ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGSOLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUIA BSTRACT. In this paper we are concerned with certain invariants of modules, called reducing invariants,
which have been recently introduced and studied by Araya - Celikbas and Araya - Takahashi. We raise thequestion whether the residue field of each commutative Noetherian local ring has finite reducing projective
dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore,
we construct new examples of modules of infinite projective dimension that have finite reducing projective
dimension, and study several fundamental properties of reducing dimensions, especially properties underlocal homomorphisms of local rings.

1. INTRODUCTION
Throughout, R denotes a local ring (commutative and Noetherian) with unique maximal ideal mand residue field k, and each R-module is assumed to be finitely generated. For unexplained basic
terminology, such as the definition of the Gorenstein dimension, we refer the reader to [4, 11, 14].
The classical homological dimensions play a significant role in the characterizations of local rings.
For example, a celabrated result of Auslander-Buchsbaum-Serre [11, 2.2.7] states that a local ring R
is regular if and only if the residue field k has finite projective dimension. It is known that similar
important characterizations of local rings via other homological dimensions of the residue field, such as
via the Gorenstein dimension, also hold: a local ring R is Gorenstein ring if and only if the Gorensteindimension of the residue field k is finite; see, [4, 4.20].
Reducing versions of the classical homological dimensions have been recently introduced and studied by Araya and Celikbas [1], and subsequently by Araya and Takahashi [3]; see Definition 2.1 and Remark 2.3 for the details. These dimensions are finer than the classical dimensions: examples of modules
that have finite reducing projective dimension but have infinite projective dimension are abundant; see
2.7 and Examples 2.10, 2.13 and 2.15. In general, unlike the aforementioned characterizations via classical dimensions, the residue field of a non-Gorenstein local ring can have finite reducing Gorenstein,
or finite reducing projective dimension; see, for example, 2.5. However, in addition to these examples,
there are several noteworthy characterizations of local rings via reducing dimensions that make theseinvariants viable notations. One such result [12] states that a local ring R is Gorenstein if and only ifeach R-module has finite reducing Gorenstein dimension; this generalizes the well-known fact that [4,4.20] a local ring R is Gorenstein if and only if each R-module has finite Gorenstein dimension.
It is interesting for us that the research that has been done so far concerning the reducing dimensions
has not yet produced an example of a local ring whose residue field has infinite reducing projective
dimension. Therefore, based on the examples and results we are aware of about these dimensions, it
seems reasonable to us to ask the following:
Question 1.1. Does the residue field of each local ring have finite reducing projective dimension?
Key words and phrases. G-regular rings, minimal multiplicity, reducing projective and reducing Gorenstein dimension.
2020 Mathematics Subject Classification. Primary 13D07; Secondary 13C12, 13D05, 13H10.
Souvik Dey was partly supported by Charles University Research Center program No.UNCE/SCI/022 and a grant GA ČR 2305148S from the Czech Science Foundation; Toshinori Kobayashi was partly supported by JSPS Grant-in-Aid for JSPS Fellows
18J20660; Hiroki Matsui was partly supported by JSPS Grant-in-Aid for Early-Career Scientists 22K13894.
1

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OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUIThe main aim of this paper is to give support and obtain an affirmative answer for Question 1.1 for alarge class of local rings. More precisely, we prove:
Theorem 1.2. Let (S, n) be a local ring such that the n-adic completion Sb of S equals R/xR, where(R, m) is a Cohen-Macaulay local ring of minimal multiplicity and {x} ⊆ m is an R-regular sequence.
If k is infinite, then it has finite reducing projective dimension as an S-module.
We give a proof of Theorem 1.2 in Section 3, where we also prove several preliminary results andstudy fundamantal properties of reducing dimensions, especially under local homomorphisms of local
rings; see Theorem 3.11 and Remark 3.12. It is worth noting that, if a module over a local ring as in
Theorem 1.2 has finite Gorenstein dimension, then it does not need to have finite projective dimension;see Example 3.17. An interesting property of these rings is that they do not admit such modules as longas reducing dimensions are considered. For example, if S is a non-Gorenstein complete local ring asin Theorem 1.2, then we prove in Proposition 3.19 that each (finitely generated) S-module has finite
reducing Gorenstein dimension if and only if it has finite reducing projective dimension.
Another focus of this paper is to give new and nontrivial examples of modules of finite reducing
projective dimension; we obtain such examples in Section 2. One of these examples deserves to bementioned here: we construct an example of a local ring where the reducing projective dimension of its
residue field is two, and we determine a reducing projective dimension sequence of that residue field
explicitly; see Example 2.10.
The last section of this paper is devoted to the (involved) proof of Proposition 3.8; this proposition
plays an important role in the proof of Theorem 1.2. It also shows that, like the classical homologicaldimensions, the reducing dimension of a module is not affected by taking direct sum with a free module.
2. P RELIMINARIES , EXAMPLES , AND REMARKSIn this section we give new examples of modules that have finite reducing, but infinite, projective
dimension; see Examples 2.10, 2.13 and 2.15. We also record several important facts that are neededfor our argument; see 2.8. We start by recalling the definition of reducing invariant of modules.
In the following, I denotes an invariant of modules over local rings, that is, a family of functions IR
from isomorphism classes of R-modules to Z ∪ {±∞} as R varies over all local rings. Classical examples
of such an invariant we use in this paper are the projective dimension I = pd, and the Gorensteindimension I = G-dim.
Definition 2.1. ([3, 2.5]; see also [1, 2.1]) Let R be a local ring, M be an R-module.
(1) We write red-IR (M) < ∞ provided that there exist integers r ≥ 1, ai ≥ 1, bi ≥ 1, ni ≥ 0, and short
exact sequences of R-modules of the form
(2.1.1)⊕ai⊕bi
0 → Ki−1→ Ki → ΩnRi Ki−1
→ 0,

for each i = 1, . . . r, where K0 = M and IR (Kr ) < ∞. If a sequence as in (2.1.1) exists, then we call
{K0 , . . . , Kr } a reducing I-sequence of M.
We set that the reducing invariant red-IR (M) of M is zero if and only if IR (M) < ∞. Moreover,
if IR (M) = ∞, then we define:red-IR (M) = inf{r ∈ N : there is a reducing I-sequence K0 , . . . , Kr of M}.
Therefore 0 ≤ red-IR (M) ≤ ∞ for each R-module M.
(2) If we consider I = pd (resp. I = G-dim), we write red-pdR (M) (resp. red-G-dimR (M)) and call it
the reducing projective dimension (resp. reducing Gorenstein dimension) of M.
We will give various examples of modules that have finite reducing homological dimension, but first
we record several remarks. The first one, which follows directly from Definition 2.1, is used throughout
the paper, for example, in the proofs of Proposition 3.1, Proposition 3.9, and Theorem 3.11.

ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

3

Remark 2.2. Let R be a local ring, N be an R-module.
(i) If 1 ≤ red-IR (N) = r < ∞, then there is an exact sequence 0 → N ⊕a → K → ΩnR N ⊕b → 0, wherea ≥ 1, b ≥ 1, and n ≥ 0 are integers, and K is an R-module such that red-IR (K) = r − 1.
(ii) If 0 → N ⊕a → K → ΩnR N ⊕b → 0 is an exact sequence of R-modules, where a ≥ 1, b ≥ 1, and n ≥ 0
are integers, then red-IR (N) ≤ red-IR (K) + 1.
Remark 2.3. The definition of the reducing dimension we use in this paper is taken from [3], eventhough this notion was originally defined in [1]. The difference between the definitions of the reducing
dimension given [1] and [3] is that [3] only requires the integers ni in Definition 2.1 to be nonnegative,
while [1] requires these numbers to be positive; see also Remark 2.12. Hence, if a module has finite
reducing invariant with respect to the definition given in [1], then it has also finite reducing invariant
with respect to the definition we use in this paper.
Remark 2.4. The definition of reducing projective dimension [1] was initially originated from reducible
complexity, a notion introduced by Bergh; see [9, 10] for the details. In general, modules having
reducible complexity also have finite reducing projective dimension, but not vice versa. In other words,
reducing projective dimension is a finer invariant than reducible complexity. One way to observe thisfact is to use Theorem 1.2: if S is as in Theorem 1.2 and S is not a complete intersection, then k does nothave finite complexity as an S-module [18, 2.3], and hence it does not have reducible complexity as an
S-module [9, 2.1], but k has finite reducing projective dimension as an S-module by Theorem 1.2. One of the motivations of our work in this paper comes from the fact that the reducing projective
dimension of the residue field is finite over local rings with radical square zero:
Example 2.5. (1) Let R be a local ring. Assume R is not Gorenstein and m2 = 0. Then pdR (ΩiR k) =∞ = G-dimR (ΩiR k) and red-pdR (ΩiR k) = 1 = red-G-dimR (ΩiR k) for all i ≥ 0; see [1, 2.3].
(2) Let R = C[[x, y]]/(x2 , xy, y2 ). Then m2 = 0 so that 2.5 implies red-pdR (k) = 1.

The rings in Example 2.5 are zero-dimensional. Next is an example of a module – over a local ringof dimension two – that has finite reducing projective, but infinite projective, dimension.
Example 2.6. ([1, 2.7]) Let R = C[[x3 , x2 y, xy2 , y3 ]] be the 3rd Veronese subring of the ring C[[x, y]] andlet M = (x2 , xy, y2 ). Then red-pdR (M) = red-G-dimR (M) = 1 < ∞ = G-dimR (M). In fact, one can checkthat there is an exact sequence of R-modules 0 → M ⊕2 → R⊕3 → M → 0 so that {M, R⊕3 } gives areducing pd-sequence of M.
In general, unlike the classical homological dimensions, the reducing projective dimension of anonzero module over a local ring, if finite, may not be bounded by the depth of the ring in question;see, for example, Example 2.5. In fact, we do not know whether or not there is a uniform upper boundfor reducing projective dimension of modules, even over Gorenstein rings. On the other hand, a consequence of what we know from [12] yields a satisfactory answer over complete intersection rings, andalso allows us to determine modules that have sufficiently large finite reducing projective dimension:
2.7. Let R be a local complete intersection ring of codimension c, for example, one can pick the ring R =k[[x1 , . . . , xc , y1 , . . . , yc ]]/(x1 y1 , . . . , xc yc ). If M is an R-module, then red-pdR (M) ≤ c since red-pdR (M) =cxR (M), where cxR (M) is the complexity of M; see [12] for the details. So, for each given r with0 ≤ r ≤ c, one can use [8, 5.7] and construct an R-module M such that red-pdR (M) = r.

We proceed to give an example of a module that has reducing projective dimension equal two andsubsequently determine a reducing sequence of it explicitly; see Example 2.10. First we record some
auxiliary results that are used for the argument of the example, as well as in section 3.
2.8. Let (R, m) → (S, n) be a homomorphism of local rings.
(i) If M is an R-module and TorRi (M, S) = 0 for each i ≥ 1, then pdS (M ⊗R S) = pdR (M) and S ⊗R
ΩnR M ∼
= ΩnS (M ⊗R S) for all n ≥ 0; see [5, 1.2.3].

4

OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUI(ii) Assume S has finite flat dimension over R. Let M be an R-modules such that G-dimR (M) = n < ∞and TorRi (M, S) = 0 for all i = 1, . . . , n. Then [14, 5.4.4] implies that TorRi (M, S) = 0 for all i ≥ 1.
Note that M is reflexive as an R-complex; see, for example, [26, 2.7]. Therefore, we conclude
from [15, 5.10] that G-dimS (M ⊗R S) = n.
(iii) If S is a finitely generated R-module such that pdR (S) = n < ∞ and N is a finitely generated Smodule, then there is some free R-module F such that ΩaR ΩbS N ∼= F ⊕ Ωa+b
R N for each a ≥ n andb ≥ 0; this fact extends [23, 4.2] and can be proved similarly.
The next observation is used for Example 2.10 and also in the proof of Proposition 3.9.
2.9. Let R be a local ring and let M be a nonzero R-module. Assume there is a non zero-divisor x ∈ m
on R such that xM = 0. Then there is a short exact sequence of R/xR-modules:(2.9.1)

0 → M → ΩR M ⊗R (R/xR) → ΩR/xR (M) → 0.

