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arXiv:2503.24186v1 [math.AC] 31 Mar 2025
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION OF
SINGULARITY CATEGORIES
SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Abstract. Let Rbe a commutative Noetherian ring. We establish a close relationship between the
(strong) generation of the singularity category of R, the nonvanishing of the annihilator of the singularity
category of R, and the nonvanishing of the cohomological annihilator of modules. As an application,
we prove that the singularity category of Rhas a strong generator if and only if the annihilator of
the singularity category of Ris nonzero when Ris a Noetherian domain with Krull dimension at most
one. Furthermore, we relate the generation of the singularity category and the extension generation
of the module category. Additionally, we introduce the notion of the co-cohomological annihilator of
modules. If the category of finitely generated R-modules has a strong generator, we show that the
infinite injective dimension locus of a finitely generated R-module Mis closed, with the defining ideal
given by the co-cohomological annihilator of M.
1. Introduction
Let Rbe a commutative Noetherian ring. In [28], Iyengar and Takahashi introduced the notion of the
cohomological annihilator of a ring R, denoted ca(R); cf. 3.3. They observed a close relationship between
the condition ca(R)6= 0 and the strong generation of the category mod(R) consisting of finitely generated
R-modules; cf. 2.9. Specifically, they prove: For a reduced ring R, if mod(R) has a strong generator,
then ca(R) contains a nonzero divisor, and hence ca(R)6= 0; For a ring Rwith finite Krull dimension, if
there exists an integer ssuch that the s-th cohomological annihilator of the ring R/p, denoted cas(R/p),
is nonzero for each prime ideal pof R, then mod(R) has a strong generator. It is established in [28] that
mod(R) has a strong generator when Ris a localization of a finitely generated algebra over a field or an
equicharacteristic excellent local ring.
For a commutative Noetherian domain R, Elagin and Lunts [19] observed that ca(R)6= 0 if the bounded
derived category of R, denoted Df(R), has a strong generator; cf. 2.10. Recently, the first author, Lank,
and Takahashi [16] established that the condition ca(R/p)6= 0 for each prime ideal pof Ris equivalent
to that mod(R/p) has a strong generator for each prime ideal pof R, which in turn is equivalent to that
Df(R/p) has a strong generator for each prime ideal pof R. If these conditions hold, they show that
mod(R) has a strong generator. As an application, it is proved in [16] that mod(R) has a strong generator
for any quasi-excellent ring Rwith finite Krull dimension.
The singularity category of R, denoted Dsg(R), was introduced by Buchweitz [12] and Orlov [38] as
the Verdier quotient of the bounded derived category by the full subcategory of perfect complexes. We
are motivated by the natural question:
Question. How can the (strong) generation of the singularity category be characterized?
Interestingly, Iyengar and Takahashi [28] observed that the singularity category Dsg(R/p) has a gen-
erator for each prime ideal pof Rif and only if the bounded derived category Df(R/p) has a generator
for each prime ideal pof R, and this is also equivalent to that the module category mod(R/p) has a
generator for each prime ideal pof R.
We investigate the above question in Section 4. It turns out that there is a close connection between
the (strong) generation of the singularity category of Rand the nonvanishing of the annihilator of the
singularity category of R, as well as the nonvanishing of the cohomological annihilator of modules.
The annihilator of the singularity category of R, denoted annRDsg(R), is an ideal of Rconsisting of
elements in Rthat annihilate all homomorphisms of complexes in Dsg (R); cf. 4.1. The ideal annRDsg (R)
2020 Mathematics Subject Classification. 13D09 (primary); 13C60, 13D05, 13D07, 18G80 (secondary).
Key words and phrases. (strong) generator, module category, singularity category, annihilator of the singularity category,
cohomological annihilator, infinite projective/injective dimension locus.
1
2 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
measures the singularity of Rin the sense that Ris regular if and only if annRDsg(R) = R. The
annihilator of the singularity category has recently attracted increasing interest and has been studied in
[20,21,35,36]. For a commutative Noetherian domain R, we prove that if the singularity category of R
has a strong generator, then the annihilator annRDsg (R) is nonzero; see Corollary 4.3. We do not know
whether the converse holds in general. However, if, in addition, the Krull dimension of Ris less or equal
to one, the converse holds. This is a consequence of Theorem 1.1 (2); see Corollary 4.21.
Theorem 1.1 is the main result in Section 4regarding the characterization of the (strong) generation of
singularity categories. The key input is the notion of the cohomological annihilator of a finitely generated
R-module M, denoted caR(M), in Section 3. By definition, caR(M) = [
n≥0
can
R(M), where can
R(M)
consists of elements rin Rsuch that r·Exti
R(M, N ) = 0 for each i≥nand each N∈mod(R). For each
Min mod(R), we observe that caR(M) coincides with the annihilator annDsg(R)(M) of the endomorphism
of Min Dsg(R); see Proposition 4.2. The importance of the ideal caR(M) lies in its role in defining the
infinite projective dimension locus of M; see Proposition 3.10.
Theorem 1.1. (See 4.2,4.17, and 4.25) Let Rbe a commutative Noetherian ring. Then:
(1) The following three conditions are equivalent.
(a) Dsg(R/p)has a generator for each prime ideal pof R.
(b) \
M∈mod(R/p)qannDsg (R/p)(M)6= 0 for each prime ideal pof R.
(c) \
M∈mod(R/p)qcaR/p(M)6= 0 for each prime ideal pof R.
(2) Assume Rhas isolated singularities. Then the following four conditions are equivalent.
(a) Dsg(R)has a strong generator.
(b) R/ annRDsg(R)is either 0or Artinian.
(c) R/(\
M∈mod(R)
caR(M)) is either 0or Artinian.
(d) mod(R)has a point-wise strong generator; cf. 4.16.
The key ingredient in proof of Theorem 1.1 (2) is the observation that annRDsg(R) coincides with
\
M∈mod(R)
caR(M); see Corollary 4.3. It is worth noting that if mod(R) has a strong generator, then
annRDsg(R) coincides with the cohomological annihilator ca(R), up to the radical; see Proposition 4.9.
When Ris, in addition, a Gorenstein local ring, Theorem 1.1 (2) was proved by Bahlekeh, Hakimian,
Salarian, and Takahashi [4] in terms of ca(R); see Remark 4.18. Moreover, Corollary 4.19 provides a new
class of rings satisfying the equality Sing(R) = V(ca(R)) which is not covered in [4]; see Remark 4.20.
Our second result studies the connection between the generator of the singularity category and the
extension generator of the module category; cf. 5.5. Theorem 1.2 (1) and (2) establish the relationship
between the existence of the extension generator of the module category and the finiteness of the Krull
dimension. Moreover, Theorem 1.2 (2) extends a recent result of Araya, Iima, and Takahashi [1] on the
generation of syzygy modules out of a single module by only taking direct summands and extensions; see
Remark 5.16. Theorem 1.2 (3) and (4) provide analogs of the results in [16,28] in terms of extension
generators.
Theorem 1.2. (See 5.15,5.18, and 5.19) Let Rbe a commutative Noetherian ring. Then:
(1) If mod(R)has an extension generator, then the Krull dimension of Ris finite.
(2) If the Krull dimension of Ris finite and the singular locus of Ris a finite set, then mod(R)has
an extension generator.
(3) The following are equivalent.
(a) Rhas finite Krull dimension and Dsg (R/p)has a generator for each prime ideal pof R.
(b) mod(R/p)has an extension generator for each prime ideal pof R.
(4) If any of the equivalent conditions in (3) holds, then mod(R/I )has an extension generator for
each ideal Iof R.
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 3
For an excellent ring R, Greco and Marinari [23] observed that IID(R) is closed in the spectrum
Spec(R) endowed with the Zariski topology; IID(M) = {p∈Spec(R)|idRp(Mp) = ∞} is the infinite
injective dimension locus of an R-module M, cf. 5.3. Takahashi [41] extended this result by proving that,
for an excellent ring Rand a finitely generated R-module M, IID(M) is closed in Spec(R).
We hope to determine the defining ideal of IID(M) when it is closed. To this end, we introduce
in Section 5the concept of the co-cohomological annihilator of an R-module M, denoted cocaR(M).
By definition, cocaR(M) = [
n≥0
cocan
R(M), where cocan
R(M) consists of elements rin Rsuch that r·
Exti
R(N, M ) = 0 for each i≥nand N∈mod(R). The main result in this section concerns the relation
between the co-cohomological annihilator and the infinite injective dimension locus.
Theorem 1.3. (See 5.7) Let Rbe a commutative Noetherian ring and Mbe a finitely generated R-
module. Then:
(1) If mod(R)has an extension generator, then IID(M)is closed in Spec(R).
(2) If mod(R)has a strong generator, then
IID(M) = V(cocaR(M)).
For a quasi-excellent ring Rwith finite Krull dimension and a finitely generated R-module M, it follows
from [16] that mod(R) has a strong generator. Consequently, by Theorem 1.3, IID(M) is closed with the
defining ideal cocaR(M).
Additionally, we compare the cohomological annihilator with the co-cohomological annihilator of a
finitely generated module M. It turns out that these annihilators coincide when Ris Gorenstein with
finite Krull dimension and Mis maximal Cohen–Macaulay; see Proposition 5.21.
Acknowledgments. We would like to thank Kaito Kimura and Ryo Takahashi for their helpful dis-
cussions and valuable comments related to this work. The first author was partially supported by the
Charles University Research Center program No.UNCE/SCI/022 and a grant GA ˇ
CR 23-05148S from
the Czech Science Foundation. The second author was supported by the National Natural Science Foun-
dation of China (No. 12401046) and the Fundamental Research Funds for the Central Universities (No.
CCNU24JC001). The fourth author was partly supported by Grant-in-Aid for JSPS Fellows 23KJ1119.
2. Notation and Terminology
Throughout this article, Rwill be a commutative Noetherian ring. For each R-module M, let pdR(M)
denote the projective dimension of Mover R, and idR(M) denote the injective dimension of Mover
R. We write mod(R) to be the category of finitely generated R-modules. Let dim(R) denote the Krull
dimension of R, and gl.dim(R) denote the global homological dimension of R.
2.1. (Strongly) Gorenstein rings. A commutative Noetherian ring Ris called strongly Gorenstein
provided that idR(R)<∞. For instance, regular local rings and complete intersection rings are both
examples of strongly Gorenstein rings.
A commutative Noetherian ring Ris said to be Gorenstein provided that Rpis strongly Gorenstein
(i.e., idRp(Rp)<∞) for each prime ideal pof R. Note that a commutative Noetherian local ring is
Gorenstein if it is strongly Gorenstein.
For a Gorenstein ring R, idR(R) = dim(R); this can be proved by combining [11, Theorem 3.1.17] and
[6, Corollary 2.3]. In particular, a commutative Noetherian ring Ris strongly Gorenstein if and only if it
is Gorenstein with finite Krull dimension.
