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 ∼ = Mn(K) for K the quotient field of R. Symmetric–Λ∗ := HomR(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 = EndR(M) where M is a reflexive R-module. Recall that an R-order Λ is called non-singular if gldim Λp = dim Rp for all p ∈ Spec R. When R is equicodimensional, this is equivalent to R being homologically homogeneous

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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 ∼ = EndR(M) for some reflexive R-module M. gldim Λ < ∞ Λ is a maximal Cohen-Macaulay R-module. Then Λ is non-singular.

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The previous theorem follows from a few key facts:

1 For R Gorenstein, Exti R(M, R) = 0 for any MCM R-module M

and i > 0.

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The previous theorem follows from a few key facts:

1 For R Gorenstein, Exti R(M, R) = 0 for any MCM R-module M

and i > 0.

2 For R a normal domain, Λ = EndR(M) is symmetric.

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The previous theorem follows from a few key facts:

1 For R Gorenstein, Exti R(M, R) = 0 for any MCM R-module M

and i > 0.

2 For R a normal domain, Λ = EndR(M) is symmetric. 3 There is a spectral sequence

Extp

Λ(B, Extq R(Λ, C)) ⇒ Extp+q R

(B, C) for all B ∈ mod Λ and C ∈ mod R. In view of (1) this yields an isomorphism for all i Exti

Λ(B, HomR(Λ, R)) ∼

= Exti

R(B, R)

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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 = EndR(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.

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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 Exti

R(M, R) = Exti 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.

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Theorem Let (R, m, k) be a CM local ring, and Λ a module-finite R-algebra such that HomR(Λ, R) has finite injective dimension as a left Λ-module and Exti

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.

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The proof

Since we have assumed Exti

R(Λ, R) = 0 for all i > 0 we get an

isomorphism as before Exti

Λ(B, HomR(Λ, R)) ∼

= Exti

R(B, R).

By assumption, the injective dimension of HomR(Λ, R) is finite, so Exti

Λ(B, HomR(Λ, R)) and hence Exti R(B, R) are zero

for i >> 0. Apply this to the B = Λ/mΛ, which is a finite dimensional k-vector space. This gives Exti

R(k, R) = 0 for some i, which implies R is

Gorenstein.

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Producing Ext Vanishing-Method 2

Another way to produce Ext vanishing is to replace C with ω in Extp

Λ(B, Extq R(Λ, C)) ⇒ Extp+q R

(B, C). Then again the spectral sequence collapses to give an isomorphism Exti

Λ(B, HomR(Λ, ω)) ∼

= Exti

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 ωΛ := HomR(Λ, ω) is a projective Λ-module.

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Theorem (Iyama-Wemyss) Let R be a CM ring with canonical module ωR 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 EndR(M) a Gorenstein order for a reflexive module M?”

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Example

Let R = k[[x2, xy, xz, y2, yz, z2]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = (x2, xy, xz) M := syz(ωR). A := EndR(R ⊕ ω ⊕ M) has global dimension 3

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Example

Let R = k[[x2, xy, xz, y2, yz, z2]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = (x2, xy, xz) M := syz(ωR). A := EndR(R ⊕ ω ⊕ M) has global dimension 3 , but it is not

• MCM. It is not a non-commutative crepant resolution.

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Example

Let R = k[[x2, xy, xz, y2, yz, z2]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = (x2, xy, xz) M := syz(ωR). A := EndR(R ⊕ ω ⊕ M) has global dimension 3 , but it is not

• MCM. It is not a non-commutative crepant resolution.

Λ := EndR(R ⊕ ω) is a noncommutative crepant resolution. It is MCM over R and isomorphic to the twisted group ring k[[z, y, z]] ∗ Z2 which is known to have global dimension 3.

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Example

Let R = k[[x2, xy, xz, y2, yz, z2]]. Then, R is known to have finite CM type with indecomposable MCM modules R ω ∼ = (x2, xy, xz) M := syz(ωR). A := EndR(R ⊕ ω ⊕ M) has global dimension 3 , but it is not

• MCM. It is not a non-commutative crepant resolution.

Λ := EndR(R ⊕ ω) is a noncommutative crepant resolution. It is MCM over R and isomorphic to the twisted group ring k[[z, y, z]] ∗ Z2 which is known to have global dimension 3. It is easy to check HomR(Λ, ωR) is a projective Λ-module.

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With this example in mind, we change our question: “When is EndR(R ⊕ ω) a Gorenstein R-order?” Theorem Suppose R is a CM henselian generically Gorenstein ring with canonical module ωR. Then EndR(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.

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The proof

(⇒) Suppose End(R ⊕ ω) is Gorenstein. Then we examine EndR(R ⊕ ω) ∼ = R ⊕ R ⊕ ω ⊕ ω∗ EndR(R ⊕ ω)v ∼ = ω ⊕ ω ⊕ R ⊕ HomR(ω∗, ω) Comparing these, it is necessary that ω ∼ = ω∗.

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The proof

(⇒) Suppose End(R ⊕ ω) is Gorenstein. Then we examine EndR(R ⊕ ω) ∼ = R ⊕ R ⊕ ω ⊕ ω∗ EndR(R ⊕ ω)v ∼ = ω ⊕ ω ⊕ R ⊕ HomR(ω∗, ω) Comparing these, it is necessary that ω ∼ = ω∗. (⇐) We pick an isomorphism ϕ : ω∗ − → ω and note Λ =

• R

ω∗ ω R

• as a ring.

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Then we show that the element f =

• ϕ

1

• is a basis for the Λ-module

Λv =

• ω

HomR(ω∗, ω) R ω

• .

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In other words, we choose a map g =

• g1

g2 g3 g4

• ∈ Λv

and produce a λ =

• λ1

λ2 λ3 λ4

• ∈ Λ

so that g(η) = λ · f (η) = f (η · λ) for all η ∈ Λ.

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