To establish the sequence in (2.9.1), we consider a syzygy sequence of M, namely an exact sequence
0 → ΩR M → F → M → 0, where F is a free R-module. Then we obtain, by tensoring the syzygy
sequence with R/xR, an exact sequence of R-modules of the form
0 → TorR1 (M, R/xR) → ΩR M ⊗R (R/xR) → F ⊗R (R/xR) → M/xM → 0.
Note that TorR1 (M, R/xR) ∼= M since x is a non zero-divisor on R and xM = 0. Hence (2.9.2) yields theexact sequence in (2.9.1).
(2.9.2)

Example 2.10. Let R = S/aS, where (S, n) is a one-dimensional local hypersurface ring and a ∈ n2 is anonzero divisor on S. For example, we can set R = C[[x, y]]/(x2 , y2 ), S = C[[x, y]]/(x2 ), and a = y2 . Then
R is an Artinian complete intersection ring of codimension two so that red-pdR (k) = 2; see 2.7. Now
we proceed and find a reducing pd-sequence of k over R.
Let us first note, by making use of (2.9.1) with the S-module k over S, we have a short exact sequenceof R-modules of the form:(2.10.1)

0 → k → R ⊗S n → ΩR k → 0.

As we know red-pdR (k) = 2, it follows that red-pdR (R ⊗S n) = 1; see Remark 2.2(ii). So it is enough tofind a reducing pd-sequence of R ⊗S n over R.
In general, if T is a local ring and X and Y are (finitely generated) T -modules such that pdT (X) = 1
and Y is torsion-free, then TorTi (X,Y ) = 0 for al i ≥ 1; see, for example, [13, 2.7]. So, as pdS (R) = 1,
and n and ΩS n are torsion-free S-modules, we have:(2.10.2)

TorSi (ΩS n, R) = 0 = TorSi (n, R) for all i ≥ 1.

Note that, as S is a one-dimensional hypersurface ring and n is a maximal Cohen-Macaulay S-modulewith no free summand, we have n ∼= Ω2S n. Moreover, since n is generated by two elements, there is a
short exact sequence of S-modules of the form:(2.10.3)

0 → n → S⊕2 → ΩS n → 0.

We obtain, by using (2.10.2) and tensoring (2.10.3) with R over S, the exact sequence of R-modules:(2.10.4)

0 → R ⊗S n → R⊕2 → R ⊗S ΩS n → 0.

Next note that R ⊗S ΩS n ∼= ΩR (n ⊗S R); see 2.8(i) and (2.10.2). Therefore, (2.10.1) and (2.10.4) showthat a reducing pd-sequence of k over R is {k, R ⊗S n, R⊕2 }.

Recall that an m-primary ideal I of a local ring R is called Ulrich if I/I 2 is a free R/I-module andI 2 = qI for some parameter ideal q of R, where q is a reduction of I; see [17, 1.1] for the details. Next
we point out that Ulrich ideals, when exist, provide nontrivial examples of modules of finite reducing
projective dimension.

ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

5

2.11. Let R be a Cohen-Macaulay local ring and let I be an Ulrich ideal of R that is not a parameter
ideal. Then pdR (R/I) = ∞ and red-pdR (R/I) = 1.
To observe this, note, if red-pdR (R/I) = 0, that is, pdR (R/I) < ∞, then, as I/I 2 is a free R/I-module,
and so [11, 2.2.8] implies that I is a parameter ideal. Hence we conclude pdR (R/I) = ∞. Next we
consider a parameter ideal q of R such that I 2 = qI and q $ I ⊆ m. Then, by [17, 2.3](2)(c), it follows that
I/q is a free R/I-module. So there is an exact sequence of R-modules 0 → (R/I)⊕a → R/q → R/I → 0
for some a ≥ 1, which shows that red-pdR (R/I) ≤ 1, that is, red-pdR (R/I) = 1.
Remark 2.12. It is worth mentioning that, if we adopt the definition of the reducing dimension givenin [1], then the argument of 2.11 does not necessarily yield that red-pdR (R/I) = 1.
Example 2.13. Let R = C[[x, y, z]]/(x3 − y2 , z2 − x2 y) and let I = (x, y). Then R/I ∼= C[[z]]/(z2 ) and I is
an Ulrich ideal of R which is not a parameter ideal; see [17, 2.7(1)]. Therefore, by 2.11, it follows thatpdR (R/I) = ∞ and red-pdR (R/I) = 1; see also [17, 2.7, 6.7 and 6.8] for similar examples.
Recall that a Cohen-Macaulay local ring R is said to have minimal multiplicity if the codimension of
R is one less than the multiplicity of R; see [11, 4.5.14]. In the following we obtain an extension of thefact recorded in Example 2.5:
2.14. Let R be a non-regular Cohen-Macaulay local ring such that R has minimal multiplicity and|k| = ∞. Then m2 = qm for some parameter ideal q that is a minimal reduction of m; see, for example,[24]. Then m is an Ulrich ideal of R that is not a parameter ideal. Therefore 2.11 yields that pdR (k) = ∞and red-pdR (k) = 1.
Example 2.15. Let R = C[[x, y]]/(x, y)2 , or R = C[[t 3 ,t 4 ,t 5 ]], or R = C[[t 4 ,t 5 ,t 6 ,t 7 ]]. Then R is a nonregular Cohen-Macaulay local ring with minimal multiplicity so that 2.14 implies that pdR (k) = ∞ andred-pdR (k) = 1.
Remark 2.16. We now have several examples of modules that have finite reducing projective dimension. Let us also mention that modules of infinite reducing projective dimension also exist: Jorgensen
and Şega [20, 1.7] constructed a local Artinian ring T and a T -module X of infinite Gorenstein dimension such that ExtiT (X, T ) = 0 for all i ≥ 1. On the other hand, Araya and Celikbas [1, 1.3] proved that, if
R is a local ring and M is an R-module such that red-G-dimR (M) < ∞ and ExtiR (M, R) = 0 for all i ≫ 0,then M has finite Gorenstein dimension. Therefore the aforementioned module X constructed by Jorgensen and Şega has infinite reducing Gorenstein dimension, and hence has infinite reducing projective
dimension, over the ring T .