2.2. Regular rings. A commutative Noetherian local ring is called regular if its maximal ideal can be
generated by a system of parameter. Auslander, Buchsbaum, and Serre observed that a commutative
Noetherian local ring is regular if and only if its global homological dimension is finite; see [11, Theorem
2.2.7].
A commutative Noetherian ring Ris called regular if Rpis regular for each prime ideal pof R.
Furthermore, a commutative Noetherian ring is regular if and only if every finitely generated module
has finite projective dimension; see [7, Lemma 4.5] for the forward direction, and the backward direction
follows from the criterion of Auslander, Buchsbaum, and Serre.
For a regular ring, gl.dim(R) = dim(R); see [34, Theorem 5.94]. In particular, a commutative Noe-
therian ring is regular with finite Krull dimension if and only if gl.dim(R) is finite.
4 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
2.3. Annihilator and support of modules. For each R-module M, let annR(M) denote the annihi-
lator of Mover R. That is, annR(M):={r∈R|r·M= 0}.
The set of all prime ideals of Ris denoted by Spec(R). It is endowed with the Zariski topology; the
closed subset in this topology is of the form V(I):={p∈Spec(R)|p⊇I}for each ideal Iof R.
For each R-module M, the support of Mis
SuppRM:={p∈Spec(R)|Mp6= 0},
where Mpis the localization of Mat p. Note that SuppR(M)⊆V(annR(M)); the equality holds if, in
addition, Mis finitely generated.
2.4. Syzygy modules. For a finitely generated R-module Mand n≥1, we let Ωn
R(M) denote the n-th
syzygy of M. That is, there is a long exact sequence
0→Ωn
R(M)→P−(n−1) → ···P−1→P0→M→0,
where P−iare finitely generated pro jective R-modules for 0 ≤i≤n−1. By Schanuel’s Lemma, Ωn
R(M)
is independent of the choice of the projective resolution of Mup to projective summands. By convention,
Ω0
R(M) = Mfor each finitely generated R-module M. Let Cbe a full subcategory of mod(R), we denote
Ωn
R(C):={Ωn
R(M)|M∈ C}.
We say a finitely generated R-module Mis an infinite syzygy if there exists an exact sequence 0 →
M→Q0→Q1→Q2→ ··· ,where Qiare finitely generated pro jective R-modules for i≥0.
2.5. Maximal Cohen–Macaulay modules. For a finitely generated R-module M, it is said to be
maximal Cohen–Macaulay provided that depth(Mp)≥dim(Rp) for each prime ideal pof R, where
depth(Mp) represents the depth of Mpover Rp; see details in [11]. Let CM(R) denote the full subcategory
of mod(R) consisting of maximal Cohen–Macaulay R-modules. A commutative Noetherian ring is Cohen–
Macaulay if Ris in CM(R).
Assume Ris a Gorenstein ring. Note that a finitely generated R-module Mis maximal Cohen–
Macaulay if and only if Exti
R(M, R) = 0 for all i > 0; this can be proved by using Ischebeck’s formula
(see [11, Exercise 3.1.24]). Then it follows from [12, Theorem B.1.6] that, for each M∈mod(R), Ωs
R(M) is
maximal Cohen–Macaulay for some s≥0. On the other hand, any maximal Cohen–Macaulay R-module
is an infinite syzygy; see [12, Theorem B.1.3].
2.6. Thick subcategories of Abelian categories. Let Abe an Abelian category. A full subcategory
Cof Ais called thick if Cis closed under direct summands, and it contains objects that fit into a short
exact sequence such that the other two objects are in C.
For each object Xin A. Let thickA(X) denote the smallest thick subcategory of Acontaining X; the
notation thickAwill have no confusion with thickTin 2.7 for a triangulated category T. This can be
constructed inductively; it is an analogous construction of a thick subcategory in a triangulated category
(see 2.7).
Set thick0
A(X):={0}. Let thick1
A(X) be the full subcategory of Aconsisting of ob jects which are
direct summands of any finite copies of X. For n≥2, let thickn
A(X) denote the full subcategory of A
consisting of direct summands of any object in Athat fits into a short exact sequence
0→Y1→Y2→Y3→0,
where the other two objects satisfying: one is in thickn−1
A(X), and the other one is in thick1
A(X). Note
that thickA(X) = [
n≥0
thickn
A(X).
2.7. Thick subcategories of triangulated categories. Let Tbe a triangulated category with a
suspension functor [1]. A full category Cof Tis called thick if it is closed under suspensions, cones, and
direct summands.
For each object Xin T, let thickT(X) denote the smallest thick subcategory of Tcontaining X.
thickT(X) can be constructed inductively as below; see [2, Section 2] for more details.
First, thick0
T(X):={0}. Let thick1
T(X) be the smallest full subcategory of Tcontaining X, and it is
closed under suspensions, finite direct sums, and direct summands. Inductively, thickn
T(X) is denoted to
be the full subcategory of Tconsisting of objects Yin Tthat appear in an exact triangle
Y1→Y⊕Y′→Y2→Y1[1],
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 5
where Y1∈thickn−1
T(X) and Y2∈thick1
T(X). Note that thickT(X) = [
n≥0
thickn
T(X).
2.8. Derived categories and singularity categories. Let D(R) denote the derived category of R-
modules. This is a triangulated category with a suspension functor [1]; for each complex X, (X[1])i=
Xi+1 and ∂X[1] =−∂X. The bounded derived category, denoted Df(R), is the full subcategory of D(R)
consisting of complexes Xsuch that its total cohomology, denoted M
i∈Z
Hi(X), is finitely generated over
R; note that Df(R) is a thick subcategory of D(R).
For each complex Xin D(R), it is called perfect if X∈thickD(R)(R). It turns out that a complex
in D(R) is perfect if and only if it is isomorphic to a bounded complex of finitely generated projective
R-modules; this can be proved by using [12, Lemma 1.2.1]. Note that thickD(R)(R)⊆Df(R).
The singularity category of Ris defined to be the Verdier quotient
Dsg(R):=Df(R)/thickD(R)(R).
This category was introduced by Buchweitz [12, Definition 1.2.2] under the name “stable derived category”
and later also found by Orlov [38, 1.2]. This category detects the singularity of Rin the sense that Dsg (R)
is trivial if and only if Ris regular.
The following definition of “strong generator” in 2.9 differs from that of Iyengar and Takahashi [27,
4.3]. However, for a commutative Noetherian ring R, mod(R) has a strong generator in the sense of 2.9
if and only if mod(R) has a strong generator in the sense of Iyengar and Takahashi; see [27, Corollary
4.6].
2.9. (Strong) generators of Abelian categories. Let Abe an Abelian category and Gbe an object
in A. The object Gis called a generator of Aif thickA(G) = A, and Gis called a strong generator of A
if thickn
A(G) = Afor some n≥0.
For example, if Ris an Artinian ring R,R/J (R) is a strong generator of mod(R), where J(R) is the
Jacobson radical of R. Moreover, mod(R) = thickℓℓ(R)
mod(R)(R/J (R)), where ℓℓ(R):= inf {n≥0|J(R)n=
0}is the Loewy length of R.
2.10. (Strong) generators of triangulated categories. Let Tbe a triangulated category and G
be an object in T. The ob ject Gis called a generator of Tif thickT(G) = T, and Gis called a strong
generator of Tif thickn
T(G) = Tfor some n≥0; see details in [9] and [39]. Note that Thas a strong
generator if and only if the Rouquier dimension (see [39]) of Tis finite.
If Ris an Artinian ring, then R/J(R) is a strong generator of Df(R); indeed, Df(R) =
thickℓℓ(R)
D(R)(R/J(R)) by [39, Proposition 7.37].
We end this section by recording the (strong) generators of the regular rings.
2.11. Let Rbe a commutative Noetherian ring. Note that, for each G∈mod(R), Gis a generator of
Df(R) if Gis a generator of mod(R). Combining with this, the following are equivalent.
(1) Ris regular.
(2) Ris a generator of mod(R).
(3) Ris a generator of Df(R).
2.12. Let Rbe a commutative Noetherian ring and Gbe a module in mod(R). If mod(R) =
thickn
mod(R)(G), then Df(R) = thick2n
D(R)(G) by [27, Lemma 7.1]. In particular, Gis a strong genera-
tor of Df(R) if Gis a strong generator of mod(R). This yields (2) ⇒(3) below. Indeed, all the following
conditions are equivalent.
(1) Ris regular with dim(R)<∞, equivalently gl.dim(R)<∞(see 2.2).
(2) Ris a strong generator of mod(R).
(3) Ris a strong generator of Df(R).
(1) ⇒(2): if gl.dim(R) = dis finite, then mod(R) = thickd+1
mod(R)(R). (3) ⇒(1): if Df(R) = thickn
D(R)(R),
then [13, Proposition 4.5 and 4.6] yields that pdR(M)≤n−1 for each M∈mod(R), and hence
gl.dim(R)≤n−1.
Moreover, if the above conditions hold, then mod(R) = thickd+1
mod(R)(R) and Df(R) = thickd+1
D(R)(R) (see
[13, Theorem 8.3] ), where d= dim(R).
6 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
3. Cohomological annihilators for modules
The following definition of cohomological annihilators for modules is inspired by the definition of
cohomological annihilators for rings introduced by Iyengar and Takahashi [27]; see their definition in 3.3.
For a subset I⊆Nand full subcategories C,Dof mod(R), set ExtI
R(C,D):=L
i∈IL
M∈C,N ∈D
Exti
R(M, N ).
3.1. For each finitely generated R-module Mand n≥0, we define the n-th cohomological annihilator of
Mto be
can
R(M):= annRExt≥n
R(M, mod(R)).
That is, can
R(M) consists of elements r∈Rsuch that r·Exti
R(M, N ) = 0 for each i≥nand N∈mod(R).
Consider the ascending chain of ideals
annR(M) = ca0
R(M)⊆ca1
R(M)⊆ca2
R(M)⊆ ··· ,
the cohomological annihilator of Mis defined to be the union of these ideals
caR(M):=[
n≥0
can
R(M).
Since Ris Noetherian, caR(M) = can
R(M) for n≫0.
Note that can
R(M) = Rif and only if pdR(M)< n. Thus, caR(M) = Rif and only if pdR(M)<∞.
3.2. Let Cbe a full subcategory of mod(R), Bahlekeh, Salarian, Takahashi, and Toosi [5, Definition 5.7]
introduced the concept of cohomological annihilator of C, denoted caR(C). It is defined as
caR(C):=\
X∈C
caR(X).
By definition, if Cconsists of only one module X, then caR(C) = caR(X).
3.3. For each n≥0, following [27, Definition 2.1], the n-th cohomological annihilator of the ring Ris
defined to be
can(R):= annRExt≥n
R(mod(R),mod(R)).
