arXiv:1703.06460v1 [] 19 Mar 2017

Almost Buchsbaumness of some rings arising from
complexes with isolated singularities
Connor Sawaske
Department of Mathematics
University of Washington
Seattle, WA 98195-4350, USA
sawaske@math.washington.edu

March 21, 2017
Abstract
We study properties of the Stanley–Reisner rings of simplicial complexes with
isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz
proved that if a complex has homologically isolated singularities, then its Stanley–
Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the
case of non-homologically isolated singularities, providing many examples in which
the Stanley–Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but
not Buchsbaum ring.

1

Introduction

Many combinatorial, algebraic, and topological statements about polytopes and triangulations of spheres or manifolds have been proven through the study of their Stanley–Reisner
rings. These rings are well-understood, and the translation of their algebraic properties
into combinatorial and topological invariants has a storied and celebrated past. The usefulness of this approach is made apparent by its presence in decades of continued progressive
research (excellent surveys may be found in [Sta96] and [KN16]).
In contrast, the main objects considered in this paper are simplicial complexes with
isolated singularities. A simplicial complex ∆ has isolated singularities if it is pure and the
link in ∆ of every face of dimension at least 1 is Cohen-Macaulay (more precise definitions
will be provided later). Common examples are provided by triangulations of a pinched
torus, the suspension of a manifold, or more generally by pseudomanifolds that fail to
be manifolds at finitely many points (see [GM80] for an in-depth discussion). The gap
between pseudomanifolds and manifolds is well understood from a topological viewpoint,
but there are powerful tools available to the Stanley–Reisner rings of triangulations of
1

manifolds that presently lack any meaningful extension to the world of pseudomanifolds.
For instance, unlike results due to Stanley ([Sta75]) and Schenzel ([Sch81]), we do not
know the Hilbert series of a generic Artinian reduction of the Stanley–Reisner ring of such
a complex, even when considering a triangulation of the suspension of a manifold that is
not a homology sphere.
The central obstruction in extending the pre-existing knowledge of triangulations of
manifolds to the singular case is the Stanley–Reisner ring failing to be Buchsbaum. Miller,
Novik, and Swartz were able to circumvent this roadblock in the case that the singularities
in question are homologically isolated. By showing that a certain quotient of the associated
Stanley–Reisner ring is in fact Buchsbaum, they established some enumerative theorems
related to f - and h-vectors in [MNS11]. Novik and Swartz were then able to calculate the
Hilbert series of a generic Artinian reduction of the Stanley–Reisner ring as well as prove
singular analogs of the Dehn-Sommerville relations in [NS12].
These established results were the inspiration for this paper. However, the ultimate
purpose here is twofold; for one, we intend for the algebraic implications of isolated singularities to become as equally well-understood as the topological ones. To this end, we will
investigate with some precision how the topological properties of singular vertices translate to the algebraic setting of Stanley–Reisner rings. This, in particular, leads to a notion
of generically isolated singularities, defined in Section 2. This notion plays a crucial role
in our main results described below. We will exhibit similarities and differences between
the singular and non-singular cases, examine some special examples, and provide alternate
interpretations of some classical results. In doing so, we will see that these rings have
interesting properties that are worth studying in their own right; these properties provide
the second purpose for this paper. In particular, we present new findings demonstrating
that some quotients of Stanley-Reisner rings of simplicial complexes with isolated singularities are very near to being Buchsbaum, and we characterize when and to what degree
this occurs.
Theorem 1.1. Let ∆ be a connected simplicial complex with isolated singularities on vertex
set V , let k be an infinite field, and denote by A the polynomial ring k[xv : v ∈ V ] and by
m the irrelevant ideal of A. If θ1 , θ2 is a generic regular sequence for the Stanley-Reisner
ring k[∆], then the local cohomology module Hmi (k[∆]/(θ1 , θ2 )k[∆]) satisfies
m · Hmi (k[∆]/(θ1 , θ2 )k[∆]) = 0
for all i if and only if the singularities of ∆ are generically isolated.
Theorem 1.2. In the setting of Theorem 1.1, the canonical maps of graded modules
ϕi : Ext iA (k, k[∆]/(θ1 , θ2 )k[∆]) → Hmi (k[∆]/(θ1 , θ2 )k[∆])
are surjective in all degrees except (possibly) 0 if and only if the singularities of ∆ are
generically isolated.
2