We prove Theorem 1.2 in the next section; we should note that Theorem 1.2, besides giving anaffirmative answer for Question 1.1 for a large class of local rings, also yields an extension of theobservation stated in 2.12.
3. P ROOF OF THE MAIN THEOREM
In this section we give a proof of Theorem 1.2; see the paragraph following Proposition 3.9. Alongthe way we obtain several new results and study further properties of reducing dimensions, especiallyproperties under local homomorphisms of local rings.
Throughout, I denotes an invariant of modules over local rings; see the paragraph preceding Definition 2.1. Our first result yields a generalization of [2, 2.9].
Proposition 3.1. Let (R, m) → (S, n) be a local homomorphism of local rings. Assume that IS (M ⊗RS) ≤ IR (M) for each R-module M for which TorRi (M, S) = 0 for all i ≥ 1. If N is an R-module such that
TorRi (N, S) = 0 for all i ≥ 1, then red-IS (N ⊗R S) ≤ red-IR (N).
Proof. Let N be an R-module such that red-IR (N) = r < ∞. We proceed by induction on r to prove
red-IS (N ⊗R S) ≤ r. We may assume N ⊗R S 6= 0.

6

OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUIAssume r = 0. Then, by Definition 2.1, we have that IR (N) < ∞. This implies, by the hypothesis,that IS (N ⊗R S) ≤ IR (N) < ∞, and hence red-IS (N ⊗R S) = 0 = r.
Next we assume r ≥ 1. Then, by Remark 2.2(i), there exists a short exact sequence of R-modules
0 → N ⊕a → K → ΩcR N ⊕b → 0, where red-IR (K) = r − 1, and a ≥ 1, b ≥ 1, and c ≥ 0 are integers. Note,since TorRi (N, S) = 0 for all i ≥ 1, we have the vanishing of TorRi (K, S) for all i ≥ 1. Hence the induction
hypothesis on r yields:(3.1.1)

red-IS (K ⊗R S) ≤ red-IR (K) = r − 1.

Once again we make use of the hypothesis that TorRi (N, S) vanishes for all i ≥ 1, tensor the short exact
sequence 0 → N ⊕a → K → ΩcR N ⊕b → 0 with S over R, and obtain the short exact sequence of S-modules
0 → (N ⊗R S)⊕a → K ⊗R S → ΩcS (N ⊗R S)⊕b → 0; see also 2.8(i). So, by Remark 2.2(ii), we deduce:(3.1.2)red-IS (N ⊗R S) ≤ red-IS (K ⊗R S) + 1.

Consequently (3.1.1) and (3.1.2) show that red-IS (N ⊗R S) ≤ r = red-IR (N).



Corollary 3.2. Let (R, m) → (S, n) be a local homomorphism of local rings and let N be an R-module
such that TorRi (N, S) = 0 for all i ≥ 1. Then the following hold:(i) red-pdS (N ⊗R S) ≤ red-pdR (N).
(ii) If S has finite flat dimension over R, then red-G-dimS (N ⊗R S) ≤ red-G-dimR (N).
Proof. The first and the second conclusion, in view of Proposition 3.1, follow from 2.8(i) and 2.8(ii),respectively.
Corollary 3.3. Let R be a local ring and let x ∈ m be a non zero-divisor on R. Assume that IR/xR (N/xN) ≤IR (N) for each R-module N such that x is a non zero-divisor on N. If M is an R-module and x is a nonzero-divisor on M, then red-IR/xR (M/xM) ≤ red-IR (M).
Proof. The result follows from Proposition 3.1 by using the ring homomorphism R → S = R/xR.



The following special case of Corollary 3.3 is worth recording separately:
Corollary 3.4. Let R be a local ring and let M be an R-module. If x ∈ m is a non zero-divisor on R andM, then red-pdR/xR (M/xM) ≤ red-pdR (M) and red-G-dimR/xR (M/xM) ≤ red-G-dimR (M).

Next we point out that the inequality in Corollary 3.2(ii) can be strict; we give an example of a local
homomorphism of local rings R → S and an R-module N such that S has finite flat dimension over R,
TorRi (N, S) = 0 for all i ≥ 1, and red-G-dimS (N ⊗R S) < ∞ = red-G-dimR (N).
Example 3.5. Let R = k[[t 3 ,t 7 ,t 8 ]] and let N = R + Rt + Rt 5. Then R is a one-dimensional domain and
N is a maximal Cohen-Macaulay R-module.
Let S = R/q, where q = (t 3 ). Then R ∼= k[[x, y, z]]/(x5 − yz, y2 − zx2 , z2 − x3 y) and S ∼= k[[y, z]]/(y, z)2 .
⊕3 . Therefore red-pd (N ⊗Moreover, we have that qN = Rt 3 + Rt 4 + Rt 8 = mN and hence N/qN ∼k=RSS) = 1 = red-G-dimS (N ⊗R S); see Example 2.5.
Note that N is reflexive if and only if N = (R :Q(R) (R :Q(R) N)); see, for example, [21, 2.4]. One
can check the equalities R :Q(R) (R :Q(R) N) = R :Q(R) (Rt 6 + Rt 7 + Rt 8 ) = R + Rt + Rt 2 hold andconclude that N is not reflexive.
As R is a one-dimensional domain, an R-module is reflexive if and only if it is the syzygy of a
maximal Cohen-Macaulay R-module; see [19, 2.4]. Hence N is not the syzygy of a maximal CohenMacaulay R-module. It is proved in [12] that, if a maximal Cohen-Macaulay module over a local ringhas finite reducing Gorenstein dimension, then it a syzygy of a maximal Cohen-Macaulay module.
Consequently, we conclude that red-G-dimR (N) = ∞. Finally we note, since pdR (S) = 1 and t 3 is a nonzero-divisor on N, that TorRi (N, S) = 0 for all i ≥ 1.
3.6. Given a local ring R, we consider the following conditions for the invariant I in question:

ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

7(i) IR (M) ≤ pdR (M) for each R-module M.
(ii) IR (ΩR M) ≤ IR (M) for each R-module M.
(iii) If IR (N) < ∞, and 0 → F → N → M → 0 is an exact sequence of R-modules, where F is free, thenIR (M) < ∞.
(iv) For every R-modules M and F, where F is free, one has IR (M ⊕ F) < ∞ if and only if IR (M) < ∞.
3.7. If I equals the projective dimension pd, or the Gorenstein dimension G-dim, then it satisfies all theconditions stated in 3.6; see, for example, [14, 1.2.9, 1.2.10] for the case where I = G-dim.
We need the following result for the proof of Theorem 3.11.
Proposition 3.8. Let R be a local ring and let IR be an invariant of modules over local rings which
satisfy the conditions (iii) and (iv) of 3.6. Then red-IR (M ⊕ F) = red-IR (M) for each R-module M andeach free R-module F.
The conclusion of Proposition 3.8 seems natural, but its proof is involved; therefore we defer the
proof of the lemma until the next section so it does not interfere with the flow of the paper.
Proposition 3.9. Let R be a local ring and let S = R/xR, where x = x1 , . . . , xn ⊆ m is an R-regularsequence. Assume the following hold:(i) I satisfy the conditions (iii) and (iv) stated in 3.6.
(ii) If N is an R-module such that x1 is a non zero-divisor on N, then IR/x1 R (N/x1 N) ≤ IR (N).
If M is a (finitely generated) S-module, then red-IS (M) ≤ red-IR (ΩnR M) + n.
Proof. We proceed by induction on n. Assume first n = 1 and= x. Then, since x is a non set x1 ΩR M
≤ red-IR (ΩR M).
zero-divisor on ΩR M, it follows from Corollary 3.3 that red-IS
x · ΩR M
Note, by (2.9.1), we have the following short exact sequence of S-modules:(3.9.1)0→M→ΩR M→ ΩS (M) → 0.
x · ΩR M

Therefore Remark 2.2(ii) applied to (3.9.1) gives red-IS (M) ≤ red-ISΩR M
x · ΩR M

+ 1. This establishes

the base case n = 1 of the induction.
Next we assume n ≥ 2 and set T = R/x′ R, where x′ = x1 , . . . , xn−1 . Then xn is a non zero-divisor on
T and S = T /xn T . So, by the case where n = 1, we have:(3.9.2)red-IS (M) ≤ red-IT (ΩT M) + 1.

We make use of the induction hypothesis with the module ΩT M and conclude that:(3.9.3)red-IT (ΩT M) ≤ red-IR (Ωn−1
R ΩT M) + n − 1.

Hence (3.9.2) and (3.9.3) yield:(3.9.4)red-IS (M) ≤ red-IR (Ωn−1
R ΩT M) + n.

∼ n
Next, note that, Ωn−1
R ΩT M = ΩR M ⊕ F for some free R-module F; see 2.8(iii). Thus the induction
argument is now complete from (3.9.4) and Proposition 3.8.

We need the following observation for the proof of Theorem 3.11.
p

3.10. Let R be a ring (not necessarily local and Noetherian) and let 0 → L → M −→ N ⊕ F → 0 be a short
exact sequence of R-modules, where F is free. Then we have the following pullback diagram, that is, a

8

OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUI

commutative diagram with exact rows and columns:
0

0
N

0

N⊕F

0

0

L

M′

0

L

M

p

F

F

0

0

This diagram yields a short exact sequence of R-modules 0 → L → M ′ → N → 0, where M ∼= M ′ ⊕F. 
The next theorem plays a key role for the proof of Theorem 1.2.
Theorem 3.11. Let (R, m) → (S, n) be a local ring homomorphism, where S is a finitely generated
R-module and pdR (S) ≤ n < ∞ for some integer n. Assume I satisfies all the conditions stated in 3.6.
Assume further that IR (N) < ∞ whenever N is a (finitely generated) S-module such that IS (N) < ∞.
Then red-IR (ΩnR N) ≤ red-IS (N) for each finitely generated S-module N.
Proof. Let N be a nonzero finitely generated S-module such that red-IS (N) = r < ∞. We proceed byinduction on r to show that red-IR (ΩnR N) ≤ r.
Assume first r = 0. Then red-IS (N) = 0, i.e., IS (N) < ∞. Hence, by the hypotheses, we conclude
that IR (ΩR N) ≤ IR (N) < ∞. This shows that red-IR (ΩnR M) = 0 = r.
Next assume r ≥ 1. Then it follows from Definition 2.1 that there exists a short exact sequence of(finitely generated) S-modules(3.11.1)

0 → N ⊕a → K → ΩcS N ⊕b → 0,

where a ≥ 1, b ≥ 1 and c ≥ 0 are integers, and red-IS (K) = red-IS (N) − 1 = r − 1. So, by the induction
hypothesis, we have:(3.11.2)

red-IR (ΩnR K) ≤ red-IS (K) = r − 1.

We obtain, by applying ΩnR (−) to (3.11.1), the following exact sequence of R-modules:(3.11.3)

0 → ΩnR N ⊕a → ΩnR K ⊕ G → ΩnR ΩcS N ⊕b → 0,

where G is a free R-module; see [16, 2.2(1)].
⊕b ⊕ F for some free R-module F; see 2.8(iii). Then, by using 3.10,
Note that ΩnR ΩcS N ⊕b ∼= Ωn+cR N
we obtain the following exact sequence of R-modules:(3.11.4)⊕b
0 → ΩnR N ⊕a → K ′ → Ωn+c
→ 0,
R N

where ΩnR K ⊕ G ∼= K ′ ⊕ F. Now we observe the following (in)equalities which complete the proof:
red-IR (ΩnR N) ≤ red-IR (K ′ ) + 1(3.11.5)= red-IR (ΩnR K) + 1≤(r − 1) + 1 = r.