By definition, can(R) = \
M∈mod(R)
can
R(M) and there is an ascending chain of ideals
0 = ca0(R)⊆ca1(R)⊆ca2(R)⊆ ··· .
The cohomological annihilator of the ring Ris defined to be the ideal
ca(R):=[
n≥0
can(R).
Also, ca(R) = can(R) for n≫0 as Ris Noetherian.
Example 3.4. Let Rbe an equicharacteristic complete Cohen–Macaulay local ring with Krull dimension
d. Wang [43, Theorem 5.3] observed that cad+1(R) contains the Jacobian ideal of R; see also [29, Theorem
1.2].
Lemma 3.5. (1) ca(R)⊆caR(mod(R)).
(2) caR(mod(R)) = Rif and only if Ris regular.
(3) ca(R) = Rif and only if Ris regular with finite Krull dimension.
Proof. (1) For each n≥0, we have
can(R) = \
M∈mod(R)
can
R(M)⊆\
M∈mod(R)
caR(M) = caR(mod(R)).
Thus, ca(R)⊆caR(mod(R)) as ca(R) = can(R) for n≫0.
(2) By definition, caR(mod(R)) = Rif and only if caR(M) = Rfor each M∈mod(R). This is
equivalent to pdR(M)<∞for each M∈mod(R); see 3.1. This is equivalent to that Ris regular; see
2.2.
(3) By [27, Example 2.5], ca(R) = Rif and only if gl.dim(R)<∞. The desired result now follows
immediately from 2.2.
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 7
In general, ca(R)6= caR(mod(R)); see the example below. They are equal if, in addition, Ris strongly
Gorenstein; see Proposition 3.13.
Example 3.6. The inclusion in Lemma 3.5 (1) can be proper. In [37, Appendix, Example 1], Nagata
constructed a commutative Noetherian regular ring Rwith infinite Krull dimension. In this case, ca(R)$
caR(mod(R)) = Rby Lemma 3.5.
3.7. Let M, N be R-modules. If Mis finitely generated, then it follows from [44, Proposition 3.3.10]
that there is a natural isomorphism
Extn
R(M, N )p∼
=Extn
Rp(Mp, Np)
for each p∈Spec(R) and n≥0.
Lemma 3.8. Let Mbe a finitely generated R-module and n≥1. Then:
(1) can
R(M) = annRExtn
R(M, Ωn
R(M)) = annREndR(Ωn−1
R(M)).
(2) {p∈Spec(R)|pdRp(Mp)≥n}=V(can
R(M)) = SuppRExtn
R(M, Ωn
R(M)).
(3) can
R(M) = annRExtn
R(M, mod(R)).
(4) can+i
R(M) = can
R(Ωi
R(M)) for each i≥1. Hence, caR(M) = caR(Ωi
R(M)) for each i≥1.
(5) Let Sbe a multiplicatively closed subset of R. Then S−1can
R(M) = can
S−1R(S−1M). In particular,
S−1caR(M) = caS−1R(S−1M).
Proof. (1) This follows from [27, Lemma 2.14] and [18, Lemma 3.8].
(2) Combining with 3.7, pdRp(Mp)≥nif and only if p∈SuppRExtn
R(M, Ωn
R(M)). Since
Extn
R(M, Ωn
R(M)) is finitely generated over R,
SuppRExtn
R(M, Ωn
R(M)) = V(annRExtn
R(M, Ωn
R(M)).
Combining this with (1), the desired statement follows.
(3) It is clear that
can
R(M)⊆annRExtn
R(M, mod(R)) ⊆annRExtn
R(M, Ωn
R(M)).
By (1), these inclusions are equal.
(4) For each X∈mod(R), consider the isomorphism
Extn
R(Ω1
R(M), X)∼
=Extn+1
R(M, X ).
Combining with (3), we have can+1
R(M) = can
R(Ω1
R(M)). Then the first statement of (4) can be obtained
by induction on the number i. Thus, for each i≥1,
caR(M) = [
n≥0
can+i
R(M) = [
n≥0
can
R(Ωi
R(M)) = caR(Ωi
R(M)).
This completes the proof.
(5) The second statement follows from the first one. For the first statement, combining with (4), it
suffices to prove the case of n= 1. By (1), we get an inclusion S−1ca1
R(M)⊆ca1
S−1R(S−1M). For the con-
verse, assume r/s ∈ca1
S−1R(S−1M), where s∈Sand r∈R. It remains to prove that r/s ∈S−1ca1
R(M).
By (1), we get that the multiplication r/s :S−1M→S−1Mfactors through a finitely generated pro jective
S−1R-module, and hence r/s :S−1M→S−1Mfactors through a finitely generated free S−1R-module.
Combining with the fact that the isomorphism S−1HomR(X, Y )∼
=HomS−1R(S−1X, S−1Y) holds for
X, Y ∈mod(R), we conclude that there is a factorization
S−1M S−1M,
S−1F
r/s
α/s1β/s2
where Fis a finitely generated free R-module, s1, s2∈S,α:M→F, and β:F→M. From the
factorization, it follows that there exists t∈Ssuch that ts2s1r=tsβα. Note that r/s = (ts2s1r)/(ts2s1s)
and ts2s1r=tsβα factors through F. By (1), ts2s1r∈ca1
R(M), and hence r/s ∈S−1ca1
R(M).
8 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
3.9. Let Mbe a finitely generated R-module.
(1) Following [42, Definition 2.9], the infinite projective dimension locus of Mis defined to be
IPD(M):={p∈Spec(R)|pdRp(Mp) = ∞}.
(2) sup{pdRp(Mp)|p/∈IPD(M)}is finite. Indeed, it follows from the proof of [7, Lemma 4.5] that
{p∈Spec(R)|pdRp(Mp)<∞} ={p∈Spec(R)|pdRp(Mp)≤n}
for some n≥0, and hence sup{pdRp(Mp)|p/∈IPD(M)} ≤ n; see also [3, Theorem 1.1].
Proposition 3.10. Let Mbe a finitely generated R-module. Then
IPD(M) = V(caR(M)) = V(cad+1
R(M)),
where d= sup{pdRp(Mp)|p/∈IPD(M)}.
Proof. The first equality follows from the following:
IPD(M) = \
n≥1
SuppRExtn
R(M, Ωn
R(M))
=\
n≥1
V(can
R(M))
=V([
n≥1
can
R(M))
=V(caR(M)),
where the first two equalities follow from Lemma 3.8; the first equality also follows directly from the
proof of [28, Lemma 2.3].
Next, we prove the second equality. The inclusion V(caR(M)) ⊆V(cad+1
R(M)) is clear. Combining
with the first equality, it remains to prove V(cad+1
R(M)) ⊆IPD(M). If not, let pbe a prime ideal
containing cad+1
R(M) and p/∈IPD(M). By assumption, pdRp(Mp)≤d. However, it follows from Lemma
3.8 that pdRp(Mp)≥d+ 1. This is a contradiction.
Recall that CM(R) is denoted to be the category of maximal Cohen–Macaulay R-modules (see 2.5).
Set
ΩCM×(R):={M∈Ω1
R(CM(R)) |Mhas no nonzero projective summands};
see the definition of Ω1
R(CM(R)) in 2.4. Next, we calculate the uniform degree of the cohomological
annihilators. It turns out that the uniform degree behaves well when the ring is strongly Gorenstein; see
Proposition 3.13.
Lemma 3.11. (1) Let Mbe a finitely generated R-module such that Exti
R(M, R) = 0 for all 0≤m≤
i≤n, then cam
R(M) = ···= can+1
R(M)
(2) If (R, m)is a d-dimensional Cohen–Macaulay local ring with a canonical module ω. If ΩCM×(R)
is closed under HomR(−, ω), then cad+1
R(M) = caR(M)for each M∈mod(R). In particular, cad+1(R) =
ca(R) = caR(mod(R)).
Proof. (1) For each j≥0, consider the short exact sequence
0→Ωj+1
R(M)→P→Ωj
R(M)→0,
where Pis a finitely generated projective R-module. Applying HomR(M, −) to this short exact sequence,
we get an exact sequence
Extj
R(M, P )→Extj
R(M, Ωj
R(M)) →Extj+1
R(M, Ωj+1
R(M)).
Set j=m. The assumption yields that Extm
R(M, P ) = 0. Thus, the above exact sequence yields that
cam+1
R(M)⊆cam
R(M) (see Lemma 3.8), and hence cam
R(M) = cam+1
R(M). The desired result follows
similarly by choosing j=m+ 1,...,n.
(2) For each M∈mod(R) and i≥d, Ωi
R(M) is in CM(R). Choose a minimal free resolution of Ωi
R(M):
0→Ωi+1
R(M)→F→Ωi
R(M)→0,where Fis a finitely generated free R-module. In this case, Ωi+1
R(M)
does not have a nonzero pro jective summand (see [24, Lemma 1.4]), and hence Ωi+1
R(M)∈ΩCM×(R).
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 9
By assumption, HomR(Ωi+1
R(M), ω)∈ΩCM(R). It follows from [33, Proposition 5.2] that there exists a
short exact sequence
0→K→ω⊕s→Ωi+1
R(M)→0
in mod(R). Note that Ext>0
R(Ωi
R(M), ω) = 0 as Ωi
R(M)∈CM(R). Applying HomR(Ωi
R(M),−)
to the above short exact sequence, Ext1
R(Ωi
R(M),Ωi+1
R(M)) ∼
=Ext2
R(Ωi
R(M), K).We conclude
from this that Exti+1
R(M, Ωi+1
R(M)) ∼
=Exti+2
R(M, K ). This isomorphism implies cai+2
R(M)⊆
annRExti+1
R(M, Ωi+1
R(M)) = cai+1
R(M),where the second equality follows from Lemma 3.8. Hence,
cai+1
R(M) = cai+2
R(M) for each i≥d. This implies caR(M) = cad+1
R(M).
For the second statement, it follows from Lemma 3.5 that cad+1(R)⊆ca(R)⊆caR(mod(R)). The
desired result now follows from the following:
cad+1(R) = \
M∈mod(R)
cad+1
R(M) = \
M∈mod(R)
caR(M) = caR(mod(R)),
This finishes the proof.
3.12. For a finitely generated R-module M, let RfdR(M) denote the (large) restricted flat dimension of
M. That is,
RfdR(M) = sup{depth(Rp)−depth(Mp)|p∈Spec R}.
By [3, Theorem 1.1] with [14, Proposition 2.2 and Theorem 2.4], RfdR(M)∈N∪{−∞}, and RfdR(M) =
−∞ if and only if M= 0.
Proposition 3.13. Let Rbe a Gorenstein ring. Then:
(1) For each M∈CM(R),ca1
R(M) = caR(M).
(2) For each nonzero finitely generated R-module M,cal+1
R(M) = caR(M), where l= RfdR(M).