The condition m · Hmi (k[∆]/(θ1 , θ2 )k[∆]) = 0 in Theorem 1.1 establishes a strong necessary condition for a ring to be Buchsbaum (see [GS84, Corollary 1.5]), known as quasiBuchsbaumness. These rings were first introduced by Goto and Suzuki ([GS84]). As with
Buchsbaum rings, the properties and various characterizations of quasi-Buchsbaum rings
have long been of some interest (see, e.g., [Suz87] and [Yam09]). The usefulness of the
property is evidenced, for example, by its equivalence to Buchsbaumness in some special
cases ([SV86, Corollary 3.6]).
The maps ϕi being surjective in Theorem 1.2 is a complement to the quasi-Buchsbaum
property in that it is incredibly near to one characterization of Buchsbaumness (see Theorem 4.1). As we will see in Corollary 4.7, if Γ is a Buchsbaum (but not Cohen-Macaulay)
complex and ∆ is an arbitrary triangulation of the geometric realization of the suspension of Γ, then k[∆]/(θ1 , θ2 )k[∆] is never Buchsbaum. Though Vogel ([Vog81]) and Goto
([Got84]) provided initial examples of quasi-Buchsbaum but not Buchsbaum rings, here
we exhibit an infinite family of quasi-Buchsbaum rings of arbitrary dimensions and varying depths which fail in a geometrically tangible way to be fully Buchsbaum in only the
slightest sense.
The structure of the paper is as follows. In Section 2 we provide definitions and
foundational results, allowing for some initial computations in Section 3. We prove our
main results in Section 4, and in Section 5 we will use some properties of quasi-Buchsbaum
rings to calculate the Hilbert series of a certain Artinian reduction of the Stanley–Reisner
ring of a complex with isolated singularities. We will close with comments and open
problems in Section 6.

2

Preliminaries

This paper has been largely influenced by the works of Miller, Novik, and Swartz in
[MNS11] and Novik and Swartz in [NS12]. In order to retain consistency with these
references, much of their notation will be adopted for our uses as well.

2.1

Combinatorics and topology

A simplicial complex ∆ with vertex set V is a collection of subsets of V that is closed
under inclusion. The elements of ∆ are called faces, and the dimension of a face F is
dim F := |F | − 1. The 0-dimensional faces in ∆ are called vertices, and the maximal
faces under inclusion are called facets. We say that ∆ is pure if all facets have the same
dimension. The dimension of the complex ∆ is dim ∆ := max{dim F : F ∈ ∆}. For the
remainder of this paper, unless stated otherwise we will assume that a simplicial complex
∆ is pure of dimension d − 1 with vertex set V .
The link of a face F is the subcomplex of ∆ defined by
lk∆ F = {G ∈ ∆ : F ∪ G ∈ ∆, F ∩ G = ∅},
3

and the contrastar of a face F is defined by
cost∆ F = {G ∈ ∆ : F 6⊂ G}.
In the case that F = {v} is a vertex, we write lk∆ v and cost∆ v instead of lk∆ {v} and
cost∆ {v}, respectively. If W ⊂ V , then the induced subcomplex ∆W is the simplicial
complex {F ∈ ∆ : F ⊂ W }. For 0 ≤ i ≤ d − 1, the complex ∆(i) := {F ∈ ∆ : |F | ≤ i + 1}
is the i-skeleton of ∆.
Given a field k, denote by Hi (∆) and H i (∆) the ith (simplicial) homology and cohomology groups of ∆ computed over k, respectively (definitions and further resources may
be found in [Hat02]). If F is a face of ∆, we denote by HFi (∆) the relative (simplicial)
cohomology group H i (∆, cost∆ F ) (in line with other conventions, we write Hvi (∆) for
i
H{v}
(∆)). It will often be helpful to identify HFi (∆) with H̃ i−|F |(lk∆ F ) (see, e.g., [Grä84,
Section 1.3]); in particular, note that H∅i (∆) = H̃ i (∆), the reduced cohomology group of
∆. Finally, let
ιiF : HFi (∆) → H∅i (∆)
be the map induced by the inclusion (∆, ∅) → (∆, cost F ).
If H∅0 (∆) = 0, then we call ∆ connected. We say that a face F of ∆ is singular if
HFi (∆) 6= 0 for some i < d − 1. Conversely, if HFi (∆) = 0 for all i < d − 1, we call F a
nonsingular face. We call ∆ Cohen-Macaulay over k if every face of ∆ (including ∅)
is nonsingular, and we call ∆ Buchsbaum (over k) if it is pure and every face aside from
∅ is nonsingular.
If ∆ contains a singular face, then we define the singularity dimension of ∆ to be
max{dim F : F ∈ ∆ and F is singular}. If the singularity dimension of ∆ is 0, then we say
that ∆ has isolated singularities. Such complexes will be our main objects of study. As
a special case, if the images of the maps ιiv : Hvi (∆) → H∅i (∆) are linearly independent as
vector subspaces of H∅i (∆), then we call the singularities of ∆ homologically isolated.
Equivalently, the singularities of ∆ are homologically isolated if the kernel of the sum of
maps
!
X
M
ιiv :
Hvi (∆) → H∅i (∆)
v∈V

