Orders and Non-commutative Crepant Resolutions arXiv: 1604.01748 - PowerPoint PPT Presentation

Orders and Non-commutative Crepant Resolutions arXiv: 1604.01748 Josh Stangle Syracuse University August 24, 2016 ICRA 2016 Stangle 1

Non-commutative crepant resolutions In 2004, Van den Bergh proposed the following qualities for a non-commutative crepant resolution , Λ, of a commutative ring R . Λ should be an R -algebra which is An order –Maximal Cohen-Macaulay as an R -module. Birational − Λ ⊗ R K ∼ = M n ( K ) for K the quotient field of R . Symmetric –Λ ∗ := Hom R (Λ , R ) ∼ = Λ as Λ-Λ-bimodules. non-singular –gldim Λ < ∞ . ICRA 2016 Stangle 2

Non-commutative crepant resolutions Let R be a Gorenstein normal domain of dimension d . Definition A non-commutative crepant resolution of R is a non-singular R -order of the form Λ = End R ( M ) where M is a reflexive R -module. Recall that an R -order Λ is called non-singular if gldim Λ p = dim R p for all p ∈ Spec R . When R is equicodimensional, this is equivalent to R being homologically homogeneous ICRA 2016 Stangle 3

The Gorenstein Case When R is Gorenstein requiring non-singularity is not necessary. Theorem (Van den Bergh ’04) Suppose R is a Gorenstein normal domain. Let Λ be an R-algebra satisfying Λ ∼ = End R ( M ) for some reflexive R-module M. gldim Λ < ∞ Λ is a maximal Cohen-Macaulay R-module. Then Λ is non-singular. ICRA 2016 Stangle 4

The previous theorem follows from a few key facts: 1 For R Gorenstein, Ext i R ( M , R ) = 0 for any MCM R -module M and i > 0. ICRA 2016 Stangle 5

The previous theorem follows from a few key facts: 1 For R Gorenstein, Ext i R ( M , R ) = 0 for any MCM R -module M and i > 0. 2 For R a normal domain, Λ = End R ( M ) is symmetric. ICRA 2016 Stangle 5

The previous theorem follows from a few key facts: 1 For R Gorenstein, Ext i R ( M , R ) = 0 for any MCM R -module M and i > 0. 2 For R a normal domain, Λ = End R ( M ) is symmetric. 3 There is a spectral sequence Ext p Λ ( B , Ext q R (Λ , C )) ⇒ Ext p + q ( B , C ) R for all B ∈ mod Λ and C ∈ mod R . In view of (1) this yields an isomorphism for all i Ext i = Ext i Λ ( B , Hom R (Λ , R )) ∼ R ( B , R ) ICRA 2016 Stangle 5

In light of the theorem we can consider the following equivalent definition Definition A non-commutative crepant resolution of a Gorenstein normal domain R is an R -order of the form Λ = End R ( M ) where M is a reflexive R -module, such that gldim Λ < ∞ . When R is only assumed to be Cohen-Macaulay, this definition is not equivalent to the previous one, such a Λ need not be non-singular. ICRA 2016 Stangle 6

Producing Ext Vanishing-Method 1 We will now let R be a Cohen-Macaulay normal domain of dimension d with canonical module ω . Recall that an R -module M is called totally reflexive if M is reflexive and Ext i R ( M , R ) = Ext i R ( M ∗ , R ) = 0 for all i > 0. Definition A strong NC resolution of R is an R -algebra Λ which has finite global dimension and is totally reflexive over R . ICRA 2016 Stangle 7

Theorem Let ( R , m , k ) be a CM local ring, and Λ a module-finite R-algebra such that Hom R (Λ , R ) has finite injective dimension as a left Λ -module and Ext i R (Λ , R ) = 0 for all i > 0 . Then R is Gorenstein. From this we deduce the following corollary Corollary If R is a CM local normal domain possessing a strong NC resolution Λ , then R is Gorenstein. ICRA 2016 Stangle 8

The proof Since we have assumed Ext i R (Λ , R ) = 0 for all i > 0 we get an isomorphism as before Ext i = Ext i Λ ( B , Hom R (Λ , R )) ∼ R ( B , R ) . By assumption, the injective dimension of Hom R (Λ , R ) is finite, so Ext i Λ ( B , Hom R (Λ , R )) and hence Ext i R ( B , R ) are zero for i >> 0. Apply this to the B = Λ / m Λ, which is a finite dimensional k -vector space. This gives Ext i R ( k , R ) = 0 for some i , which implies R is Gorenstein. ICRA 2016 Stangle 9