(3) If, in addition, Rhas finite Krull dimension d, then there are equalities
cad+1(R) = ca(R) = caR(mod(R)) = \
M∈CM(R)
annREndR(M).
Proof. (1) For each M∈CM(R), we have Ext>0
R(M, R) = 0; see 2.5. The desired result now follows from
Lemma 3.11.
(2) Note that 3.12 yields that l= RfdR(M) is finite. By [11, 1.3.7], Ωl
R(M) is maximal Cohen–
Macaulay over R. Let i≥1 be an integer. By Lemma 3.8, we have
cal+i
R(M) = annREndR(Ωl+i−1
R(M)).
Note that EndR(Ωl+i−1
R(M)) ∼
=EndR(Ωl
R(M)) as Ωl
R(M)∈CM(R); see [12, Theorem B.1.8]. Thus,
cal+i
R(M) = cal+1
R(M), and hence caR(M) = cal+1
R(M).
(3) In [20, Lemma 2.3], Esentepe observed that ca(R) = \
M∈CM(R)
annREndR(M).By the same ar-
gument in the proof of Lemma 3.11, we have cad+1(R) = ca(R) = caR(mod(R)). This completes the
proof.
Remark 3.14. (1) Keep the same assumption as Proposition 3.13 (3), the first author and the second
author proved ca(R) = cad+1(R) in [17, Proposition 3.4]. As mentioned in the proof, the equality
ca(R) = \
M∈CM(R)
annREndR(M) was due to Esentepe [20, Lemma 2.3].
(2) For a finitely generated R-module M, we define
nR(M) = inf{m≥0|cam+1
R(M) = caR(M)}.
Since Ris Noetherian, nR(M) is always finite. For instance, if pdR(M) is finite, then nR(M) = pdR(M).
For a full subcategory Cof mod(R), we set nR(C) = sup{nR(M)|M∈ C}. If nR(C)<∞, then
caR(C) = \
M∈C
caR(M) = \
M∈C
cak+1
R(M) for each k≥nR(C). Hence, if nR(mod(R)) <∞is finite, then
caR(mod(R)) = [
k≥nR(mod(R))
(\
M∈mod(R)
cak+1
R(M)) = ca(R).
10 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Note that the ring R, under the assumption of Lemma 3.11 (2) or Proposition 3.13 (2), satisfies
nR(mod(R)) is finite.
(3) Let (R, m) be a commutative Noetherian local ring with m2= 0. For each M∈mod(R), the
assumption that m2= 0 implies that Ω1
R(M) is a direct sum of some finite copies of Rand k. By Lemma
3.8 (1), we conclude that ca2
R(M) = caR(M) for each M∈mod(R), and hence nR(mod(R)) ≤1.
Let Max(R) denote the subset of Spec(R) consisting of all maximal ideals of R. In view of Remark
3.14 (2), the following result may be regarded as a version of the local-global principle for projective
dimensions.
Proposition 3.15. Let Rbe a commutative Noetherian ring. Then:
(1) For each M∈mod(R),nR(M) = sup{nRp(Mp)|p∈Spec(R)}= sup{nRm(Mm)|m∈Max(R)}.
In particular, nR(mod(R)) = sup{nRp(mod(Rp)|p∈Spec(R)}= sup{nRm(mod(Rm)|m∈
Max(R)}.
(2) nR(mod(R)) ≥dim(R)−1. In particular, if nR(mod(R)) is finite, then Rhas finite Krull
dimension.
Proof. (1) The second statement follows from the first one and the fact that the localization functor
mod(R)→mod(Rp) is dense for each p∈Spec(R). For the first statement, let n= nR(M) + 1. It follows
that can
R(M) = caR(M). By Lemma 3.8 (5), for each p∈Spec(R), we have
can
Rp(Mp) = can
R(M)p= caR(M)p= caRp(Mp).
It follows form this that nR(M)≥sup{nRp(Mp)|p∈Spec(R)} ≥ sup{nRm(Mm)|m∈Max(R)}.
Assume m= sup{nRm(Mm)|m∈Max(R)}+ 1. Again by Lemma 3.8 (5), we conclude that the
localization of the inclusion cam
R(M)⊆caR(M) at each maximal ideal of Ris equal, and hence cam
R(M) =
caR(M). It follows from this that nR(M)≤m= sup{nRm(Mm)|m∈Max(R)}.
(2) Let p∈Spec(R), and let Jbe an ideal of Rpgenerated by a maximal regular sequence of Rp. Since
Jhas finite projective dimension over Rp, we obtain the first equality below
nRp(Rp/J) = pdRp(Rp/J ) = depth(Rp)
where the second equality is by our choice of J. Combining with (1), we conclude that nR(mod(R)) ≥
sup{depth(Rp)|p∈Spec(R)} ≥ dim(R)−1,where the second inequality is by [14, Lemma 1.4].
4. (Strong) generation of singularity categories
In this section, we investigate the (strong) generation of the singularity category. The main result of
this section is Theorem 1.1 from the introduction; see Theorems 4.17 and 4.25.
4.1. Let Xbe a complex in the singularity category Dsg(R). The annihilator of Xover Dsg(R), denoted
annDsg(R)(X), is defined to be the annihilator of HomDsg (R)(X, X ) over R. That is,
annDsg(R)(X):={r∈R|r·HomDsg (R)(X, X ) = 0}.
The annihilator of Dsg (R) is defined to be
annRDsg(R):=\
X∈Dsg(R)
annDsg(R)(X);
see [35] for more details about the annihilator of the singularity category.
Proposition 4.2. Let Mbe a finitely generated R-module. Then
caR(M) = annDsg(R)(M).
Proof. First, we prove the caR(M)⊆annDsg (R)(M). Let r∈ca1
R(M). Note that ca1
R(M) =
annREndR(M); see Lemma 3.8. Thus, the multiplication map r:M→Mfactors through a projec-
tive R-module. In particular, the multiplication map r:M→Min the singularity category Dsg(R) is
zero. This yields that r∈annDsg (R)(M), and hence ca1
R(M)⊆annDsg(R)(M). For each n > 1,
can
R(M) = ca1
R(Ωn−1
R(M)) ⊆annDsg(R)(Ωn−1
R(M)),
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 11
where the equality is by Lemma 3.8, and the inclusion follows from the argument above. By the isomor-
phism Ωn−1
R(M)[n−1] ∼
=M, we have
annDsg(R)(Ωn−1
R(M)) = annDsg(R)(M).
Thus, can
R(M)⊆annDsg(R)(M) for all n > 1, and hence caR(M)⊆annDsg (R)(M).
Now, we prove the converse direction. Assume a∈annDsg(R)(M). That is, the multiplication map
a:M→Min Dsg(R) is zero. This implies the multiplication map a:M→Min Df(R) factors through
a perfect complex P. That is, in Df(R), there is a commutative diagram
M M.
P
a
Since Pis perfect, there exists n > 0 such that
HomDf(R)(P, N [i]) = 0
for all i≥nand N∈mod(R). Applying HomDf(R)(−, N [i]) on the above commutative diagram, we get
that the multiplication map
a: HomDf(R)(M, N [i]) →HomDf(R)(M, N [i])
is zero for all i≥nand N∈mod(R). Combining with the isomorphism
Exti
R(M, N )∼
=HomDf(R)(M, N [i]),
we conclude that a·Exti
R(M, N ) = 0 for all i≥nand N∈mod(R). That is, a∈can
R(M), and hence
a∈caR(M). It follows that annDsg (R)(M)⊆caR(M).
Corollary 4.3. Let Rbe a commutative Noetherian ring. Then
annRDsg(R) = caR(mod(R)).
Proof. Let Xbe a complex in Dsg (R). By choosing a projective resolution of X, we may assume Xis
a bounded above complex of finitely generated pro jective R-modules with finitely many nonzero coho-
mologies. Then by taking brutal truncation, we conclude that X[n] is isomorphic to a finitely generated
R-module in Dsg (R) for n≪0. Assume X[n]∼
=Mfor some n≪0 and M∈mod(R). Note that
annDsg(R)(X) = annDsg (R)(M). This yields the second equality below:
annRDsg(R) = \
X∈Dsg(R)
annDsg(R)(X)
=\
M∈mod(R)
annDsg(R)(M)
=\
M∈mod(R)
caR(M),
where the third equality follows from Proposition 4.2.
Let Sing(R) denote the singular locus of R. That is, Sing(R):={p∈Spec(R)|Rpis not regular}.
Lemma 4.4. For each finitely generated R-module M,V(caR(M)) ⊆Sing(R).The equality holds if, in
addition, Mis a generator of Dsg(R).
Proof. It follows from Proposition 3.10 that V(caR(M)) = IPD(M). It is clear that
IPD(M)⊆Sing(R).
By [27, Lemma 2.6], the equality holds if Mis a generator of Dsg(R).
12 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
4.5. For each X∈mod(R), Dao and Takahashi [15, Defintion 5.1] introduced a construction of an
ascending chain of full subcategories built out of Xas follows:
Set |X|R
0={0}, and denote |X|R
1to be the full subcategory of mod(R) consisting of modules which are
direct summands of any finite copies of X. Inductively, for n≥2, denote |X|R
nto be the full subcategory
of mod(R) consisting of modules Ythat appear in a short exact sequence
0→Y1→Y⊕Y′→Y2→0,
where Y1∈ |X|R
n−1and Y2∈ |X|R
1. If there is no confusion, we will use |X|nto denote |X|R
n.
In [27, Corollary 4.6], Iyengar and Takahashi observed that mod(R) has a strong generator if and only
if there exist s, t ≥0 and G∈mod(R) such that
Ωs
R(mod(R)) ⊆ |G|t.
4.6. Let (R, m) be a commutative Noetherian local ring. Assume there exist s, t ≥0 and G∈mod(R)
such that Ωs
R(R/mi)⊆[G]R
tfor all i≥0; see the definition of [G]R
t(i.e., the ball of radius tcentered at G)
in [15, Definition 2.1]. With the same argument in the proof of [40, Proposition 3.2], one has dim(R)≤s.
The following proposition characterizes the Krull dimension of Rin terms of strong generation of
mod(R).
Proposition 4.7. Let Rbe a commutative Noetherian ring. Then:
dim(R)≤inf{s≥0|there exist t≥0, G ∈mod(R)such that Ωs
R(mod(R)) ⊆ |G|R
t}.
In particular, if mod(R)has a strong generator, then dim(R)is finite. In addition, if Ris quasi-excellent
and either admits a dualizing complex or is a local ring, then the equality holds.
Proof. Let s, t be nonnegative integers and Gbe a finitely generated R-module. Assume Ωs
R(mod(R)) ⊆
|G|R
t. Since |G|R
t⊆[G]R
t, Ωs
R(mod(R)) ⊆[G]R
t; c.f. 4.6. Let pbe a prime ideal of R. We claim that
Ωs
Rp(mod(Rp)) ⊆[Gp]Rp
t.Combining with this, 4.6 yields that dim(Rp)≤s, and hence dim(R)≤sas p
is arbitrary.
