-Yau (d-sCY) if the isomorphism holds for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in K^b(\text{proj}\Lambda)\)

When \(\Lambda=R\), it is known that \(R\) is \(d\)-sCY iff \(R\) is Gorenstein and equi-codimensional with dim \(R=d\). Notice that purely commutative statements are proved in a commutative setting.

Now, let \(R\) be an equi-codimensional CM ring of dimension \(d\) with canonical module \(\omega_R\). CM \(R\) is the category of CM \(R\)-modules, \(\underline{\text{CM}} R\) is the stable category, and \(\overline{CM}R\) is the costable category. The AR stranslation is \(\tau=\text{Hom}_R(\Omega^d\text{Tr}(-),\omega_R):\underline{\text{CM}} R\rightarrow\overline{CM}R\).

When \(R\) is an isolated singularity, one of the fundamental properties of the category CM \(R\) is the existence of AR duality: \(\underline{\text{Hom}}(X,Y)\equiv D_0\text{Ext}^1_R(Y,\tau X)\) for all \(X,Y\in\) CM \(R\). Letting \(D_1:=\text{Ext}^{d-1}_R(-,\omega_R)\) be the duality on the category of CM-modules of dimension 1, the authors prove that AR duality generalizes to mildly non-isolated singularities, in a precise sense. This generalised AR-duality also implies a generalized \((d-1)\)-Calabi-Yau property of the triangulated category \(\underline{\text{CM}} R\), and the symmetry in the Hom groups gives the tool needed to move from the CT level to the MM level. It is analogues to the symmetry given in cluster theory.

Recall that if \(R\) is a CM ring and \(\Lambda\) a module finite \(R\)-algebra, then \(\Lambda\) is an \(R\)-order if \(\Lambda\in\text{CM}R\), an \(R\)-order \(\Lambda\) is non singular if gl.dim\(\Lambda_{\mathfrak p}=\text{dim}R_{\mathfrak p}\) for all primes \(\mathfrak p\subset R\), and an \(R\)-order \(\Lambda\) has isolated singularities if \(\Lambda\) is a non-singular \(R_{\mathfrak p}\)-order for all non-maximal primes \(\mathfrak p\subset R\). Then for \(R\) CM, a noncommutative crepant resolution (NCCR) of \(R\) is \(\Gamma:=\text{End}_R(M)\) where \(M\) is reflexive and non-zero such that \(\Gamma\) is a non-singular \(R\)-order.

The article gives the following classification of cluster tilting (CT) modules: For a field of characteristic zero, \(S=k[x_1,\dots,x_d]\), and \(G\) a finite linear group. Let \(R=S^G\), then \(S\) is a CT \(R\)-module. The main result for CT modules is that for \(R\) a normal \(3-s\)CY ring, (1) CT modules are those reflexive generators which give NCCRs, (2)\(R\) has a NCCR\(\Longleftrightarrow R\) has a NCCR given by a CM generator \(\Longleftrightarrow R\) has a CT module.

After giving precise definitions of modifying and maximal modifying models the authors prove that when a NCCR exists, then the Maximal modifying algebras are exactly the same as NCCRs.

The authors explain their results on derived equivalences. They show that any algebra derived equivalent to a modifying algebra also has the form \(\text{End}_R(M)\). They give a result detailing the relationship between modifying and maximal modifying modules on the level on categories, and a relationship between maximal modifying modules and tilting modules. A main issue is the construction of modifying modules by mutations, by a method using exchange sequences. Defining left and right mutations allows the mutation of any NCCR of any dimension, at any direct summand, and gives another NCCR together with a derived equivalence, and this process together with its main results are covered in the article.

This is a long and advanced article. However, it is rather explicit with nice examples, and this makes it a good introduction and survey of AR-duality and its generalization. The article rather self contained, and illustrates the development of the categorical representation theory.

Reviewer: Arvid Siqveland (Kongsberg)

##### MSC:

14A22 | Noncommutative algebraic geometry |

16G20 | Representations of quivers and partially ordered sets |

14B99 | Local theory in algebraic geometry |

##### Keywords:

noncommutative Crepant resolutions; Auslander Reiten theory; Auslander Reiten duality; Auslander Reiten algebra; cluster tilting; mutations; right mutation; left mutation; exchange seqences##### References:

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