decomposes as the direct sum
M
v∈V

v∈V


Kerιiv : Hvi (∆) → H∅i (∆) .

Lastly, we call the singularities of ∆ generically isolated if for sufficiently generic choices
of coefficients {αv : v ∈ V } and {γv : v ∈ V }, the two maps θα and θγ defined by
!
!
X
M
X
M
αv ιiv :
Hvi (∆) → H∅i (∆)
and
γv ιiv :
Hvi (∆) → H∅i (∆),
v∈V

v∈V

v∈V

4

v∈V

respectively, satisfy
(Kerθα ) ∩ (Kerθγ ) =

M

Kerιiv : Hvi (∆) → H∅i (∆)

v∈V



for all i.

2.2

The connection to algebra

For the remained of the paper, let k be a fixed infinite field. Define A := k[xv : v ∈ V ]
and let m = (xv : v ∈ V ) be the irrelevant ideal of A. If F ⊂ V , let
Y
xF =
xv
v∈F

and define the Stanley-Reisner ideal I∆ by
I∆ = (xF : F 6∈ ∆).
The ring k[∆] := A/I∆ is the Stanley-Reisner ring of ∆. We will usually consider k[∆]
as an A-module that is graded with respect to Z by degree.
Given any Z-graded A-module M of Krull dimension d, we denote by Mj the collection
of homogeneous elements of M of degree j. A sequence Θ = (θ1 , θ2 , . . . , θd ) of linear forms
in A is called a homogeneous system of parameters (or h.s.o.p.) for M if each θi is
a homogeneous element of A and M/ΘM is a finite-dimensional vector space over k. In
the case that each θi is linear, we call Θ a linear system of parameters (or l.s.o.p.)
for M. If
(θ1 , . . . , θi−1 )M :M θi = (θ1 , . . . , θi−1 )M :M m
for 1 ≤ i ≤ d for any choice of Θ, then we call M Buchsbaum. The reasoning behind the
geometric definition of a Buchsbaum simplicial complex is made apparent in the following
theorem due to Schenzel ([Sch81]):
Theorem 2.1. A pure simplicial complex ∆ is Buchsbaum over
Buchsbaum A-module.

k if and only if k[∆] is a

Two collections of objects that are most vital to our results are the A-modules
Ext iA (k, k[∆]) and Hmi (k[∆]). An excellent resource for their construction and basic
properties is [ILL+ 07]. In the case of Stanley-Reisner rings, the Z-graded structure of
Ext iA (k, k[∆]) is provided by [Mi89, Theorem 1]:
Theorem 2.2 (Miyazaki). Let ∆ be a simplicial complex. Then as vector spaces over k,

0
j < −1 or j > 0


 i−1
H∅ (∆)
j=0
M
Ext iA (k, k[∆])j ∼
=
i−2

H∅ (cost∆ v)
j = −1.


v∈V

5

The local cohomology modules Hmi (k[∆]) may be computed as the direct limit of the
Ext iA (A/ml , k[∆]) modules (see [Mi89, Corollary 1]). However, we will also need a thorough
understanding of the A-module structure of the local cohomology modules of k[∆]. The
necessary details are provided by [Grä84, Theorem 2]:
Theorem 2.3 (Gräbe). Let ∆ be a (d − 1)-dimensional
singularities and let 0 ≤ i ≤ d. Then

0


 i−1
H
∅ (∆)
M
Hmi (k[∆])j ∼
=

Hvi−1 (∆)



simplicial complex with isolated
j>0
j=0
j<0

v∈V

as k-vector spaces. If αv ∈ Hvi−1(∆), then the A-module structure on Hmi (k[∆]) is given
by
·xu : Hmi (k[∆])j−1 −−−−→ 
Hmi (k[∆])j
αv
j < 0 and u = v

i−1
ι
(α
)
j=0
αv
7−−−−→
v
 v
0
otherwise.