Now, we proceed to prove the claim. Let Xbe an ob ject in Ωs
Rp(mod(Rp)). Since the functor
(−)p: mod(R)→mod(Rp) is dense, there exists M∈mod(R) such that X∼
=Ωs
Rp(Mp). On the other
hand, there exist n, m ≥0 such that Ωs
Rp(Mp)⊕R⊕n
p∼
=(Ωs
R(M))p⊕R⊕m
p. Hence,
X⊕R⊕n
p∼
=(Ωs
R(M))p⊕R⊕m
p∼
=Ωs
R(M)⊕R⊕mp∈(Ωs
R(mod(R)))p⊆[Gp]Rp
t;
the inclusion here uses (Ωs
R(mod(R)))p⊆([G]R
t)p⊆[Gp]Rp
t.As [Gp]Rp
tis closed under direct summands,
one has X∈[Gp]Rp
t.
Suppose that Ris a quasi-excellent ring of finite Krull dimension dand either admits a dualizing
complex or is a local ring. Combining [27, Theorem 5.2] and [31, Corollary 2.6], there exist t≥0 and
G∈mod(R) such that Ωd
R(mod(R)) ⊆ |G|R
t. This implies that the converse of the inequality stated in
the proposition holds.
Remark 4.8. Let Rbe a commutative Noetherian ring, and let Max(R) denote the set of all maximal
ideals of R. By a similar argument as in the previous proposition, the following three values coincide.
(1) The Krull dimension of R.
(2)
inf (s≥0
for every p∈Spec(R),there exist t≥0, G ∈mod(c
Rp)
such that Ωs
c
Rp
(mod( c
Rp)) ⊆ |G|
c
Rp
t).
(3)
inf (s≥0
for every m∈Max(R),there exist t≥0, G ∈mod( c
Rm)
such that Ωs
d
Rm
(mod( c
Rm)) ⊆ |G|
d
Rm
t).
Additionally, in (2), the values obtained by replacing Ωs
c
Rp
(mod( c
Rp)) ⊆ |G|
c
Rp
twith
•Ωs
c
Rp
(mod( c
Rp)) ⊆[G]
c
Rp
t,
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 13
•Ωs
c
Rp
(c
Rp/pic
Rp)∈ |G|
c
Rp
tfor all i≫0, or
•Ωs
c
Rp
(c
Rp/pic
Rp)∈[G]
c
Rp
tfor all i≫0
are all equal. The same holds for (3).
The equality Sing(R) = V(ca(R)) in (3) below was established by Iyengar and Takahashi [27, Theorem
1.1] under the additional assumption that the finitistic global dimension is finite. Proposition 4.7 shows
this assumption is unnecessary.
Proposition 4.9. Let Rbe a commutative Noetherian ring. Then:
(1) If Dsg (R)has a generator, then there exists M∈mod(R)such that Mis generator in Dsg(R).
Moreover, for any M∈mod(R)which is a generator in Dsg(R),
Sing(R) = V(annDsg(R)(M)) = V(caR(M)) = IPD(M).
(2) If Dsg (R)has a strong generator, then
Sing(R) = V(annRDsg(R)) = V(caR(mod(R))).
(3) If mod(R)has a strong generator, then
Sing(R) = V(ca(R)) = V(annRDsg(R)) = V(caR(mod(R))).
(4) ca(R)⊆annRDsg(R) = caR(mod(R)). If, in addition, mod(R)has a strong generator, then they
are equal up to radical.
Proof. (1) Assume Gis a generator of Dsg (R). With the same argument in the proof of Corollary 4.3,
there exists n∈Zsuch that G[n]∼
=Min Dsg(R) for some M∈mod(R). Then Mis also a generator of
Dsg(R). Combining with Lemma 4.4 and Proposition 4.2,
Sing(R) = V(caR(M)) = V(annDsg (R)(M)).
The desired result follows as V(caR(M)) = IPD(M); see Proposition 3.10.
(2) By Corollary 4.3, annRDsg (R) = caR(mod(R)). It remains to prove the first equality. Assume
Dsg(R) = thicks
Dsg(R)(G) for some G∈Dsg(R) and s≥0. By definition, annRDsg(R)⊆annDsg (R)(G).
Combining with the assumption, it follows from [20, Lemma 2.1] that
(annDsg(R)(G))s⊆annRDsg (R).
Thus, annRDsg(R) is equal to annDsg (R)(G) up to radical. This yields that
V(annRDsg(R)) = V(annDsg (R)(G)).
The desired result now follows immediately from (1); note that G[n]∼
=Min Dsg(R) for some M∈mod(R)
and n∈Z.
(3) The strong generation of mod(R) implies the strong generation of Dsg (R); see 2.12. Combining
with (2), it remains to show the first equality, and this follows immediately by combining Proposition 4.7
and [27, Theorem 1.1].
(4) The first statement follows from Corollaries 3.5 and 4.3. The second statement follows from (3).
Remark 4.10. (1) The equality Sing(R) = IPD(M) in Proposition 4.9 (1) was due to Iyengar and
Takahashi [28, Lemma 2.9]. The new input in (1) is the defining ideal of Sing(R) via the annihilator over
the singularity category.
(2) The first equality of Proposition 4.9 (2) was established by the second author in [35, Theorem 4.6]
through the localization of the singularity category. In contrast, our proof takes a different approach
by employing the cohomological annihilators of modules and their connection to the annihilators of the
singularity category. The inclusion ca(R)⊆annRDsg (R) was observed in [35, Proposition 1.2].
(3) Assume Dsg(R) has a strong generator. In [5, Corollary 5.8], Bahlekeh, Salarian, Takahashi, and
Toosi observed that
dim(R/ caR(mod(R))) ≤dim Sing(R).
Indeed, this inequality is an equality by Proposition 4.9 (2).
Corollary 4.11. Let Rbe a commutative Noetherian domain. If Dsg(R)has a strong generator, then
annRDsg(R)6= 0.
14 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Proof. By Proposition 4.9, annRDsg (R) defines the singular locus of R. It follows from this that
annRDsg(R)6= 0. If not, the zero ideal of R, denoted (0), is in Sing(R). This contradicts that R(0)
is a field. Hence, annRDsg(R)6= 0.
Remark 4.12. (1) Corollary 4.11 is an analog of a result by Elagin and Lunts [19, Theorem 5] concerning
the derived category. Specifically, for a commutative Noetherian domain R, they observed that ca(R)6= 0
if Df(R) has a strong generator.
In general, if Df(R) has a strong generator, then Dsg (R) also possesses one. However, the converse is
not true, even when Ris a domain; for instance, consider a regular domain with infinite Krull dimension
constructed by Nagata [37, Appendix, Example 1] (see 2.12). Additionally, ca(R)⊆annRDsg (R) (see
Proposition 4.9), and the inclusion can be proper, as illustrated by Nagata’s example (see Lemma 3.5 and
Corollary 4.3). Therefore, there seems to be no implication between Corollary 4.11 and the aforementioned
result of Elagin and Lunts.
(2) There exists a one-dimensional Noetherian domain with annRDsg (R) = 0; see Example 4.26.
Example 4.13. Let Rbe a finitely generated algebra over a field or an equicharacteristic excellent local
ring. Iyengar and Takahashi [27, Theorem 1.3] established that mod(R) has a strong generator. Recently,
this result was extended by the first author, Lank, and Takahashi [16, Corollary 3.12], who demonstrated
that mod(R) has a strong generator for any quasi-excellent ring with finite Krull dimension.
Corollary 4.14. Let Rbe a quasi-excellent ring with finite Krull dimension. Then
pca(R) = qannRDsg(R) = pcaR(mod(R)).
Proof. This follows immediately from Proposition 4.9 and Example 4.13.
Remark 4.15. Assume Ris a localization of a finitely generated algebra over a field or an equicharac-
teristic excellent local ring. Then Ris excellent with finite Krull dimension, and hence Corollary 4.14
implies that ca(R) is equal to annRDsg (R) up to radical; in this case, this was proved in [35, Proposition
1.2].
4.16. Inspired by the observation of Iyengar and Takahashi in 4.5, we say that mod(R) has a point-wise
strong generator provided that there exist G∈mod(R) and t≥0 such that, for each M∈mod(R),
Ωs
RM∈ |G|tfor some s≥0.
We say that Rhas isolated singularities if Sing(R)⊆Max(R), where Max(R) consists of all maximal
ideals of R.
Theorem 4.17. Let Rbe a commutative Noetherian ring with isolated singularities. Then the following
are equivalent.
(1) Dsg(R)has a strong generator.
(2) R/ annRDsg(R)is either 0or Artinian.
(3) R/ caR(mod(R)) is either 0or Artinian.
(4) mod(R)has a point-wise strong generator.
Proof. (1) ⇒(2): By Proposition 4.9, one has Sing(R) = V(annRDsg (R)). Since Rhas isolated sin-
gularities, Sing(R) consists of maximal ideals. Hence, V(annRDsg (R)) consists of maximal ideals. This
implies (2).
(2) ⇒(3): This follows from Corollary 4.3.
(3) ⇒(4): If R/ caR(mod(R)) = 0, then Ris regular; see Lemma 3.5. Thus, for each M∈mod(R),
Ωi
R(M) is finitely generated projective for some i≥0, and hence Ωi
R(M)∈ |R|1. This implies that
mod(R) has a point-wise strong generator. Next, we assume that R/ ca(mod(R)) is Artinian.
Set I:= caR(mod(R)) and assume x=x1,...,xnis a generating set of I. Let Mbe a finitely
generated R-module. Then
I= caR(mod(R)) ⊆caR(M) = cai+1
R(M) = ca1
R(Ωi
R(M))
for some i≥0, where the second equality is by Lemma 3.8. This implies that
I⊆annRExt1
R(Ωi
R(M),Ωi+1
R(M)).
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 15
Combining with this, [4, Theorem 3.1 (1)] shows that there exists L∈mod(R/I) such that Ωi
R(M)∈
|
n
M
j=1
Ωj
R(L)|R
n+1.Since R/I is Artinian, L∈ |N|R/I
ℓℓ(R/I), where ℓℓ(R/I) is the Loewy length of R/I and
Nis the quotient of R/I by its Jacobson radical. Restriction scalars along the map R→R/I, one has
L∈ |N|R
ℓℓ(R/I), and hence Ωj
R(L)⊆ |Ωj
R(N)|R
ℓℓ(R/I)for each j≥0. It follows from this that
n
M
j=1
Ωj
R(L)∈
|
n
M
j=1
Ωj
R(N)|R
ℓℓ(R/I). By [16, Lemma 3.8], Ωi
R(M)∈ |
n
M
j=1
Ωj
R(N)|R
(n+1)ℓℓ(R/I). Hence,
n
M
j=1
Ωj
R(N) is a
point-wise strong generator of mod(R).
