Singular derived categories of \(\mathbb{Q}\)-factorial terminalizations and maximal modification algebras.(English)Zbl 1326.14033
The authors are going into this work to understand certain aspects of the minimal model program in dimension three using categorical and noncommutative techniques. It is reasonable to think that in dimension 3, the theory of noncommutative minimal models controls the geometry of commutative minimal models in the same feature that noncommutative crepant resolutions control the geometry of (commutative) crepant resolutions.
To say something about the above, the authors give an extension of the characterisation of smoothness by the singular derived category to be zero to a characterization of \(\mathbb Q\)-factoriality. This involves moving from smooth schemes to singular ones, and it is not clear what form such a derived category characterization should take.
From the noncommutative side, in the study of maximal modification algebras (MMAs) one homological condition calledrigid-freeness completely characterizes when maximality has been reached. In this article, the authors prove that this condition gives a necessary condition for a scheme with isolated Gorenstein singularities to be \(\mathbb Q\)-factorial. Also, the condition is sufficient if the singularities are hypersurfaces, as in the case with Gorenstein terminal singularities. Of course, an overall goal is to highlight that the noncommutative side gives new results in the pure sense of algebraic geometry.
A normal scheme \(X\) is \(\mathbb Q\)-factorial if for every Weil divisor \(D\), there exists \(n\in\mathbb N\) such that \(nD\) is Cartier. If \(n=1\), \(X\) is called locally factorial.
A scheme \(X\) satisfies (ELF) if \(X\) is separated, Noetherian, of finite Krull dimension, such that coh\(X\) has enough locally free sheaves.
Let \(\mathcal T\) be a triangulated category with suspension functor. An object \(a\in\mathcal T\) is calledrigid if \(\text{Hom}_{\mathcal T}(a,a[1])=0\), and \(\mathcal T\) is calledrigid-free if every rigid object is isomorphic to the zero object.
The article’s main result on \(\mathbb Q\)-factoriality are:
Theorem 1.1. Suppose that \(X\) is a Gorenstein scheme of dimension \(d\) satisfying (ELF), with isolated singularities \(\{x_1,\dots, x_n\}.\) Then there is a triangle equivalence \[D_{sg}\hookrightarrow\oplus_{i=1}^n\underline{\text{CM}}\mathcal O_{X,x_i}\] up to direct summands.
Theorem 1.2. Suppose that \(X\) is a normal 3-dimensional Gorenstein scheme over a field \(k\), satisfying (ELF), with isolated singularities \(\{x_1,\dots,x_n\}\).
(1) If \(D_{sg}(X)\) is rigid free, then \(X\) is locally factorial.
(2) If \(\overline{D_{sg}(X)}\) is rigid free, then \(X\) is complete locally factorial.
(3) If \(\mathcal O_{X,x_i}\) are hypersurfaces for all \(1\leq i\leq n\), then the following are equivalent:
(a) \(X\) is locally factorial.
(b) \(X\) is \(\mathbb Q\)-factorial.
(c) \(D_{sg}(X)\) is rigid-free.
(d) \(\underline{CM}\mathcal O_{X,x}\) is rigid free for all closed points \(x\in X\).
(4) If \(\hat{\mathcal O}_{X,x_i}\) are hypersurfaces for all \(1\leq i\leq n\), then the following are equivalent.
(a) \(X\) is complete locally factorial.
(b) \(X\) is complete locally \(\mathbb Q\)-factorial.
(c) \(\overline{D_{sg}(X)}\) is rigid-free.
(d) \(\underline{CM}\hat{\mathcal O}_{X,x}\) is rigid-free for all closed points \(x\in X\).
Let \(R\) be a commutative ring with \(\dim R=d\) and let \(\Lambda\) be a module-finite \(R\)-algebra. For \(X\in\text{mod}\Lambda\), \(E(X)\) denotes the injective hull of \(X\), \(E=\bigoplus_{m\in\text{Max}R}E(R/\mathfrak m)\), \(D=\text{Hom}_R(-,E)\), also \(D^b_{fl}\) denote all bounded complexes with finite length cohomology. Then for \(n\in\mathbb Z\), \(\Lambda\) is called \(n\)-CY if there is a functorial isomorphism \[\text{Hom}_{D(\text{Mod}\Lambda)}(X,Y[n])\cong D\text{Hom}_{D(\text{Mod}\Lambda)}(Y,X)\] for all \(X\in D^b_{fl}(\Lambda)\) and \(Y\in D^b(\text{mod}\Lambda)\). If this holds for \(X\in D^b_{fl}(\Lambda)\) and \(Y\in K^b(\text{proj}\Lambda)\), \(\Lambda\) is calledsingular \(n\)-CY. Here \(K^b(\text{proj})\) is the homotopy category of bounded chain complexes, and this makes the definition of thesingular derived category of \(\Lambda\) possible: \(D_{sg}(\Lambda)=D^b(\text{mod}\Lambda)/K^b(\text{proj}\Lambda)\).