Here, in (3.11.5), the first inequality follows from (3.11.4) and Remark 2.2(ii), and the second one
follows from (3.11.2); moreover, the first equality is due to Proposition 3.8.


ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

9Remark 3.12. Let (R, m) → (S, n) be a local ring homomorphism, where S is a finitely generated Rmodule and pdR (S) < ∞. Then the hypothesis on the ring map considered in Theorem 3.11, namely thecondition that IR (N) < ∞ whenever N is a finitely generated S-module such that IS (N) < ∞, holds whenI equals the projective dimension. This condition also holds for certain ring homomorphisms when I
equals the Gorenstein dimension. For example, if the closed fibre of the ring map (R, m) → (S, n) is
Gorenstein, then G-dimS (N) < ∞ if and only if G-dimR (N) < ∞ for each finitely generated S-module N;see [6, 4.2] and [7, 7.11]. Therefore, if we consider the natural surjection R → S = R/(x) for some Rregular sequence x ⊆ m, then G-dimS (N) < ∞ if and only if G-dimR (N) < ∞ for each finitely generated
S-module N.

Next is a consequence of Theorem 3.11; this result, for the case where I is the projective dimension, or the Gorenstein dimension of modules, can also be deduced from [1]. However, as mentioned
in Remark 2.3, the definition given in [1] is slightly more restrictive than the definition of reducing
dimensions we adopt in this paper; see Definition 2.1.
Corollary 3.13. Let R be a local ring. Assume I satisfies all the conditions stated in 3.6. Then
red-IR (ΩiR M) ≤ red-IR (M) for each R-module M and for each i ≥ 0.
Proof. The corollary follows from Theorem 3.11 by setting R = S.

d denotes the completion
We need a few more results to prove Theorem 1.2; in the following (−)
functor.
b = IR (M) for each R-module M. If N is an R-module,
3.14. Let R be a local ring. Assume that IRb(M)b ≤ red-IR (N).
then it follows from Definition 2.1 that red-I b(N)R

b = red-IR (N) if N is an R-module which
We can show, under the hypothesis of 3.14, that red-IRb(N)is locally free on the punctured spectrum of R, that is, if Np is free over Rp for all non-maximal prime
ideals p of R.

b = IR (M) for each R-module M. If N is an
Proposition 3.15. Let R be a local ring. Assume that IRb(M)b = red-IR (N).
R-module that is locally free on punctured spectrum of R, then red-IRb(N)Proof. Let N be an R-module which is locally free on punctured spectrum of R. Then, in view of 3.14,
b Notice we may assume red-I b(N)b < ∞ as otherwise there
it is enough to prove red-IR (N) ≤ red-IRb(N).
Rb
is nothing to prove. We set r = red-IRb(N) and proceed by induction on r.
b < ∞; see Definition 2.1. Hence our assumption yields IR (N) = I b(N),b
If r = 0, then we have IRb(N)Rband so shows that red-IR (N) = 0 = red-IRb(N).
c N ⊕b → 0,\d⊕a → K → Ωb
Now let r ≥ 1. Then there exists a short exact sequence of R-modules:
0→NR

where red-IRb(K) = r − 1, and a ≥ 1, b ≥ 1, and c ≥ 0 are integers; see Remark 2.2(i). Note that, since
Ext1R (ΩcR N ⊕b , N ⊕a ) is a finite length R-module, in the following the natural maps are isomorphisms:c N ⊕b , Nd⊕a )Ext1R (ΩcR N ⊕b , N ⊕a ) ∼= Ext1R (ΩcR N ⊕b , N ⊕a ) ⊗R Rb ∼= Ext1Rb(Ω\R

As the composition of these two natural maps is given by the completion of short exact sequences, webdeduce that there is a short exact sequence of R-modules 0 → N ⊕a → X → ΩcR N ⊕b → 0, where K ∼= X;
see also [22, 2.7(i)].
Note that an R-module T is locally free on the punctured spectrum of R if and only if Tb is locally freeb Therefore, since K ∼on the punctured spectrum of R.
= Xb and K is locally free on the punctured spectrumbb = red-I b(K) = r − 1,of R, we conclude that X is locally free on punctured spectrum of R. As red-IRb(X)Rb
we see by the induction hypothesis that red-IR (X) ≤ red-IRb(X). Hence, in view of Remark 2.2(ii), theexact sequence 0 → N ⊕a → X → ΩcR N ⊕b → 0 implies that red-IR (N) ≤ red-IR (X) + 1 ≤ (r − 1) + 1 = r.
This completes the induction and the proof.
10

OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUI

Corollary 3.16. Let R be a local ring and let N be an R-module which is locally free on puncturedb = red-pdR (N) and red-G-dim b(N)b = red-G-dimR (N).
spectrum of R. Then red-pdRb(N)R

b = pdR (M) and G-dim b(M)b = G-dimR (M) for each R-module M; see, for
Proof. Note that pdRb(M)R
example, 2.8(i) and [15, 5.10(c)]. Therefore the result follows from Proposition 3.15.

We are now ready to prove Theorem 1.2 as advertised in the introduction.
Proof of Theorem 1.2. Note that we have:
red-pdSb(k) ≤ red-pdR (ΩnR k) + n ≤ red-pdR (k) + n < ∞.