(4) ⇒(1): This can be deduced by [5, Proposition 5.3].
Remark 4.18. (1) The implication (2) ⇒(1) of the above theorem follows from the third author’s work
[36, Corollary 4.3].
(2) Let Rbe a commutative Noetherian ring with isolated singularities, if Dsg (R) has a strong generator
and n(mod(R)) <∞, then mod(R) has a strong generator. Indeed, this follows from the same argument
as in the proof of Theorem 4.17 (3) ⇒(4).
Combining the above with Proposition 3.13, if Ris a Gorenstein ring with isolated singularities and
Ris of finite Krull dimension, then the following are equivalent.
•Dsg(R) has a strong generator.
•R/ ca(R) is either 0 or Artinian.
•mod(R) has a strong generator.
The local case of this was established by Bahlekeh, Hakimian, Salarian, and Takahashi [4, Theorem 3.2].
Recall that a Cohen–Macaulay local ring (R, m) with a canonical module ωis said to be nearly Goren-
stein if the trace ideal of ωcontains the maximal ideal m; see [25, Definition 2.2].
Corollary 4.19. Let Rbe a one-dimensional Cohen–Macaulay ring. Assume that, for each maximal
ideal mof R, the local ring Rmhas minimal multiplicity. Additionally, assume that, for each maximal
ideal mof height 1,Rmis nearly Gorenstein. If Dsg(R)has a strong generator, then we have:
(1) Sing(R) = V(ca(R)).
(2) If, in addition, Rhas isolated singularities, then mod(R)has a strong generator.
Proof. (1) We claim that nR(mod(R)) <∞. By Proposition 3.15, this is equivalent to showing that
sup{nRm(mod(Rm)|m∈Max(R)}<∞. If mis a maximal ideal of height 0, the by hypothesis
m2Rm= 0, and hence nRm(mod(Rm)) ≤1 by Remark 3.14 (3). If mis a maximal ideal height 1, then we
conclude by Lemma 3.11 and the equivalence (3) ⇐⇒ (9) in [33, Theorem 5.9] that nRm(mod(Rm)) ≤1.
Therefore, nR(mod(R)) ≤1. Combining this with Remark 3.14 (2), we have ca(R) = caR(mod(R)).
Since Dsg(R) has a strong generator, it follows from Proposition 4.9 that Sing(R) = V(ca(R)).
(2) By the proof of (1), nR(mod(R)) <∞. Remark 4.18 now shows that mod(R) has a strong
generator.
Remark 4.20. For a Gorenstein local ring R, Bahlekeh, Hakimian, Salarian, and Takahashi [4, Theroem
3.3] observed that if Dsg (R) has a strong generator, then Sing(R) = V(ca(R)). Thus, Corollary 4.19
provides a new class of rings satisfying the equality Sing(R) = V(ca(R)). Moreover, the rings satisfying
Corollary 4.19 (2) form a new class of rings for which all the conditions of [4, Theorem 3.2] are equivalent.
The following result is a direct consequence of Theorem 4.17.
Corollary 4.21. Let Rbe a commutative Noetherian domain with dim R≤1. Then the following are
equivalent.
(1) Dsg(R)has a strong generator.
(2) annRDsg(R)6= 0.
(3) caR(mod(R)) 6= 0.
(4) mod(R)has a point-wise strong generator.
16 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Corollary 4.22. Let Rbe a commutative Noetherian ring with dim R≤1. Then the following are
equivalent.
(1) Dsg(R/p)has a strong generator for each prime ideal pof R.
(2) annR/pDsg(R/p)6= 0 for each prime ideal pof R.
(3) caR/p(mod(R/p)) 6= 0 for each prime ideal pof R.
Proof. For each prime ideal pof R,R/pis a domain and dim R/p≤1. Thus, the equivalence (1) ⇐⇒ (2)
follows from Corollary 4.21. The equivalence (2) ⇐⇒ (3) is a direct consequence of Corollary 4.3.
Remark 4.23. Due to Iyengar and Takahashi [28, Theorem 1.1], the following three conditions are
equivalent for a commutative Noetherian ring.
(1) mod(R/p) has a generator for each prime ideal pof R.
(2) Df(R/p) has a generator for each prime ideal pof R.
(3) Dsg(R/p) has a generator for each prime ideal pof R.
Moreover, if these conditions hold, then mod(R), Df(R), and Dsg (R) all have generators.
Recently, the first author, Lank, and Takahashi [16, Theorem 1.1] showed that the following three
conditions are equivalent for a commutative Noetherian ring.
(1) mod(R/p) has a strong generator for each prime ideal pof R.
(2) Df(R/p) has a strong generator for each prime ideal pof R.
(3) ca(R/p)6= 0 for each prime ideal pof R.
Furthermore, if these conditions hold, then dim(R)<∞, and both mod(R) and Df(R) have strong
generators.
A natural question arises: do Corollaries 4.21 and 4.22 still hold if we remove the assumption about
the Krull dimension? Theorem 4.25 is a result related to this question. Keep the same assumption as
Corollary 4.22, we don’t know whether Dsg (R) has a strong generator when the conditions of Corollary
4.22 hold.
For a full subcategory Cof mod(R), we define √caR(C):=\
X∈C pcaR(X).
Lemma 4.24. Let Rbe a commutative Noetherian ring. Then:
(1) ca(R)⊆ca(mod(R)) ⊆√caR(mod(R)).
(2) Sing(R)⊆V(√caR(mod(R))) ⊆V(ca(mod(R))) ⊆V(ca(R)).
Proof. (1) This follows from Lemma 3.5.
(2) By (1), it remains to prove Sing(R)⊆V(√caR(mod(R))). Let pbe a prime ideal of Rsuch that
p/∈V(√caR(mod(R))). Since the inclusion relation V(pcaR(R/p)) ⊆V(√caR(mod(R))) holds, the
prime ideal pdoes not belong to V(pcaR(R/p)) = V(caR(R/p)). This implies that p/∈IPD(R/p); see
Proposition 3.10. Hence, the Rp-module Rp/pRphas finite projective dimension. This implies that p
does not belong to Sing R, and hence Sing(R)⊆V(√caR(mod(R))).
Regarding the generation of the singularity category and the vanishing of the cohomology annihilator,
we have the following.
Theorem 4.25. Let Rbe a commutative Noetherian ring. Then Dsg (R/p)has a generator for each prime
ideal pof Rif and only if √caR/p(mod(R/p)) 6= 0 for each prime ideal pof R.
Proof. For the forward direction, assume that Dsg (R/p) has a generator for each prime ideal pof R. To
obtain the conclusion, it suffices to show √caR(mod(R)) 6= 0 under the assumption that Ris a domain.
By assumption and Proposition 4.9 (1), there exists M∈mod(R) such that Mis a generator of Dsg (R)
and Sing(R) = V(caR(M)). With the same argument in the proof of Corollary 4.11, caR(M)6= 0, and
hence there exists f6= 0 in can
R(M) for some n≥0. Combining this with Lemma 3.8 (1), we get that the
projective dimension of Mfover Rfis finite. Note that Mfis a generator of Dsg (Rf) as Mis a generator
of Dsg(R). Hence, Rfis a regular ring. Let Xbe a finitely generated R-module. Since the Rf-module
Xfhas finite projective dimension, there exists m≥0 such that
Extm
R(X, Ωm
R(X))f∼
=Extm
Rf(Xf,(Ωm
R(X))f) = 0.
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 17
It follows that fr·Extm
R(X, Ωm
R(X)) = 0 for some r≥0. This yields that f∈pcam
R(X)⊆pcaR(X).
Thus,
f∈\
X∈mod(R)pcaR(X) = √caR(mod(R)).
For the backward direction, assume that √caR/p(mod(R/p)) 6= 0 for each prime ideal pof R. Then 0 ∈
Spec(R/p)\V(√caR/p(mod(R/p))) ⊆Reg(R/p) in Spec(R/p), where Reg(R/p) = Spec(R/p)\Sing(R/p);
the inclusion is from Lemma 4.24. This implies that Reg(R/p) contains a nonempty open subset. By
virtue of [28, Theorem 1.1], Dsg(R/p) has a generator for each prime ideal pof R.
We end with this section by showing that there exists a one-dimensional Noetherian domain satisfying
annRDsg(R) = 0.
Example 4.26. In [26, Example 1], Hochster constructed a one-dimensional Noetherian domain R
whose regular locus (i.e., Spec(R)\Sing(R)) does not contain a nonempty open subset. According to
[28, Theorem 1.1], there exists a prime ideal pof Rsuch that Dsg(R/p) does not have a generator. This
implies that pmust be the zero ideal as Ris a one-dimensional domain. Therefore, Dsg (R) does not have
a generator, and by Corollary 4.21, we conclude that annRDsg(R) = 0.
5. co-cohomological annihilators for modules
In this section, we introduce the notion of a co-cohomological annihilator of modules. The main result
is Theorem 1.3 from the introduction.
5.1. For each R-module Mand n≥0, we define the n-th co-cohomological annihilator of Mto be
cocan
R(M):= annRExt≥n
R(mod(R), M ).
In words, cocan
R(M) consists of elements r∈Rsuch that r·Exti
R(N, M ) = 0 for each i≥nand
N∈mod(R). By the dimension shifting, we have cocan
R(M) = annRExtn
R(mod(R), M ). Consider the
ascending chain of ideals
annRM= coca0
R(M)⊆coca1
R(M)⊆coca2
R(M)⊆ ··· ,
the co-cohomological annihilator of Mis defined to be the ideal
cocaR(M):=[
n≥0
cocan
R(M).
Since Ris Noetherian, cocaR(M) = cocan
R(M) for n≫0.
Combining with Baer’s criterion for injectivity, we conclude that, for each R-module Mand n≥0,
cocan
R(M) = Rif and only if idR(M)< n. Thus, cocaR(M) = Rif and only if idR(M)<∞.
Lemma 5.2. For each R-module Mand n > 0,
{p∈Spec(R)|idRp(Mp)≥n}= SuppRExtn
R(mod(R), M ).
Proof. Note that idRp(Mp)< n if and only if Extn
Rp(mod(Rp), Mp) = 0; this can be proved by using
Baer’s criterion for injectivity. This yields the first equality below:
{p∈Spec(R)|idRp(Mp)≥n}={p∈Spec(R)|Extn
Rp(mod(Rp), Mp)6= 0}
={p∈Spec(R)|Extn
R(mod(R), M )p6= 0}
= SuppRExtn
R(mod(R), M ),
where the second one follows from 3.7 and the fact that mod(R)→mod(Rp) is dense. This finishes the
proof.