It is immediate from Theorem 2.3 that a pure simplicial complex ∆ is Buchsbaum if
and only if it is pure and Hmi (k[∆]) is concentrated in degree 0 for all i 6= d. We note
at this point that an A-module M is called quasi-Buchsbaum if m · Hmi (M) = 0 for
all i. Evidently, the above theorem implies that k[∆] is Buchsbaum if and only if it is
quasi-Buchsbaum; this is certainly not the case for general modules, as we will see!
Elements and homomorphisms related to Hmi (k[∆]) will usually be represented and
referenced according to the topological identifications above; these identifications will also
be expanded as we progress. As a motivating example, consider the k-vector space
M

i−1
i−1
K i :=
Kerιi−1
:
H
(∆)
→
H
(∆)
.
v
v
∅
v∈V

This space is central to the definition of homological isolation of singularities. Sometimes,
it will be easier to identify K i with its counterpart in local cohomology. If we denote by
Hmi (k[∆])v the direct summand of Hmi (k[∆])−1 corresponding to Hvi−1 (∆) in Theorem 2.3,
then K i is identified with the submodule
M

Ker ·xv : Hmi (k[∆])v → Hmi (k[∆])
v∈V

of Hmi (k[∆]). In light of how intertwined these two objects are, we will use the notation
K i interchangeable between the two settings. In local cohomology, the equality
!
X
M
K i = Ker
·xv :
Hmi (k[∆])v → Hmi (k[∆])0
v∈V

v∈V

6

is equivalent to the singularities of ∆ being homologically isolated. In the same way, if
θ1 and θ2 are generic linear forms, then the generic isolation of the singularities of ∆ is
equivalent to the equality
!
!
M
M
Ker ·θ1 :
Hmi (k[∆])v → Hmi (k[∆])0 ∩Ker ·θ2 :
Hmi (k[∆])v → Hmi (k[∆])0 = K i
v∈V

v∈V

holding for all i.
Remark 2.4. In their original statements, the above structure theorems are written with
respect to a Z|V | -grading and are proved by examining the chain complex Hom A (K·l , k[∆]),
where K·l is the Koszul complex of A with respect to ml . When coarsening to a Z-grading,
this chain complex becomes much larger. However, an argument similar to Reisner’s
original proof that Hmi (k[∆])0 ∼
= H∅i−1 (∆) (see [Rei76], pp. 41-42) shows that the only
potentially non-acyclic components of Hom A (K·l , M) under a Z-grading are those also
appearing in the Z|V | -graded complex.

3

Auxiliary calculations

Unless stated otherwise, we will always assume that ∆ is a connected (d − 1)-dimensional
simplicial complex with isolated singularities. Let V be the vertex set of ∆ and set A :=
k[xv : v ∈ V ]. We will always consider R := k[∆] as an A-module. If θ1 , . . . , θd is a
homogeneous system of parameters for ∆, we denote Ri := k[∆]/(θ1 , . . . , θi )k[∆].
Since ∆ is connected, the 1-skeleton ∆(1) of ∆ is Cohen-Macaulay and the depth of
R is at least 2 (see [Hib91, Corollary 2.6]). Hence, there exists a homogeneous system of
parameters θ1 , . . . , θd for ∆ in which θ1 , θ2 are linear and form the beginning of a regular
sequence for R, i.e., θ1 is a non-zero-divisor on R and θ2 is a non-zero-divisor on R1 .
Generically, we may assume that θ1 and θ2 both have non-zero coefficients on all xv ’s.
Unless stated otherwise, we will always work with such a system of parameters for ∆.
Our results primarily depend upon an understanding of the A-modules Hmi (Rj ). We begin by computing their dimensions over k when j = 1 or 2 and discussing some connections
to the topology of ∆.

3.1

Local cohomology

Consider the exact sequence of A-modules
·θ

π

1
0 → R −→
R−
→ R1 → 0.

