Our main theorem of this chapter is that if an order Λ is n-canonical and has only finitely many nonsiomorphic indecomposable modules in ΩnCM Λ, then Λ has finite global dimension
on the punctured spectrum of R. Here we give such orders a name and investigate some of their properties. These are direct generalizations of isolated singularities and non-singular orders which have been investigated by various authors; see e.g., [27,29,32].
Definition 4.2.1. Let Λ be an order over a CM ring R. We call Λ an n-isolated singularity if
gldim Λp6 n + dim Rp
for all non-maximal prime ideals p ∈ Spec R. We say Λ is n-nonsingular if gldim Λp =
n + dim Rp for all p ∈ Spec R.
The following category is of central importance here, so we endow it with its own notation. Notation 4.2.2. Denote by S the additive closure of the full subcategory of nth syzygies of maximal Cohen-Macaulay Λ-modules, i.e., S = add{ΩnCM(Λ)}
The rest of this Chapter focuses on the class of modules S. Generally, over n-canonical orders, modules in S exhibit similar behavior to MCM modules over Gorenstein local rings. We begin by noting that Λ is an n-isolated singularity if and only if modules in S are projective on the punctured spectrum. To do this, we need the following lemma.
Lemma 4.2.3. Let Λ be an n-isolated singularity over a CM local ring R. Then if M ∈ CM(Λ) we have
projdim Mp 6 n
for all non-maximal primes p ∈ Spec R.
Proof. Let M ∈ CM Λ. It follows that Mp ∈ CM Λp. Pick a maximal Mp-regular sequence
x1, . . . , xt ∈ pRp. We have an exact sequence
0 −→ Mp x1
−→ Mp−→ Mp/x1Mp−→ 0
which induces an exact sequence ExtiΛp(M, −) x1 −→ Exti Λp(M, −) −→ Ext i+1 Λp (M/x1M, −) −→ Ext i+1 Λp (M, −).
It follows from this that projdimΛpMp/x1Mp= projdimΛpMp+ 1, and by induction we get
projdimΛpMp/(x1, . . . , xt)Mp = projdimΛpMp+ dim(Rp). Since gldim Λp = n + dim(Rp), it
must be that projdimΛpMp 6 n.
From this we get the following useful characterization of n-isolated singularities.
Corollary 4.2.4. Let Λ be an order over a CM local ring R. Then Λ is an n-isolated singularity if and only if for all X ∈ S, Xp is a projective Λp-module for all non-maximal
primes p ∈ Spec R.
Proof. (⇒): This follows at once from the previous lemma.
(⇐): This is clear by the Depth Lemma. Since any dth syzygy of a Λ-module is in CM Λ,
we have nth syzygies of modules in CM Λ are projective on the punctured spectrum. It
follows that (n + d)th syzygies of arbitrary (finitely generated) Λ-modules are projective on
the punctured spectrum. This implies Λ is an n-isolated singularity.
Lemma 4.2.5. Let R be a CM local ring with canonical module ω. Let Λ be an R-order. Then Λ is an n-isolated singularity if and only if `R(Ext1Λ(N, M )) < ∞ for all M, N ∈ S,
where `R(−) denotes the length of an R-module.
Proof. (⇒): Recall that an R-module M has finite length if and only if `R(M ) < ∞for all
non-maximal p ∈ Spec R. Since N ∈ S, by Corollary 4.2.4 we know Np is projective on the
punctured spectrum. It follows that ExtΛ(N, M )p= ExtΛp(Np, Mp) = 0 for all non-maximal
p∈ Spec R.
(⇐): Suppose `(Ext1
Λ(N, M )) < ∞for all M, N ∈ S. Suppose there is a prime ideal p for
which gldim Λp > n + dim(Rp). Then there must exist a maximal Cohen-Macaulay module
M so that projdim X > 1 for X = ΩnM. Consider the short exact sequence
0 −→ ΩX −→ F −→ X −→ 0, where F is a free module since X, ΩX ∈ S, by assumption Ext1
Λ(X, ΩX) = 0. This means
the above sequence splits, and hence X is projective, a contradiction. Thus it must be that gldim Λp 6 n + dim(Rp).
The next proposition illustrates that nth syzygies (of MCM modules) over an n-isolated
singularity behave like MCM modules over an isolated singularity. This is shown for the n = 0case in [27, Theorem 1.3.1]; the proof is largely the same except the d = 2 case of part (1).
