Noncommutative resolutions using syzygies

Given a noether algebra with a noncommutative resolution, a general construction of new noncommutative resolutions is given. As an application, it is proved that any finite length module over a regular local or polynomial ring gives rise, via suitable syzygies, to a noncommutative resolution.

The focus of this article is on constructing endomorphism rings with finite global dimension. This problem has arisen in various contexts, including Auslander's theory of representation dimension [1], Dlab and Ringel's approach to quasi-hereditary algebras in Lie theory [4,6], Rouquier's dimension of triangulated categories [10], cluster tilting modules in Auslander-Reiten theory [8], and Van den Bergh's noncommutative crepant resolutions in birational geometry [12].
For a noetherian ring R which is not necessarily commutative, and a finitely generated faithful R-module M , the ring End R (M ) is a noncommutative resolution (abbreviated to NCR) if its global dimension is finite; see [5]. When this happens, M is said to give an NCR of R. We give a method for constructing new NCRs from a given one. Theorem 1. Let R be a noether algebra, and let M, X ∈ mod R. If M is a dtorsionfree generator giving an NCR, and gldim End R (X) is finite, then for any integer 0 ≤ c < min{d, grade R X}, the following statements hold.
(1) The R-module M ⊕ Ω c X is a c-torsionfree generator.
(2) There is an inequality In particular, M ⊕ Ω c X gives an NCR of R.
A commutative ring is equicodimensional if every maximal ideal has the same height. Typical examples of equicodimensional regular rings are polynomial rings over a field, and regular local rings.
Corollary 2. Let R be an equicodimensional regular ring, and N a finite length R-module such that gldim End R (N ) is finite. Given non-negative integers c 1 , . . . , c n with In particular, M gives an NCR of R.
For any finite length R-module X, there exists a finite length R-module Y such that End R (X ⊕ Y ) has finite global dimension [7]. In the setting of the corollary, it follows that an NCR can be constructed using any finite length R-module. In the definition of noncommutative resolution, it is sometimes required that the module be reflexive [11]. If dim R ≥ 3 in the setting of the corollary, then for any finite length R-module, by taking all c i ≥ 2 it can be ensured that the module giving the NCR is reflexive, but is not free.

Proofs
Throughout, R will be a noether algebra, in the sense that it is finitely generated as a module over its centre, and the latter is a noetherian ring. Thus R is a noetherian ring, and for any M in mod R, the category of finitely generated left R-modules, the ring End R (M ) is also a noether algebra, and hence noetherian.
The grade of M ∈ mod R is defined to be . When R is commutative, this is the length of a longest regular sequence in the annihilator of the R-module M ; see, for instance, [9,Theorem 16.7].
A finitely generated R-module M is d-torsionfree, for some positive integer d, if where Tr M be the Auslander transpose of M ; see [2]. This is equivalent to the condition that M is the d-th syzygy of an R-module N satisfying Ext i R (N, R) = 0 for 1 ≤ i ≤ d; see [2].
Given R-modules X and Y we write Hom R (X, Y ) for the quotient of Hom R (X, Y ) by the abelian subgroup of morphisms factoring through projective R-modules.
Lemma 3. Let 0 → X → Y → Z → 0 be an exact sequence of R-modules. If an R-module W satisfies Hom R (W, Z) = 0, then the following sequence is exact.
Proof. By hypothesis any morphism f : W → Z factors as W → P f ′ − → Z, where P is a projective R-module, and since f ′ lifts to Y , so does f .
As usual, we write ΩX for a syzygy of X.
Lemma 4. Let X and Y be finitely generated R-modules.

Proof. Part (1) is clear, and implies part (2) for its hypotheses yields
Hom R (Ω c X, Ω c+n Y ) ∼ = Hom R (X, Ω n Y ) and the right-hand module is zero as Hom R (X, R) = 0 implies Hom R (X, Ω n Y ) = 0, since Ω n Y is a submodule of a projective R-module. Since Hom R (X, R) = 0 one obtains the equality below while the isomorphism is obtained by repeated application of Lemma 4(1), noting that c < grade R X. Therefore, to verify the claim, it is enough to prove This justifies the assertion about Cok(g). As to the exactness, for each 0 ≤ i ≤ n set K i := Im(f i ), where f i are the maps in (B). Then there are exact sequences For each i ≥ 1, using the fact that M i is d-torsionfree, and K 0 = Ω c X, it follows by induction that K i is a (c + 1)-st syzygy. Lemma 4(2) then yields that Hom R (Ω c X, K i ) = 0 for i ≥ 1. By Lemma 3, one then obtains an exact sequence Thus the sequence (C) is exact, as desired.
Recall that a commutative ring R is regular if it is noetherian and every localization at a prime ideal has finite global dimension. When R is further equicodimensional, the global dimension of R is finite, since it equals dim R.
Set M 0 = R and for each integer 1 ≤ j ≤ n, set We prove, by an induction on j, that M j is c j -torsionfree and that gldim End R (M j ) ≤ 2 j dim R + (2 j − 1)(gldim End R (N ) + 1).
The base case j = 0 is a tautology, for R is regular and hence its global dimension equals dim R. Assume the inequality holds for j − 1 for some integer j ≥ 1.
For the induction step, set M = M j−1 , so that Since R is equicodimensional, grade R N = dim R and M j−1 is c j−1 -torsionfree, Theorem 1 applies to yield that M j is c j -torsionfree, and further that gldim End R (M j ) ≤ 2 gldim End R (M j−1 ) + gldim End R (N ) + 1.
Applying the induction hypothesis gives the desired upper bound for the global dimension of End R (M j ).