5.3. Let Mbe a finitely generated R-module, the infinite injective dimension locus of Mis defined to
be
IID(M):={p∈Spec(R)|idRp(Mp) = ∞}.
Lemma 5.4. For each R-module Mand n > 0,
{p∈Spec(R)|idRp(Mp)≥n} ⊆ V(cocan
R(M)).
In particular, IID(M)⊆V(cocaR(M)).
18 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Proof. By Lemma 5.2, we have the first equality of the following:
{p∈Spec(R)|idRp(Mp)≥n}= SuppRExtn
R(mod(R), M )
=[
N∈mod(R)
SuppRExtn
R(N, M )
⊆[
N∈mod(R)
V(annRExtn
R(N, M ))
⊆V(\
N∈mod(R)
annRExtn
R(N, M )).
Combining with cocan
R(M) = \
N∈mod(R)
annRExtn
R(N, M ), the first statement of the lemma follows.
By the first statement, we get the inclusion below:
IID(M) = \
n>0{p∈Spec(R)|idRp(Mp)≥n}
⊆\
n>0
V(cocan
R(M)).
Combining with \
n>0
V(cocan
R(M)) = V([
n>0
cocan
R(M)) = V(cocaR(M)), we conclude that IID(M)⊆
V(cocaR(M)).
5.5. (1) As mentioned in 4.5, mod(R) has a strong generator if and only if there exist s, t ≥0 and
G∈mod(R) such that
Ωs
R(mod(R)) ⊆ |G|t.
Note that if mod(R) has finite extension dimension in the sense of Beligiannis [8, Definition 1.5], then
mod(R) has a strong generator.
(2) Suggested by above, we say that mod(R) has an extension generator if there exists s≥0 and
G∈mod(R) such that
Ωs
R(mod(R)) ⊆[
t≥0|G|t;
in this case, we say Gis an extension generator of mod(R). By [4, Remark 2.11], for each G∈mod(R), the
union [
t≥0|G|tis the smallest full subcategory of mod(R) that contains Gand is closed under extensions
and direct summands.
By above, if mod(R) has a strong generator G, then Gis an extension generator of mod(R).
If mod(R) has an extension generator G, then mod(R) has a generator G⊕R. However, the next
example shows that mod(R) may not have an extension generator if mod(R) has a generator.
Example 5.6. Let Rbe a regular ring with infinite Krull dimension. By 2.11,Ris a generator of
mod(R). However, mod(R) doesn’t have an extension generator. If not, assume that there exist s≥0
and G∈mod(R) such that Ωs
R(mod(R)) ⊆[
t≥0|G|t. Since Ris regular, there exists n≥0 such that
Ωn
R(G) is projective over R. This yields the second inclusion below:
Ωn+s
R(mod(R)) = Ωn
RΩs
R(mod(R)) ⊆[
t≥0|Ωn
R(G)|t⊆ |R|1,
where the first inclusion is by the horseshoe lemma. Thus, gl.dim(R)≤t+s. By 2.12,Rhas finite Krull
dimension. This is a contradiction, and hence mod(R) doesn’t possess an extension generator.
Theorem 5.7. Let Rbe a commutative Noetherian ring and Mbe a finitely generated R-module. Then:
(1) If mod(R)has an extension generator, then IID(M)is closed in Spec(R).
(2) If mod(R)has a strong generator, then
IID(M) = V(cocaR(M)).
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 19
Proof. (1) Assume mod(R) has an extension generator G. That is, there exists s≥0 satisfying: for each
X∈mod(R), there exists t≥0 such that Ωs
R(X)⊆ |G|t. For each n > 0, Lemma 5.2 yields the first
equality below:
{p∈Spec(R)|idRp(Mp)≥n+s}= SuppRExtn+s
R(mod(R), M )
= SuppRExtn
R(Ωs
R(mod(R)), M )
=[
N∈mod(R)
SuppRExtn
R(Ωs
R(N), M ).
For each N∈mod(R), it follows from the assumption that Ωs
R(N)∈ |G|tfor some t≥0. This yields
that SuppRExtn
R(Ωs
R(N), M )⊆SuppRExtn
R(G, M ).Hence,
{p∈Spec(R)|idRp(Mp)≥n+s} ⊆ SuppRExtn
R(G, M ).
Taking intersections throughout all n > 0, IID(M)⊆\
n>0
SuppRExtn
R(G, M ).On the other hand, by
Lemma 5.2,\
n>0
SuppRExtn
R(G, M )⊆\
n>0{p∈Spec(R)|idRp(Mp)≥n}= IID(M).
Thus,
IID(M) = \
n>0
SuppRExtn
R(G, M ).
The desired result follows as SuppRExtn
R(G, M ) = V(annRExtn
R(G, M )) is closed.
(2) By Lemma 5.4, it remains to prove V(cocaR(M)) ⊆IID(M). Let n > 0 be an integer. Since
mod(R) has a strong generator, there exist s, t ≥0 and G∈mod(R) such that Ωs
R(mod(R)) ⊆ |G|t; see
5.5. Hence,
cocan+s
R(M) = annRExtn
R(Ωs
R(mod(R)), M )⊇annRExtn
R(|G|t, M ).
It is routine to check that annRExtn
R(|G|t, M )⊇(annRExtn
R(G, M ))t. Thus,
V(cocan+s
R(M)) ⊆V(annRExtn
R(G, M ))
⊆ {p∈Spec(R)|idRp(Mp)≥n},
where the second inclusion is due to V(annRExtn
R(G, M )) = SuppRExtn
R(G, M ) and 3.7; the equality
here holds as Extn
R(G, M ) is finitely generated. Combining with cocan+s
R(M)⊆cocaR(M), we get
V(cocaR(M)) ⊆ {p∈Spec(R)|idRp(Mp)≥n}.
Since n > 0 is arbitrary,
V(cocaR(M)) ⊆\
n>0{p∈Spec(R)|idRp(Mp)≥n}= IID(M).
This completes the proof.
Remark 5.8. As noted in Example 5.6, there exist rings Rsuch that mod(R) has a generator but does
not have an extension generator. Let Rbe a commutative Noetherian ring and M∈mod(R). Motivated
by Theorem 5.7 (1), a natural question arises: is IID(M) closed if mod(R) has a generator?
Corollary 5.9. Let Rbe a quasi-excellent ring with finite Krull dimension. Then
IID(M) = V(cocaR(M)).
Proof. By Example 4.13, mod(R) has a strong generator. The desired result now follows from Theorem
5.7.
Remark 5.10. (1) Let Rbe an excellent ring. Greco and Marinari [23, Corollary 1.5] observed that
IID(R) is closed. Takahashi [41] then extended this to modules, proving that IID(M) is closed for each
M∈mod(R). Recently, Kimura [32, Theorem 1.1] generalized Takahashi’s result to acceptable rings.
(2) By Proposition 4.7 and Example 4.13, for a quasi-excellent ring R, mod(R) has a strong generator
if and only if Rhas finite Krull dimension.
20 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Corollary 5.11. Let Mbe a finitely generated R-module. Assume mod(R)has a strong generator;
equivalently, there exist s, t ≥0and G∈mod(R)such that Ωs
R(mod(R)) ⊆ |G|t.Set d= sup{idRp(Mp)|
p/∈IID(M)}. Then:
(1) dis finite.
(2) IID(M) = V(cocaR(M)) = V(cocad+s+1
R(M)).
Proof. (1) This follows from Proposition 4.7 and [11, Theorem 3.1.17].
(2) By Theorem 5.7, IID(M) = V(cocaR(M)). It is clear that
V(cocaR(M)) ⊆V(cocad+s+1
R(M)).
Assume this is not equal. Combining with IID(M) = V(cocaR(M)), there exists a prime ideal pof
Rcontaining cocad+s+1
R(M) and p/∈IID(M). Thus, pcontains annRExtd+s+1
R(mod(R), M ). By the
argument in the proof of Theorem 5.7 (2),
V(cocan+s
R(M)) ⊆ {p∈Spec(R)|idRp(Mp)≥n}
for each n > 0. Thus, idRp(Mp)≥d+1. This contradicts with p/∈IID(M); by assumption, idRp(Mp)≤d
if p/∈IID(M). This completes the proof.
5.12. (1) Let Vbe a specialization closed subset of Spec(R). Namely, Vis a union of closed subsets of
Spec(R). If, in addition, Vis a finite set, then Vis a closed subset of Spec(R) and dim(V)≤1. Indeed,
the assumption yields that V=[
p∈V
V(p). Since Vis finite, Vis a finite union of closed subsets, and
hence Vis closed. It follows that V=V(I) for some ideal Iof R. If dim(V)(= dim(R/I )) ≥2, then
there exists a chain of prime ideals p0(p1(p2in V(I) = V. By [30, Theorem 144], Vis infinite. This
is a contradiction. Hence, dim(V)≤1.
(2) If Vis a closed subset of Spec(R), Ris semi-local, and dim(V)≤1, then Vis a finite set.
In fact, the assumption yields that V=V(I) for some ideal Iof R. Since R/I is semi-local and
dim(R/I) = dim(V)≤1, one has Spec(R/I) is a finite set. It follows that V(I) = Vis finite.
Example 5.13. For a commutative Noetherian ring R, the singular locus Sing(R) is finite in any of the
following cases:
(1) Ris a semi-local ring with dim(R) = 1.
(2) Ris semi-local with isolated singularities.
(3) Ris a semi-local J-0 domain with dim(R) = 2.
(4) Ris a semi-local J-1 normal ring with dim(R) = 3.
For (1), since dim(R) = 1, Ris semi-local, and the set of minimal prime ideals of Ris finite, it follows
that Sing(R) is finite. (2) holds immediately by assumption.
For (3), since Ris J-0, we have Sing(R)⊆V(I) for some nonzero ideal Iof R, and as Ris a domain, it
follows that dim(R/I)≤1. Combining this with that Ris semi-local, we conclude that V(I) = Spec(R/I)
is finite, and hence Sing(R) is finite.
For (4), since Ris J-1, we have Sing(R) = V(I) for some ideal Iof R, and as Ris normal, V(I)
cannot contain any prime ideal of height 0 or height 1. Thus, the height of Iis at least 2. It follows that
dim(R/I)≤1, and by the same reasoning as in (3), Sing(R) is finite.
5.14. Let Vbe a closed subset of Spec(R). We define the arithmetic rank of Vto be
ara(V) := inf{n≥0|V=V(x1,···, xn) for some x1,...,xn∈R}.
If we write V=V(I) for some ideal Iof R, then the value ara(V(I)) coincides with the arithmetic rank
of Ias defined in [10, Definition 3.3.2].
Proposition 5.15. Let Rbe a commutative Noetherian ring with finite Krull dimension d. Assume
Sing(R)is a finite set. Then Sing(R)is closed. Moreover,
Ωd
R(mod(R)) ⊆[
t≥0
n
M
i=0 M
p∈Sing(R)
Ωi
R(R/p)t,
where n:= ara(Sing(R)). In particular, mod(R)has an extension generator.