Here, in view of 3.7, the first inequality is due to Proposition 3.9 and the second one is due to Corollary
3.13; the third inequality is clear if R is regular, and it holds by 2.14 in case R is singular. Now,
since k is a complete S-module that is locally free on the punctured spectrum of S, it follows that
red-pdS (k) = red-pdSb(bk) = red-pdSb(k); see from Corollary 3.16. This completes the proof.
We finish this section with a further application of Proposition 3.9 and Corollary 3.13. Recall thata local ring R is called G-regular [25] provided that G-dimR (M) < ∞ if and only if pdR (M) < ∞ foreach R-module M. As, in general, over a local ring R, we have that G-dimR (M) = pdR (M) for each
R-module M with pdR (M) < ∞, it is clear that each regular local ring is G-regular. A nontrivial fact isthat a non-Gorenstein Cohen-Macaulay local ring of minimal multiplicity is G-regular; see [25, 5.1] forthe details. On the other hand, deformations of G-regular rings, that is, G-regular rings modulo ideals
generated by regular sequences, need not be G-regular as we see next:
Example 3.17. Let R = C[[t 3 ,t 4 ,t 5 ]] = C[[x, y, z]]/(x3 − yz, z2 − yx2 , y2 − xz) and let S = R/x2 R. Then x2
is a non zero-divisor on R and R is G-regular since it is a non-Gorenstein Cohen-Macaulay local ringwith minimal multiplicity. Moreover, as x2 ∈ m2 , it follows from [25, 4.6] that S is not G-regular.

Although deformations of G-regular rings are not G-regular in general, we finish this section bymaking a related observation: a deformation of a G-regular ring is reducing G-regular, that is, a ringover which each module that has finite reducing Gorenstein dimension also has finite reducing projective
dimension. First we note that each G-regular ring is reducing G-regular.
3.18. Let R be a G-regular local ring. Then, by definition, pdR (M) = G-dimR (M) for each R-module M.
Therefore it follows from Definition 2.1 that red-pdR (M) = red-G-dimR (M) for each R-module M.
Proposition 3.19. Let R be a G-regular local ring and let S = R/xR, where x = x1 , . . . , xn ⊆ m is an
R-regular sequence. If M is a (finitely generated) S-module, thenred-G-dimS (M) ≤ red-pdS (M) ≤ red-G-dimS (M) + n.
So, for each finitely generated S-module N, we have red-pdS (N) < ∞ if and only if red-G-dimS (N) < ∞.
Proof. Let M be a finitely generated S-module. Then the first inequality follows by Definition 2.1. For
the second inequality, note that we have:
red-pdS (M) ≤ red-pdR (ΩnR M) + n = red-G-dimR (ΩnR M) + n ≤ red-G-dimS (M) + n.
Here the first inequality, in view of [11, 1.3.5], follows from Proposition 3.9, the second inequality
follows from Theorem 3.11 and Remark 3.12, and the equality is due to 3.18.
ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

11

4. P ROOF OF P ROPOSITION 3.8
This section is devoted to a proof of Proposition 3.8. We start by noting that:
Remark 4.1. Let R be a local ring, M be an R-module, F be a free R-module, and let I be an invariantof modules over local rings satisfying condition (iv) of 3.6. If {K0 , . . . , Kr } is a reducing I-sequence ofM, then one can find some free R-modules G1 , G2 , . . . , Gr such that {K0 ⊕ F, K1 ⊕ G1 , . . . , Kr ⊕ Gr } is areducing I-sequence of M ⊕ F; see Definition 2.1. Therefore red-IR (M ⊕ F) ≤ red-IR (M).

Proposition 3.8, in view of Remark 4.1, is subsumed by the following result:
Proposition 4.2. Let R be a local ring and let I be an invariant of modules over local rings which
satisfies the conditions (iii) and (iv) of 3.6. Let 0 → F → N → M → 0 be an exact sequence of nonzero
R-modules, where F is free. Then red-IR (M) ≤ red-IR (N).
Proof. To prove the claim, we assume red-IR (N) ≤ r for some integer r ≥ 0, and proceed by induction
on r. Note, if r = 0, then by definition, IR (N) < ∞, and then condition (iii) of 3.6 implies IR (M) < ∞,and hence red-IR (M) = 0 ≤ red-IR (M). So we assume r ≥ 1.
χIt follows that there is a short exact sequence of R-modules 0 → N ⊕a −→ K → ΩnR N ⊕b → 0, where
red-IR (K) = red-IR (N)− 1 ≤ r − 1 and a, b ≥ 1, n ≥ 0 are integers; see Remark 2.2(ii). Now we consider
the following commutative diagram (with exact rows and columns) obtained by taking the pushout ofψ→ M → 0 is the exact sequence we consider.
the maps χ and ψ ⊕a , where 0 → F → N −

0

0

0

F ⊕a

F ⊕a

N ⊕a

χ

K

ΩnR N ⊕b

0

M ⊕a

L

ΩnR N ⊕b

0

0

0

ψ ⊕a

0

Now, since red-IR (K) ≤ r − 1, we use the induction hypothesis on r with the short exact sequence
0 → F ⊕a → K → L → 0 obtained in the above diagram, and conclude that
(4.2.1)

red-IR (L) ≤ red-IR (K) = red-IR (N) − 1.

Case 1: Suppose n ≥ 1 and consider the exact sequence 0 → M ⊕a → L → ΩnR N ⊕b → 0.
We take syzygies of the short exact sequence 0 → F → N → M → 0, and see that, for each v ≥ 1,
there exists a free R-module Hv such that ΩvR M ∼= ΩvR N ⊕ Hv . So, by adding free summands to the exact⊕an⊕bsequence 0 → M → L → ΩR N → 0, we obtain the exact sequence:
0 → M ⊕a → L ⊕ Hn⊕b → ΩnR N ⊕b ⊕ Hn⊕b → 0.
The sequence (4.2.2), in view of the isomorphism Ωn M ∼= Ωn N ⊕ Hn , gives the exact sequence:(4.2.2)R(4.2.3)R

0 → M ⊕a → L ⊕ Hn⊕b → ΩnR M ⊕b → 0.

Now we use (4.2.3) and conclude from Remarks 2.2(ii) and 4.1 that:(4.2.4)red-IR (M) ≤ red-IR (L ⊕ Hn⊕b) + 1 ≤ red-IR (L) + 1.

Thus, since red-IR (K) + 1 = red-IR (N), the proof is complete in view of (4.2.1) and (4.2.4).