Let \(R\) be a \(d\)-sCY ring. an \(M\in\text{ref}(R)\) is called arigid module if \(\text{Ext}^1_R(M,M)=0\), it is called amodifying module if \(\text{End}_R(M)\in CM(R)\), and amaximal modifying module (MM) if it is modifying and further if \(M\oplus Y\) is modifying for \(Y\in\text{ref} R\), then \(Y\in\text{add}M\).
If \(M\) is a maximal modifying \(R\)-module, then \(\text{End}_R(M)\) is amaximal modification algebra (MMA). These algebras were introduced with the aim of generalizing NCCRs to cover the more general situation when crepant resolutions don’t exist. \(Y\) is said to bederived equivalent to \(\Lambda\) of \(D^b(\text{coh}Y)\) is equivalent to \(D^b(\text{mod}Y)\).
The corresponding results om MMAs are the following:
Theorem 1.4. Let \(Y\rightarrow\text{Spec}R\) be a projective birational morphism between \(d\)-dimensional varieties. Suppose that \(Y\) is derived equivalent to some ring \(\Lambda\). If \(\Lambda\) is a reflexive \(R\)-module, then \(\Lambda\cong\text{End}_R(M)\) for some reflexive \(R\)-module \(M\).
Theorem 1.5. Let \(f:Y\rightarrow\text{Spec}R\) be a projective birational morphism between \(d\)-dimensional Gorenstein varieties. Suppose that \(Y\) is derived equivalent to some ring \(\Lambda\), then
(1) \(f\) is crepant \(\Leftrightarrow\Lambda\in\text{CM}R\)
(2) \(f\) is a crepant resolution \(\Leftrightarrow\Lambda\) is a NCCR of \(R\).
In either case, \(\Lambda\cong\text{End}_R(M)\) for some \(M\in\text{ref}R.\)
Theorem 1.6. Let \(f:Y\rightarrow\text{Spec}R\) be a projective birational morphism, where \(Y\) and \(R\) are both Gorenstein varieties of dimension 3. Assume that \(Y\) has at worst isolated singularities \(\{x_1,\dots,x_n\}\) where \(\mathbb O_{Y,x_i}\) is a hypersurface. If \(Y\) is derived equivalent to some ring \(\Lambda\), then \(f\) is crepant and \(Y\) is \(\mathbb Q\)-factorial if and only if \(\Lambda\) is a MMA of \(R\). In this situation, all MMAs of \(R\) have isolated singularities, and are all derived equivalent.
The authors apply these results to the \(cA_n\) singularities to obtain several explicit examples, which is of great value for the reader, and which illustrates the applications of the more abstract theory to pure algebraic geometry. The article is self-contained, and is an important contribution for making the category theoretic view (derived categories) of algebraic geometry more geometric.
To say something about the above, the authors give an extension of the characterisation of smoothness by the singular derived category to be zero to a characterization of \(\mathbb Q\)-factoriality. This involves moving from smooth schemes to singular ones, and it is not clear what form such a derived category characterization should take.
From the noncommutative side, in the study of maximal modification algebras (MMAs) one homological condition calledrigid-freeness completely characterizes when maximality has been reached. In this article, the authors prove that this condition gives a necessary condition for a scheme with isolated Gorenstein singularities to be \(\mathbb Q\)-factorial. Also, the condition is sufficient if the singularities are hypersurfaces, as in the case with Gorenstein terminal singularities. Of course, an overall goal is to highlight that the noncommutative side gives new results in the pure sense of algebraic geometry.
A normal scheme \(X\) is \(\mathbb Q\)-factorial if for every Weil divisor \(D\), there exists \(n\in\mathbb N\) such that \(nD\) is Cartier. If \(n=1\), \(X\) is called locally factorial.
A scheme \(X\) satisfies (ELF) if \(X\) is separated, Noetherian, of finite Krull dimension, such that coh\(X\) has enough locally free sheaves.
Let \(\mathcal T\) be a triangulated category with suspension functor. An object \(a\in\mathcal T\) is calledrigid if \(\text{Hom}_{\mathcal T}(a,a[1])=0\), and \(\mathcal T\) is calledrigid-free if every rigid object is isomorphic to the zero object.
The article’s main result on \(\mathbb Q\)-factoriality are:
Theorem 1.1. Suppose that \(X\) is a Gorenstein scheme of dimension \(d\) satisfying (ELF), with isolated singularities \(\{x_1,\dots, x_n\}.\) Then there is a triangle equivalence \[D_{sg}\hookrightarrow\oplus_{i=1}^n\underline{\text{CM}}\mathcal O_{X,x_i}\] up to direct summands.
Theorem 1.2. Suppose that \(X\) is a normal 3-dimensional Gorenstein scheme over a field \(k\), satisfying (ELF), with isolated singularities \(\{x_1,\dots,x_n\}\).
(1) If \(D_{sg}(X)\) is rigid free, then \(X\) is locally factorial.
(2) If \(\overline{D_{sg}(X)}\) is rigid free, then \(X\) is complete locally factorial.
(3) If \(\mathcal O_{X,x_i}\) are hypersurfaces for all \(1\leq i\leq n\), then the following are equivalent:
(a) \(X\) is locally factorial.