By looking at graded pieces of this sequence, there are exact sequences of vector spaces
over k of the form
·θ1
π
0 → Rj−1 −→
Rj −
→ Rj1 → 0.
(3.1)
7

These sequences induce the following long exact sequence in local cohomology, where
θ1i,j : Hmi (R)j−1 → Hmi (R)j is the map induced by multiplication, π is the map induced by
the projection R → R1 , and δ is the connecting homomorphism:
θ i,j

π

θ i+1,j

δ

1
1
−→ Hmi+1(R)j .
→ Hmi (R)j −
→ Hmi (R1 )j −
→ Hmi+1 (R)j−1 −−
Hmi (R)j−1 −−

(3.2)

In light of Theorem 2.3, we make the following conclusions. When j > 1, all terms are
zero. When j = 1, δ is an isomorphism. When j ≤ −1, each θ1i,j is an isomorphism (all
coefficients of θ1 are non-zero). When j = 0, we obtain the short exact sequence
0 → Coker θ1i,0 → Hmi (R1 )0 → Kerθ1i+1,0 → 0.

(3.3)

Hence, as k-vector spaces,



H∅i (∆)
j=1
Hmi (R1 )j ∼
Coker θ1i,0 ⊕ Kerθ1i+1,0
j=0
=

0
otherwise.

(3.4)

It will be useful to keep in mind that Coker θ1i,0 is identified with a quotient of H∅i−1 (∆)
and that Kerθ1i+1,0 is identified with a submodule of ⊕v∈V Hvi (∆). Although the short exact
sequence (3.3) above is not necessarily split, we will leverage the “geometric” A-module
structures of Coker θ1i,0 and Kerθ1i+1,0 along with (3.3) to further analyze Hmi (R1 ) in Section
4.
Now we repeat this argument; consider the short exact sequence
·θ

π

2
1
0 → Rj−1
−→
Rj1 −
→ Rj2 → 0

(3.5)

of vector spaces over k, giving rise to the long exact sequence
θ i,j

π

δ

θ i+1,j

2
2
−→ Hmi+1 (R1 )j .
→ Hmi (R1 )j −
→ Hmi (R2 )j −
→ Hmi+1 (R1 )j−1 −−
Hmi (R1 )j−1 −−

(3.6)

As in the previous computation, all terms are zero when j < 0 or j > 2, π is an
isomorphism when j = 0, and δ is an isomorphism when j = 2. When j = 1, we have the
exact sequence
0 → Coker θ2i,1 → Hmi (R2 )1 → Kerθ2i+1,1 → 0.
(3.7)
Hence, as vector spaces over k,
H∅i+1 (∆)
j=2
i,1
i+1,1
Coker
θ
⊕
Kerθ
j=1
2
2
Hmi (R2 )j ∼
=
i,0
i+1,0

Coker θ1 ⊕ Kerθ1
j=0


0
otherwise.





8

(3.8)

3.2

Local cohomology: suspensions

We will now briefly consider the special case of suspensions. Suppose ∆ is an arbitrary triangulation of the suspension of a (d−2)-dimensional manifold that is not Cohen-Macaulay,
with suspension points a and b (so that a and b are isolated singularities of ∆). In this
context, the maps ιia and ιib from Section 2 are isomorphisms. If g i is a generator for H∅i (∆),
denote gai := (ιia )−1 (g) ∈ Hai (∆) and gbi := (ιib )−1 (g) ∈ Hbi (∆). As usual, we will consider
these generators interchangeable with their corresponding elements in Hmi+1 (R).
P
Examining
the
sequence
in
(3.3)
for
this
special
case,
suppose
θ
=
1
v∈V xv and
P
i−1
i−1
θ2 =
c
x
with
c
=
6
c
and
c
,
c
=
6
0.
Given
g
a
generator
of
H
a
b
a b
v∈V v v
∅ (∆), the
i,0 i−1
i,0
i,0 i−1
i,0
i−1
map θ1 acts via θ1 (ga ) = θ1 (gb ) = g . Hence, θ1 is a surjection whose kernel is
generated as a direct sum by elements of the form (gai−1 − gbi−1 ). In particular, Hmi (R1 )0 ∼
=
i+1,0 ∼
i
Ker(θ1 ) = H∅ (∆). In summary,
 i
j=1
 H∅ (∆)
i
1
i
∼
H (∆)
j=0
Hm (R )j =
 ∅
0
otherwise,
under the aforementioned isomorphisms.
Now repeat the process above using the sequence in (3.7). If g i is a generator of H∅i (∆),
then θ2i+1,0 acts on Hmi+1 (R)−1 via θ2i+1,0 (gai ) = ca g i and θ2i+1,0 (gbi ) = cb g i . In particular,
identifying Hmi+1 (R1 )0 with the subspace Ker(θ1i+1,0 ) of Hmi+1 (R)1 , the induced action of
i+1,1
is
θ2i+1,1 is given by θ2i+1,1 (gai − gbi ) = (ca − cb )g i ∈ H∅i (∆) ∼
= Hmi+1(R1 )1 . That is, θ2
an isomorphism as long as ca 6= cb (note also that the singularities of ∆ are generically
isolated); this means that Hmi (R2 )1 = 0. In summary:
 i+1
j=2
 H∅ (∆)
i
2
i
∼
H∅ (∆)
j=0
(3.9)
Hm (R )j =

