Dominant Dimension and Orders over Cohen-Macaulay Rings 2. This - PowerPoint PPT Presentation

Maurice Auslander Distinguished Lectures and International Conference Özgür Esentepe April 25, 2019 University of Toronto Dominant Dimension and Orders over Cohen-Macaulay Rings

2. This Year 1. Last Year 1 Table of contents

Last Year

Recall from last year (!) that for a commutative Gorenstein ring, the cohomology annihilator ideal is 2 Last Year ∩ ca ( R ) = ann R End R ( M ) M ∈ MCM ( R ) • If R has finite global dimension, then ca ( R ) = R . • Under mild assumptions, V ( ca ( R )) = sing ( R ) .

If R is the complete local coordinate ring of a reduced curve singularity, then the cohomology annihilator ideal coincides with the conductor ideal. R is the integral closure of R in its total quotient ring. • In our case, the normalization is a module finite R -algebra, it is maximal Cohen-Macaulay as an R -module, and it has finite global dimension. 3 Last Year Theorem • The conductor of R is the ideal { r ∈ R : r ¯ R ⊆ R } where ¯ • The conductor is also equal to ann R End R (¯ R ) .

be a ring homomorphism. • f is a split monomorphism, With these assumptions, we have 4 Last Year Let R be a Gorenstein ring, Λ be a noncommutative ring and f : R → Λ • Λ is finitely generated as an R -module, • Λ is maximal Cohen-Macaulay as an R -module, • Λ has finite global dimension δ , Theorem [ ann R End R (Λ)] δ + 1 ⊆ ca ( R ) ⊆ ann R End R (Λ) .

This Year

• f is a split monomorphism, With these assumptions, we have 5 This year Let R be a Gorenstein ring of Krull dimension at most 2 , Λ be a noncommutative ring and f : R → Λ be a ring homomorphism. • Λ is finitely generated as an R -module, • Λ is maximal Cohen-Macaulay as an R -module, • Λ has finite global dimension δ , • Λ ∗ = Hom R (Λ , R ) has projective dimension n as a Λ -module . Theorem [ ann R End R (Λ)] n + 1 ⊆ ca ( R ) ⊆ ann R End R (Λ) . In particular, if Λ ∗ is projective, then ca ( R ) = ann R End R (Λ) .

maximal Cohen-Macaulay as an R -module. 6 Definitions and Notations • R is a Cohen-Macaulay local ring with canonical module ω R . • Λ is an R -order. That is, it is a module-finite R -algebra which is • MCM (Λ) = { X ∈ Λ -mod : X ∈ MCM ( R ) } . • D = Hom R ( − , ω R ) : MCM (Λ) → MCM (Λ op ) - it is an exact duality. • ω Λ = D Λ is the canonical module of Λ .

non-singular order. The following are equivalent [Iyama-Wemyss]: 7 1. ω Λ is projective and Λ has finite global dimension, 2. Every maximal Cohen-Macaulay Λ -module is projective. 3. gldim Λ p = dim R p for every prime ideal p of R . 4. gldim Λ m = dim R m where m is the maximal ideal of R . If Λ satisfies one of the above conditions, then it is called a

formula: Iyama-Wemyss]. [Josh Stangle] generalizes this in his PhD thesis as 8 If ω Λ is projective, then we have a version of Auslander-Buchsbaum pd Λ M + depth M = dim R for any Λ -module M of finite projective dimension [Iyama-Reiten, follows: If ω Λ has projective dimension n , then dim R ≤ pd Λ M + depth M ≤ n + dim R for every Λ -module M of finite projective dimension.

module is projective. positive projective dimension, there are non-projective maximal Cohen-Macaulay modules. • How do we understand the structure of the stable category of maximal Cohen-Macaulay modules? • For instance, how many indecomposable non-projective maximal Cohen-Macaulay modules are there? (Auslander-Roggenkamp). 9 Question • If Λ is non-singular, then every maximal Cohen-Macaulay • If Λ has finite global dimension with a canonical module ω Λ of

• Dualizing a projective resolution of the maximal MCM -relatively injectives and vice versa. injective coresolution of the maximal Cohen-Macaulay 10 Injectives in MCM ( R ) • The canonical module ω Λ is an injective object in MCM (Λ) and in fact any MCM -relatively injective Λ -module is isomorphic to a direct summand of finite direct sums of ω Λ . • The duality D = Hom R ( − , ω R ) takes projectives to Cohen-Macaulay Λ op -module DM gives a MCM -relatively Λ -module M . • The relative injective dimension of Λ is equal to the projective dimension of ω Λ .

the MCM -relative dominant dimension is at least projective. then the path algebra RQ has MCM -relative dominant dimension 1. 11 Dominant Dimension Let 0 → Λ → I 0 → I 1 → . . . → I k − 1 → I k → . . . be a minimal MCM -relatively injective coresolution of Λ . We say that Λ has MCM -relative dominant dimension at least k if I 0 , . . . , I k − 1 are • If Λ is a non-singular order, then its MCM -relative dominant dimension is ∞ . • If Λ is an order of the form End R ( M ) where M ∈ MCM ( R ) , then max { 2 , dim R − 2 } . • If R is a regular local ring and Q is a linearly directed A n quiver,

j module 12 Tilting Let 0 → Λ → I 0 → I 1 → . . . → I k − 1 → I k → . . . be a minimal MCM -relatively injective coresolution of Λ . Denote the image of I j → I j + 1 by K j + 1 . Lemma Then, K j + 1 is also a maximal Cohen-Macaulay module. Theorem If Λ has relative dominant dimension at least k and j < k, then the I j ⊕ K j + 1 ⊕ T j = i = 0 is a k-tilting Λ -module.

13 the Artinian case. Theorem Let Γ j = End Λ ( T j ) op where T j is the tilting module defined above and suppose that Λ has finite global dimension. Then, 1. Γ j is also an R-order of finite global dimension. 2. ***The projective dimension of ω Γ j is at most the projective dimension of ω Λ . Note: See [Pressland, Sauter] and [Nguyen, Reiten, Todorov, Zhu] for

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