Proposition 4.2.6. Let Λ be an n-isolated singularity over a d-dimensional CM local ring R. For X ∈ S: 1. Exti Λ(Tr Xop, Λ) = 0 for i = 1, . . . , d. 2. Exti Λ(X, Y ), Tor Λ
i (Z, X), and HomΛ(X, Y ) are all finite length over R for any Y ∈
Proof. For assertion (1), we note that if d = 0 there is nothing to show. In the case where d = 1, the fact that X is projective on the punctured spectrum implies that Ext1Λ(Tr Xop, Λ)
has finite length since Tr Xop
p = 0 for any non-maximal prime ideal p. Then the well-known
exact sequence (see, e.g., [35, Proposition 12.8])
0 −→ Ext1Λ(Tr Xop, Λ) −→ X −→ X∗∗−→ Ext2Λ(Tr Xop, Λ) −→ 0 shows that Ext1
Λ(Tr Xop, Λ) embeds in X. But, depth X > 1 since d > 1 so that X is
torsion-free. Thus, Ext1
Λ(Tr Xop, Λ) = 0.
Now suppose d > 2. We still have that Ext1
Λ(Tr Xop, Λ) = 0by the above case. Thus, we
have an exact sequence
0 −→ X −→ X∗∗−→ Ext2 Λ(Tr X
op, Λ) −→ 0.
By virtue of being a dual module, X∗∗ has depth over R at least 2, by Lemma 2.1.5. Since
Ext2Λ(Tr Xop, Λ)is of finite length, hence depth zero, the Depth Lemma implies depthRX = 1. This is a contradiction since d > 2 and X ∈ CM Λ. Thus, we must have Ext2Λ(Tr Xop, Λ) = 0. It now follows from the above exact sequence that X ∼= X∗∗.
Finally, suppose Tr Xop ∈ ⊥k−1Λ for some 3 6 k 6 d. We begin with a projective
resolution
... −→ Pk−→ Pk−1−→ . . . −→ P1 −→ P0 −→ Tr Xop −→ 0.
Dualizing the above exact sequence, and utilizing the fact that X ∼= X∗∗, we get an exact sequence
0 −→ X −→ P2∗ −→ P3∗ −→ . . . −→ Pk−1∗ −→ (ΩkX)∗ −→ Extk Λ(Tr X
op, Λ) −→ 0,
where depthR(ΩkX)∗ > 2, again by Lemma 2.1.5 since it is a dual module. Now, if
has finite length and P∗
i ∈ CM Λfor all i. This is impossible since X ∈ CM Λ. Thus, it must
be that Extk
Λ(Tr Xop, Λ) = 0. Thus part (1) is proved by induction.
The next result shows that n-nonsingular orders have trivial category of nth syzygies.
This is the analog of [32, Prop 2.17], and the proof is largely the same.
Proposition 4.2.7. Let Λ be an order over a CM ring R with canonical module ωR. The
following are equivalent: 1. Λ is n-nonsingular.
2. gldim Λm 6 n + d for all maximal ideals m ∈ Spec R.
3. CM Λ ⊂ projdim6nΛ.
4. projdimΛopωΛ 6 n and gldim Λ < ∞.
Proof. The first 3 implications are the same argument as [32], but we include them for the convenience of the reader. (1) ⇒ (2) This is immediate.
(2) ⇒ (3) This proof is nearly identical to the proof of Lemma 4.2.3.
(3) ⇒ (4) Since ωΛ ∈ CM Λ, we know it has projective dimension at most n by (3).
Further, since each dth syzygy is MCM by the depth lemma, we also have gldim Λ < ∞.
(4) ⇒ (1) Fix a non-maximal prime ideal p ∈ Spec R and let X be in CM(Λp). We will
show that projdimΛpX 6 n and the depth lemma will conclude the proof as in the previous
step. Since localization can only reduce projective dimension, we have that projdimΛopp ωΛp 6
n and gldim Λp< ∞. The result then follows from Theorem 4.1.2
Remark 4.2.8. One might ask if we can strengthen condition (3) to be a set equality. If n > 1, the answer is no: consider a regular sequence x = x1, . . . , xd on Λ, and take the
Koszul complex over Λ on x. Then this is exact, and has length d. Then Ωd−1(Λ/xΛ) has
depth d − 1 by the Depth Lemma, but the end of the Koszul complex gives a length one resolution. Thus Ωd−1(Λ/xΛ) ∈ projdim
6nΛbut is not in CM Λ. We finally note that (3) is