12

OLGUR CELIKBAS, SOUVIK DEY, TOSHINORI KOBAYASHI, AND HIROKI MATSUI

Case 2: Suppose n = 0 and consider the exact sequence 0 → M ⊕a → L → ΩnR N ⊕b → 0.
Recall that, by the hypothesis and the previous pushout diagram, we have the short exact sequencesψ ⊕b

0 → M ⊕a → L → N ⊕b → 0 and 0 → F ⊕b → N ⊕b −−→ M ⊕b → 0. Next we take the pullback of the maps
L → N ⊕b and ψ ⊕b , and obtain the following commutative diagram with exact rows and columns:

0

M ⊕a

0

0

Y

F ⊕b

0

ψ ⊕b

0

M ⊕a

L

N ⊕b

M ⊕b

M ⊕b

0

0

0

The top exact sequence in the above diagram shows that Y ∼= M ⊕a ⊕ F ⊕b as F ⊕b is a free R-module. So
the left vertical exact sequence in the above diagram gives the short exact sequence:
0 → M ⊕a ⊕ F ⊕b → L → M ⊕b → 0.

(4.2.5)

We add, if needed, free R-modules to (4.2.5) and obtain an exact sequence for some integers c, d ≥ 0:
0 → (M ⊕ F ⊕c )⊕a → L ⊕ F ⊕d → (M ⊕ F ⊕c )⊕b → 0.

(4.2.6)

Next we observe the following inequalities:
red-I(M) ≤ red-IR (M ⊕ F ⊕c ) ≤ red-IR (L ⊕ F ⊕d ) + 1 ≤ red-I(L) + 1≤ red-I(K) + 1= red-I(N)≤r.
Here we obtain the second inequality by applying Remark 2.2(ii) to (4.2.6), the third inequality is due
to Remark 4.1, the fourth inequality and the equality follow from (4.2.1), the last inequality is ourassumption, and the first inequality can be deduced by the next claim:
Claim. Let A and H be R-modules such that H is free and red-IR (A ⊕ H) ≤ r. Then it follows
red-IR (A) ≤ red-IR (A ⊕ H).
Proof of the Claim: We note, due to Definition 2.1 and Remark 2.2(ii), that there exists a short exactpsequence of R-modules 0 → (A ⊕ H)⊕a → K −→ ΩsR (A ⊕ H)⊕b → 0, where a, b ≥ 1, s ≥ 0 are integers
and red-IR (K) = red-IR (A ⊕ H) − 1 ≤ r − 1. Note also that ΩsR (A ⊕ H)⊕b ∼= ΩsR A⊕b ⊕ G for some free
R-module G. Thus we have an exact sequence of the form:(4.2.7)0 → (A ⊕ H)⊕a → K → ΩsR A⊕b ⊕ G → 0.

Now we use 3.10 with the sequence (4.2.7) and obtain the following exact sequence of R-modules:(4.2.8)where K ∼= K ′ ⊕ G.

α

0 → (A ⊕ H)⊕a −→ K ′ → ΩnR A⊕b → 0,

ON THE REDUCING PROJECTIVE DIMENSION OVER LOCAL RINGS

13

Next we consider the commutative diagram (with exact rows and columns) that is obtained by taking
the pushout of the map α from (4.2.8) with the canonical surjection (A ⊕ H)⊕a ։ A⊕a :
0

0

H ⊕a

H ⊕a

0

(A ⊕ H)⊕a

0

A⊕a

α

K′

ΩnR A⊕b0

XΩnR A⊕b000′∼Note that, since K = K ⊕ G, we have an exact sequence 0 → G → K → K ′ → 0. We use this exact
sequence and, since red-IR (K) ≤ r − 1, conclude by the induction hypothesis on r that
(4.2.9)red-IR (K ′ ) ≤ red-IR (K) ≤ r − 1.

Next we consider the exact sequence obtained from the above diagram: 0 → H ⊕a → K ′ → X → 0.
As red-IR (K ′ ) ≤ r − 1 by (4.2.9), we make use of the induction hypothesis on r and conclude that
(4.2.10)red-IR (X) ≤ red-IR (K ′ ) ≤ red-IR (K).

We also consider the exact sequence obtained from the above diagram: 0 → A⊕a → X → ΩnR A⊕b → 0.
We deduce from this short exact sequence and Remark 2.2(ii) that:(4.2.11)

red-IR (A) − 1 ≤ red-IR (X).

Finally, as red-IR (K) = red-IR (A ⊕ H) − 1, we see red-IR (A) ≤ red-IR (A ⊕ H) by (4.2.10) and (4.2.11).
This completes the proof of the Claim, and hence that of the proposition.
ACKNOWLEDGEMENTS

The authors are grateful to Roger Wiegand for discussions, and for his helpful and valuable comments
and suggestions on the manuscript. The authors also thank the anonymous referee for careful reading
and giving helpful comments.
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O LGUR C ELIKBAS : S CHOOL OF M ATHEMATICAL AND D ATA S CIENCES , W EST V IRGINIA U NIVERSITY, M ORGANTOWN ,WV 26506 U.S.A
Email address: olgur.celikbas@math.wvu.edu
S OUVIK D EY: D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF K ANSAS , 405 S NOW H ALL , 1460 JAYHAWK B LVD .,L AWRENCE , KS 66045, U.S.A.
(C URRENT ADDRESS ) C HARLES U NIVERSITY, FACULTY OF M ATHEMATICS AND P HYSICS , D EPARTMENT OF A LGEBRA ,S OKOLOVSKÁ 83, 186 75 P RAHA , C ZECH R EPUBLIC .
Email address: souvik.dey@matfyz.cuni.cz
T OSHINORI K OBAYASHI : S CHOOL OF S CIENCE AND T ECHNOLOGY, M EIJI U NIVERSITY, 1-1-1 H IGASHI -M ITA , TAMA KU , K AWASAKI - SHI , K ANAGAWA 214-8571, JAPAN

Email address: tkobayashi@meiji.ac.jp
H IROKI M ATSUI : D EPARTMENT OF M ATHEMATICAL S CIENCES , FACULTY OF S CIENCE AND T ECHNOLOGY, T OKUSHIMAU NIVERSITY, 2-1 M INAMIJYOUSANJIMA - CHO , T OKUSHIMA 770-8506, JAPAN
Email address: hmatsui@tokushima-u.ac.jp