(b) \(X\) is \(\mathbb Q\)-factorial.
(c) \(D_{sg}(X)\) is rigid-free.
(d) \(\underline{CM}\mathcal O_{X,x}\) is rigid free for all closed points \(x\in X\).
(4) If \(\hat{\mathcal O}_{X,x_i}\) are hypersurfaces for all \(1\leq i\leq n\), then the following are equivalent.
(a) \(X\) is complete locally factorial.
(b) \(X\) is complete locally \(\mathbb Q\)-factorial.
(c) \(\overline{D_{sg}(X)}\) is rigid-free.
(d) \(\underline{CM}\hat{\mathcal O}_{X,x}\) is rigid-free for all closed points \(x\in X\).
Let \(R\) be a commutative ring with \(\dim R=d\) and let \(\Lambda\) be a module-finite \(R\)-algebra. For \(X\in\text{mod}\Lambda\), \(E(X)\) denotes the injective hull of \(X\), \(E=\bigoplus_{m\in\text{Max}R}E(R/\mathfrak m)\), \(D=\text{Hom}_R(-,E)\), also \(D^b_{fl}\) denote all bounded complexes with finite length cohomology. Then for \(n\in\mathbb Z\), \(\Lambda\) is called \(n\)-CY if there is a functorial isomorphism \[\text{Hom}_{D(\text{Mod}\Lambda)}(X,Y[n])\cong D\text{Hom}_{D(\text{Mod}\Lambda)}(Y,X)\] for all \(X\in D^b_{fl}(\Lambda)\) and \(Y\in D^b(\text{mod}\Lambda)\). If this holds for \(X\in D^b_{fl}(\Lambda)\) and \(Y\in K^b(\text{proj}\Lambda)\), \(\Lambda\) is calledsingular \(n\)-CY. Here \(K^b(\text{proj})\) is the homotopy category of bounded chain complexes, and this makes the definition of thesingular derived category of \(\Lambda\) possible: \(D_{sg}(\Lambda)=D^b(\text{mod}\Lambda)/K^b(\text{proj}\Lambda)\).
Let \(R\) be a \(d\)-sCY ring. an \(M\in\text{ref}(R)\) is called arigid module if \(\text{Ext}^1_R(M,M)=0\), it is called amodifying module if \(\text{End}_R(M)\in CM(R)\), and amaximal modifying module (MM) if it is modifying and further if \(M\oplus Y\) is modifying for \(Y\in\text{ref} R\), then \(Y\in\text{add}M\).
If \(M\) is a maximal modifying \(R\)-module, then \(\text{End}_R(M)\) is amaximal modification algebra (MMA). These algebras were introduced with the aim of generalizing NCCRs to cover the more general situation when crepant resolutions don’t exist. \(Y\) is said to bederived equivalent to \(\Lambda\) of \(D^b(\text{coh}Y)\) is equivalent to \(D^b(\text{mod}Y)\).
The corresponding results om MMAs are the following:
Theorem 1.4. Let \(Y\rightarrow\text{Spec}R\) be a projective birational morphism between \(d\)-dimensional varieties. Suppose that \(Y\) is derived equivalent to some ring \(\Lambda\). If \(\Lambda\) is a reflexive \(R\)-module, then \(\Lambda\cong\text{End}_R(M)\) for some reflexive \(R\)-module \(M\).
Theorem 1.5. Let \(f:Y\rightarrow\text{Spec}R\) be a projective birational morphism between \(d\)-dimensional Gorenstein varieties. Suppose that \(Y\) is derived equivalent to some ring \(\Lambda\), then
(1) \(f\) is crepant \(\Leftrightarrow\Lambda\in\text{CM}R\)
(2) \(f\) is a crepant resolution \(\Leftrightarrow\Lambda\) is a NCCR of \(R\).
In either case, \(\Lambda\cong\text{End}_R(M)\) for some \(M\in\text{ref}R.\)
Theorem 1.6. Let \(f:Y\rightarrow\text{Spec}R\) be a projective birational morphism, where \(Y\) and \(R\) are both Gorenstein varieties of dimension 3. Assume that \(Y\) has at worst isolated singularities \(\{x_1,\dots,x_n\}\) where \(\mathbb O_{Y,x_i}\) is a hypersurface. If \(Y\) is derived equivalent to some ring \(\Lambda\), then \(f\) is crepant and \(Y\) is \(\mathbb Q\)-factorial if and only if \(\Lambda\) is a MMA of \(R\). In this situation, all MMAs of \(R\) have isolated singularities, and are all derived equivalent.
The authors apply these results to the \(cA_n\) singularities to obtain several explicit examples, which is of great value for the reader, and which illustrates the applications of the more abstract theory to pure algebraic geometry. The article is self-contained, and is an important contribution for making the category theoretic view (derived categories) of algebraic geometry more geometric.
Reviewer: Arvid Siqveland (Kongsberg)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 14A22 | Noncommutative algebraic geometry |
Keywords:
minimal models;noncommutative resolutions;crepant resolution;noncommutativ crepant resolution;modification algebra;Maximal modification modules;MMA;singular derived category;isolated singularitiesCite
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