Producing Ext Vanishing-Method 2 Another way to produce Ext vanishing is to replace C with ω in Ext p Λ ( B , Ext q R (Λ , C )) ⇒ Ext p + q ( B , C ) . R Then again the spectral sequence collapses to give an isomorphism Ext i = Ext i Λ ( B , Hom R (Λ , ω )) ∼ R ( M , ω ) for all B ∈ mod Λ. In order to use the above, we need a condition like symmetry. Definition An R -order Λ is called a Gorenstein R-order if ω Λ := Hom R (Λ , ω ) is a projective Λ-module. ICRA 2016 Stangle 10

Theorem (Iyama-Wemyss) Let R be a CM ring with canonical module ω R and Λ an R-order. TFAE: 1 Λ is homologically homogeneous. 2 gldim Λ < ∞ and Λ is a Gorenstein R-order. But, endomorphism rings are not necessarily Gorenstein orders if R is not Gorenstein. We’d like to answer the question “When is End R ( M ) a Gorenstein order for a reflexive module M ?” ICRA 2016 Stangle 11

Example Let R = k [[ x 2 , xy , xz , y 2 , yz , z 2 ]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = ( x 2 , xy , xz ) M := syz( ω R ). A := End R ( R ⊕ ω ⊕ M ) has global dimension 3 ICRA 2016 Stangle 12

Example Let R = k [[ x 2 , xy , xz , y 2 , yz , z 2 ]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = ( x 2 , xy , xz ) M := syz( ω R ). A := End R ( R ⊕ ω ⊕ M ) has global dimension 3 , but it is not MCM. It is not a non-commutative crepant resolution. ICRA 2016 Stangle 12

Example Let R = k [[ x 2 , xy , xz , y 2 , yz , z 2 ]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = ( x 2 , xy , xz ) M := syz( ω R ). A := End R ( R ⊕ ω ⊕ M ) has global dimension 3 , but it is not MCM. It is not a non-commutative crepant resolution. Λ := End R ( R ⊕ ω ) is a noncommutative crepant resolution. It is MCM over R and isomorphic to the twisted group ring k [[ z , y , z ]] ∗ Z 2 which is known to have global dimension 3. ICRA 2016 Stangle 12

Example Let R = k [[ x 2 , xy , xz , y 2 , yz , z 2 ]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = ( x 2 , xy , xz ) M := syz( ω R ). A := End R ( R ⊕ ω ⊕ M ) has global dimension 3 , but it is not MCM. It is not a non-commutative crepant resolution. Λ := End R ( R ⊕ ω ) is a noncommutative crepant resolution. It is MCM over R and isomorphic to the twisted group ring k [[ z , y , z ]] ∗ Z 2 which is known to have global dimension 3. It is easy to check Hom R (Λ , ω R ) is a projective Λ-module. ICRA 2016 Stangle 12

With this example in mind, we change our question: “When is End R ( R ⊕ ω ) a Gorenstein R -order?” Theorem Suppose R is a CM henselian generically Gorenstein ring with canonical module ω R . Then End R ( R ⊕ ω ) is a Gorenstein R-order if and only if ω ∼ = ω ∗ . In particular if R is a CM local domain, then this is further equivalent to [ ω ] having order 2 in the divisor class group of R. ICRA 2016 Stangle 13

The proof ( ⇒ ) Suppose End( R ⊕ ω ) is Gorenstein. Then we examine End R ( R ⊕ ω ) ∼ = R ⊕ R ⊕ ω ⊕ ω ∗ End R ( R ⊕ ω ) v ∼ = ω ⊕ ω ⊕ R ⊕ Hom R ( ω ∗ , ω ) Comparing these, it is necessary that ω ∼ = ω ∗ . ICRA 2016 Stangle 14

The proof ( ⇒ ) Suppose End( R ⊕ ω ) is Gorenstein. Then we examine End R ( R ⊕ ω ) ∼ = R ⊕ R ⊕ ω ⊕ ω ∗ End R ( R ⊕ ω ) v ∼ = ω ⊕ ω ⊕ R ⊕ Hom R ( ω ∗ , ω ) Comparing these, it is necessary that ω ∼ = ω ∗ . ( ⇐ ) We pick an isomorphism ϕ : ω ∗ − → ω and note � � ω ∗ R Λ = ω R as a ring. ICRA 2016 Stangle 14

Then we show that the element � � 0 ϕ f = 1 0 is a basis for the Λ-module � � ω Hom R ( ω ∗ , ω ) Λ v = . R ω ICRA 2016 Stangle 15

In other words, we choose a map � � g 1 g 2 ∈ Λ v g = g 3 g 4 and produce a � � λ 1 λ 2 λ = ∈ Λ λ 3 λ 4 so that g ( η ) = λ · f ( η ) = f ( η · λ ) for all η ∈ Λ. ICRA 2016 Stangle 16