0
otherwise.

Since Hmi (R2 )1 = 0, is is immediate that Hmi (R2 ) is quasi-Buchsbaum for all i. The specific
choice of θ1 was made for the ease of calculation. For sufficiently generic choices of θ1 and
θ2 , the same isomorphisms hold. In particular, it is evident that the singularities of ∆ are
generically isolated.

4
4.1

Results
Buchsbaumness

We now move on to showing whether or not certain modules are Buchsbaum. For this,
the following theorem ([SV86, Theorem I.3.7]) is vital.
Theorem 4.1. Let k be an infinite field, with M a Noetherian graded A-module and
d := dim M > 0. Then M is a Buchsbaum module if and only if the natural maps
ϕiM : Ext iA (k, M) → Hmi (M) are surjective for i < d.
9

Thus far we know some limited information about Ext iA (k, Rj ) and Hmi (Rj ) in terms
of the simplicial cohomology of subcomplexes of ∆. Thankfully, Miyazaki has furthered
this correspondence with an explicit description of ϕiR in [Mi91, Corollary 4.5].
Theorem 4.2. The canonical map ϕiR : Ext iA (k, R) → Hmi (R) corresponds to the identity
map on H∅i−1 (∆) in degree zero and to the direct sum of maps
M

ϕiv : H∅i−2 (cost∆ v) → H∅i−2 (lk∆ v)
v∈V

induced by the inclusions in degree −1.

For our purposes, an alternate expression for ϕiR turns out to be even more powerful
than the one above. For some fixed v, consider the long exact sequence in simplicial
cohomology for the triple (∆, cost∆ v, ∅). In our notation, it is written as
ιi−1

δ

v
· · · → H∅i−2 (∆) → H∅i−2 (cost∆ v) −
→ Hvi−1 (∆) −−
→ H∅i−1 (∆) → · · · .

(4.1)

Under the isomorphism Hvi−1 (∆) ∼
= H∅i−2 (lk∆ v), a quick check shows that the connecting
homomorphism δ in this sequence is equivalent to ϕiv in the theorem above (this is also
made apparent in examining its proof). On the other hand, if we consider the cohomology
modules above as components of Hmi (R) as in Theorem 2.3, then the ιvi−1 map in this
sequence is the same as the “multiplication by xv ” map on Hmi (R)v . These equivalences
along with the exactness of (4.1) allow us to deduce the following proposition.
Proposition 4.3. The image of the H∅i−2 (cost∆ v) component of Ext iA (k, R)−1 under the
canonical map ϕiR : Ext iA (k, R) → Hmi (R) is precisely the kernel of ιi−1
: Hvi−1 (∆) →
v
i−1
H∅ (∆). In particular,
(Im ϕiR )−1 = K i
through the identifications
Im ϕiR



−1

=

M
v∈V

 M

Im ϕiv =
= K i.
Kerιi−1
v
v∈V

Note that if θ is any linear form then the proposition immediately implies that
(Im ϕiR )−1 ⊆ Kerθi,0 . Examining this containment more closely provides a characterization of the Buchsbaumness of R1 . Before stating this characterization, we note that one
commutative diagram in particular will be used repeatedly in proving many of our results.
Here we explain its origin.
Construction 4.4. The short exact sequence (3.1) induces the following commutative
diagram of vector spaces with exact rows:
Ext iA (k, R)j−1

θ1i,j

ϕiR

Hmi (R)j−1

Ext iA (k, R)j

π

Hmi (R)j

δ

ϕi

ϕiR
θ1i,j

Ext iA (k, R1 )j
Hmi (R1 )j

10

θ1i+1,j

ϕi+1
R

R1

π

Ext i+1
A (k, R)j−1

δ

Hmi+1 (R)j−1

Ext i+1
A (k, R)j
ϕi+1
R

θ1i+1,j

Hmi (R)j .