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 21
Proof. By 5.12, Sing(R) is closed. Assume Sing(R) = V(x1,...,xn) for some elements x1,...,xnin R.
Let Mbe a finitely generated R-module. For each p/∈Sing(R), we have
Ext1
R(Ωd
R(M),Ωd+1
R(M))p= 0.
It follows that the non-free locus of Ωd
R(M), given by {p∈Spec(R)|Ωd
R(M)pis not free}, is contained
in Sing(R). By [4, Theorem 3.1 (2)], Ωd
R(M)∈[
t≥0
n
M
i=0 M
p∈Sing(R)
Ωi
R(R/p)t.
Remark 5.16. Proposition 5.15 extends a recent result of Araya, Iima, and Takahashi [1, Proposition
4.7 (1)(2)]. Specifically, if Ris either a semi-local ring with isolated singularities or a semi-local J-0
domain with dim(R) = 2, then Sing(R) is finite by Example 5.13 (2) and (3), and hence mod(R) has an
extension generator by Proposition 5.15.
The following result concerns the relationship between the generator of the singularity category and
the extension generator of the module category; compare Example 5.6.
Proposition 5.17. Let Rbe a commutative Noetherian ring with finite Krull dimension d. If Dsg(R/p)
has a generator for each prime ideal p∈Spec(R), then Ωd
R(mod(R)) ⊆[
t≥0|G|tfor some G∈mod(R).
In particular, mod(R)has an extension generator.
Proof. We prove it by induction on d= dim R. Note that when d= 0, R/pis Artinian for every p, and
hence mod(R/p) has an extension generator. So we assume d > 0.
First, we consider the case when Ris a domain. By [28, Theorem 1.1], Sing(R) = V(I) for some ideal
I. As Ris a domain, Iis a non-zero ideal, so pick 0 6=x∈I. Then Rxis regular. As every prime ideal
of R/(x) is of the form p/(x) for some p∈Spec(R), and (R/(x))/(p/(x)) ∼
=R/p, thus Dsg((R/(x))/q)
has a classical generator for each q∈Spec(R/(x)). Since Ris a domain, n:= dim(R/(x)) <dim(R) = d.
Thus, by induction hypothesis, Ωn
R/(x)(mod(R/(x)) ⊆[
t≥0|G|R/(x)
tfor some G∈mod(R/(x)). By [1,
Lemma 4.5(2)], there exists H∈mod(R) such that Ωd
R(mod(R)) ⊆[
t≥0|H|R
t. This completes the case
where Ris a domain.
The proof of the general case now proceeds similarly to part (b) of the proof in [1, Proposition
4.6(3)]; note that each R/piin loc. cit. is a domain and satisfies the same hypothesis on the singularity
category.
The following result is related with Proposition 4.7; Rfd represents the large restricted flat dimension,
see 3.12. Recall that a full subcategory of mod(R) is called resolving if it contains all finitely generated
projective modules and is closed under syzygies, extensions, and direct summands. For each M∈
mod(R), let res(M) denote the smallest resolving subcategory of mod(R) that contains M. Note that
|M|t⊆res(M) for each t≥0.
Proposition 5.18. Let R be a commutative Noetherian ring. If there exist an integer s≥0and a nonzero
finitely generated R-module Gsuch that Ωs
R(mod(R)) ⊆res(G), then R has finite Krull dimension and
dim(R)≤s+ Rfd(G) + 1.
In particular, if mod(R)has an extension generator, then R has finite Krull dimension.
Proof. By 3.12, Rfd(G) is finite. Set Rfd(G) = n. By [14, Definition 2.1 and Theorem 2.4], we conclude
that the category C:={M∈mod(R)|Rfd(M)≤n}is a resolving subcategory of mod(R) containing G,
hence it contains res(G). It follows from the hypothesis that Ωs
R(M)∈ C for each M∈mod(R). Thus,
for each M∈mod(R), Rfd(Ωs
R(M)) ≤n. Again, by [14, Definition 2.1 and Theorem 2.4], we conclude
that Rfd(M)≤s+nfor each M∈mod(R). Since depth(Rp)≤Rfd(R/p), we get sup{depth(Rp)|p∈
Spec(R)} ≤ s+n. Since prime ideals of localization are localization of prime ideals, [14, Lemma 1.4]
implies that dim(R)−1≤s+n.
In the following, we say mod(R) has a resolving generator provided that Ωs
R(mod(R)) ⊆res(G) for
some s≥0 and G∈mod(R).
22 SOUVIK DEY, JIAN LIU, YUKI MIFUNE, AND YUYA OTAKE
Corollary 5.19. Let Rbe a commutative Noetherian ring. Then the following are equivalent.
(1) Rhas finite Krull dimension and Dsg (R/p)has a generator for each prime ideal pof R.
(2) mod(R/p)has an extension generator for each prime ideal pof R.
(3) mod(R/p)has a resolving generator for each prime ideal pof R.
Moreover, if any of the above equivalent conditions holds, then mod(R/I)has an extension generator for
each ideal Iof R.
Proof. The second statement and the implication (1) ⇒(2) follows from Proposition 5.17. The implica-
tion (2) ⇒(3) follows from the definition.
(3) ⇒(1): Assume mod(R/p) has a resolving generator for each p∈Spec(R). By Proposition 5.18,
R/phas finite Krull dimension for each prime ideal pof R. Since the set of minimal prime ideals of Ris
finite, we conclude that Rhas finite Krull dimension.
Fix a prime ideal pof R, assume Gis a resolving generator of mod(R/p). Note that the modules in
res(G) are contained in thickDsg (R/p)(G). Since each complex in Dsg (R/p) is isomorphic to some shift of
a module Min mod(R/p), we conclude that Gis a generator of Dsg (R/p). This finishes the proof.
We end this section by comparing the cohomological annihilators and the co-cohomological annihila-
tors; see Proposition 5.21. Recall that Ris said to be Gorenstein in codimension n if Rpis Gorenstein
for each prime ideal pof Rwith height at most n.
Lemma 5.20. (1) Let (R, m)be a d-dimensional Cohen–Macaulay local ring with a canonical module
ω. Set (−)†= HomR(−, ω). For each M∈CM(R),ca1
R(M†)⊆cocad+1
R(M), and equality holds if, in
addition, Ris Gorenstein in codimension d−1.
(2) Let Rbe a Gorenstein ring with finite Krull dimension d. For each M∈CM(R),
cocad+1
R(M) = cocaR(M) = annREndR(M).
Proof. (1) For each X∈mod(R), since Ωd
R(X)∈CM(R), we get
ca1
R(M†)⊆annRExt1
R(M†,(Ωd
R(X))†) = annRExt1
R(Ωd
R(X), M ) = annRExtd+1
R(X, M ).
Since this is true for any X∈mod(R), we get ca1
R(M)⊆annRExtd+1
R(mod(R), M ) = cocad+1
R(M).
Assume Ris Gorenstein in codimension d−1. Note that (Ω1
R(M†))†∈CM(R). By [22, Theorem 3.8],
(Ω1
R(M†))†∼
=Ωd
R(Y) for some Y∈mod(R). This yields the first equality below:
cocad+1
R(M)⊆annRExtd+1
R(Y, M ) = annRExt1
R((Ω1
R(M†))†, M ) = annRExt1
R(M†,Ω1
R(M†)) = ca1
R(M†),
where the last equality is by Lemma 3.8. This proves the converse direction.
(2) Indeed, for each n > 0, assume r∈cocan
R(M). This yields
0 = r·Extn
R(mod(R), M ) = r·Ext1
R(Ωn−1
R(mod(R)), M ).
Since M∈CM(R) and Ris strongly Gorenstein (see 2.1), Mis an infinite syzygy. In particular, there
is a short exact sequence ξ: 0 →M→P→C→0,where Pis a finitely generated projective and
C∈Ωn−1
R(mod(R)). By above, r·ξ= 0. In particular, r:M→Mfactors through P. This implies that
r∈annREndR(M), and therefore cocan
R(M)⊆annREndR(M) for each n > 0. It follows from this that
cocaR(M)⊆annREndR(M).
By above, cocad+1
R(M)⊆cocaR(M)⊆annREndR(M). Next, we prove the inclusion annREndR(M)⊆
cocad+1
R(M). Assume a∈annREndR(M). Namely, there is a factorization
M M,
Q
a
where Qis finitely generated pro jective. Note that idR(Q)≤d; see 2.1. Hence, Extd+1
R(X, Q) = 0 for
each X∈mod(R). Applying Extd+1
R(X, −) to the above diagram, we conclude that a: Extd+1
R(X, M )→
Extd+1
R(X, M ) is zero for each X∈mod(R). That is, a∈cocad+1
R(M), and hence annREndR(M)⊆
cocad+1
R(M).
ANNIHILATION OF COHOMOLOGY AND (STRONG) GENERATION 23
Proposition 5.21. Let Rbe a commutative Noetherian ring and Mbe a finitely generated R-module.
Assume Mis an infinite syzygy, then:
(1) cocaR(M)⊆ca1
R(M) = annREndR(M)⊆caR(M). All these inclusions are equal if, in addition,
Ris Gorenstein with finite Krull dimension.
(2) If mod(R)has a strong generator, then
IPD(M) = V(caR(M)) ⊆V(cocaR(M)) = IID(M).
The inclusion is equal if, in addition, Ris Gorenstein. Moreover, in this case,
cocaR(M) = caR(M).
Proof. (1) By Lemma 3.8, it remains to prove cocaR(M)⊆annREndR(M). This follows from the same
argument in the proof of Lemma 5.20. If, in addition, Ris strongly Gorenstein, then Lemma 5.20 yields
that cocaR(M) = annREndR(M). The desired result follows by noting that ca1
R(M) = caR(M); see
Proposition 3.13.
(2) The first statement follows from Proposition 3.10, Theorem 5.7, and (1). If, in addition, Ris Goren-
stein, then Rpis strongly Gorenstein for each prime ideal p. Hence, for each p∈Spec(R), idRp(Mp)<∞
if and only if pdRp(Mp)<∞. It follows that IPD(M) = IID(M). Thus, the inclusion in the proposition
is equal. In particular, V(caR(M)) = V(cocaR(M)). Since mod(R) has a strong generator, Proposition
4.7 implies that Rhas finite Krull dimension. Thus, cocaR(M) = caR(M) by (1).
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Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovsk´
a 83, 186 75
Praha, Czech Republic
Email address:souvik.dey@matfyz.cuni.cz
School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China
Normal University, Wuhan 430079, P.R. China
Email address:jianliu@ccnu.edu.cn
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Email address:yuki.mifune.c9@math.nagoya-u.ac.jp
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Email address:m21012v@math.nagoya-u.ac.jp