Since θ1 is the beginning of a regular sequence for R, it acts trivially on Ext iA (A/ml , R)
for all i and j (see, e.g., [HH11, p. 272]). By the commutativity of the rightmost square,
i+1
the image of Ext i+1
must lie in Kerθ1i+1,j . On the other hand, the
A (k, R)j−1 under ϕR
exactness of the bottom row tells us that π : Hmi (R)j → Hmi (R1 )j factors through the
projection Hmi (R)j → Coker θ1i,j . As this does not alter the commutativity of the diagram,
we can now alter it so that the top and bottom rows are both short exact sequences as
follows
0

Ext iA (k, R)j

Ext iA (k, R1 )j

Ext i+1
A (k, R)j−1
ϕi+1
R

ϕi

R1

0

Coker θ1i,j

0

Kerθ1i+1,j

Hmi (R1 )j

0,

where the left vertical map is the composition of ϕiR with the projection Hmi (R)j →
Coker θ1i,j . We can repeat this construction starting with the short exact sequence (3.5),
yielding the same diagram as above with R, R1 , and θ1 replaced by R1 , R2 , and θ2 ,
respectively.
Our first use of this construction will be in proving the following proposition (an alternate proof of the “if” direction also appears in [NS12, Lemma 4.3(2)]).
Proposition 4.5. If ∆ has isolated singularities, then R1 is Buchsbaum if and only if the
singularities of ∆ are homologically isolated.
Proof. Construction 4.4 provides the following diagram:
0

Ext iA (k, R)0

Ext iA (k, R1 )0

Ext i+1
A (k, R)−1
ϕi+1
R

ϕi

R1

0

Coker θ1i,0

0

Kerθ1i+1,0

Hmi (R1 )0

0.

By definition, if ∆ contains singularities that are not homologically isolated then there
exists some i such that K i+1 ( Kerθ1i+1,0 . By Proposition 4.3, this implies that the ϕi+1
R
map appearing in the diagram above is not a surjection. Since ϕiR : Ext iA (k, R)0 → Hmi (R)0
is an isomorphism, the left vertical map is always a surjection. Then the snake lemma
applied to this diagram shows that ϕiR1 is not a surjection, so R1 is not Buchsbaum by
Theorem 4.1.
Conversely, if the singularities of ∆ are homologically isolated, then Kerθ1i+1,0 = K i+1
for all i, so that the ϕi+1
R map in the diagram is always a surjection. The snake lemma now
i
shows that ϕR1 is a surjection in degree 0. In degree 1, we only need to raise the degrees
in the previous diagram by one. The diagram simplifies to
0

Ext iA (k, R1 )1

Ext i+1
A (k, R)0

ϕi

ϕi+1
R

R1

0

0

Hmi (R1 )1

Hmi+1 (R)0
11

0,

because Ext iA (k, R)1 = Coker θ1i,1 = 0. Since ϕi+1
is an isomorphism is degree 0, this
R
completes the proof.
We have now seen that spaces with homologically isolated singularities are “close” to
being Buchsbaum in that R1 is Buchsbaum. It is natural to ask whether descending to R2
could always provide a Buchsbaum module, even in the non-homologically-isolated case.
This is not true, as exhibited by the following proposition.
Proposition 4.6. Suppose ∆ is a space with non-homologically-isolated singularities and
that there exists i such that H∅i−1 (∆) = 0, while Kerθ1i+1,0 6= 0 and ιiv is injective for all v.
Then R2 is not Buchsbaum.
Proof. The hypotheses combined with the exact sequence in (4.1) show that H∅i−1 (cost v) =
i,0
i−1
0, so that Ext i+1
A (k, R)−1 = 0. Also, Coker θ1 = 0 because H∅ (∆) = 0. Then the
diagram of Construction 4.4 can be filled in as follows
0

Ext iA (k, R)0

Ext iA (k, R1 )0

0

0

Kerθ1i+1,0

0.

ϕi

R1

0

Hmi (R1 )0

0

This demonstrates that ϕiR1 is the zero map in degree 0. Now repeat this argument using R1
i+1
1
and R2 instead of R and R1 . In this case, Ext i+1
A (k, R )−1 = 0 because Ext A (k, R)−1 = 0.
Since Hmi (R1 )−1 = 0, the diagram provided is
Ext iA (k, R1 )0

∼

ϕi

ϕi

R1

Hmi (R1 )0

Ext iA (k, R2 )0
R2

∼

Hmi (R2 )0 ,

showing that ϕiR2 is the zero map in degree 0 as well. Since Hmi (R2 )0 6= 0, Theorem 4.1
completes the proof.
The hypotheses of this proposition may seem fairly restrictive, but that is not necessarily the case. In fact, choosing i to be the least i such that H∅i (∆) 6= 0 when ∆ is a
triangulation of the suspension of a manifold that is not a homology sphere will always do
the trick, yielding the following Corollary:
Corollary 4.7. If ∆ is the suspension of a Buchsbaum complex that is not Cohen-Macaulay,
then R2 is not Buchsbaum.
12

4.2

Almost Buchsbaumness

Although these R2 modules are not guaranteed to be Buchsbaum when ∆ has nonhomologically-isolated singularities, they are “close” to being Buchsbaum in some interesting ways and share some of the same properties. The examples above in which R2 is not
Buchsbaum fail the criterion of Theorem 4.1 in the degree 0 piece of Hmi (R2 ). Theorem
1.2 asserts that (in the generically isolated case) this is the only possible obstruction to
satisfying Theorem 4.1, and we now present its proof.
Proof of Theorem 1.2. : By the calculations in Section 3, we only need to verify surjectivity
in degrees 1 and 2. The last diagram in the proof of Proposition 4.5 holds regardless
of whether or not the singularities of ∆ are homologically isolated. Hence, ϕiR1 is an
isomorphism in degree 1. Construction 4.4 then induces the diagram
0

Ext iA (k, R2 )2

1
Ext i+1
A (k, R )1
ϕi+1
1

ϕi

R2

R

Hmi (R2 )2

0

0

Hmi+1 (R1 )1

0,

so that ϕiR2 is always an isomorphism in degree 2.
It remains to show that ϕiR2 is a surjection in degree 1. As usual, we have the following
diagram.
1
Ext i−1
A ( k, R ) 1

0

2
Ext i−1
A (k, R )1

Ext iA (k, R1 )0

0

Kerθ2i,1

0.

ϕi−1
2
R

Coker θ2i−1,1

0

Hmi−1 (R2 )1

According to the previous paragraph, the left vertical map must be a surjection. If we can
show that the right vertical map is a surjection as well, then the proof will be complete.
The right map is obtained by restricting the range of ϕiR1 to the subspace Kerθ2i,1 of
Hmi (R1 )0 . Note that the failure of ϕiR1 to be a surjection in this degree is precisely what
made R1 fail to be Buchsbaum in the non-homologically-isolated case.
Now consider a larger commutative diagram, all of whose rows are exact. All vertical
maps are those induced by the action of θ2 , and all maps from the back “panel” to the
front are induced by the canonical maps ϕiRj .
Ext iA (k, R)0

0
Coker θ1i,0

0

0

Ext i+1
A (k, R)−1
Kerθ1i+1,0

Hmi (R1 )0
Ext iA (k, R)1

0

0

Ext iA (k, R1 )0

Ext iA (k, R1 )1
Hmi (R1 )1

0
Ext i+1
A (k, R)0

Hmi+1 (R)0

13

0

0

0.

Now consider applying the snake lemma to both the front panel and the back panel. Note
that the vertical maps on the back panel are all identically zero, since θ2 acts trivially on
all of the modules there. Denote by τ the restriction of θ2i+1,0 to Kerθ1i+1,0 , appearing as
the right vertical map in the front panel of the diagram. By the naturality of the sequence
induced by the snake lemma, we obtain maps from the “top” panel as follows:
0

Ext iA (k, R)0

Ext iA (k, R1 )0

Ext i+1
A (k, R)−1

0

0

Coker θ1i,0

Kerθ2i,0

Kerτ

0.

Since the left vertical map is a surjection, we will be done if we can show that the right
vertical map is a surjection. However, Kerτ is simply
(Kerθ1i+1,0 ) ∩ (Kerθ2i+1,2 ) := Li+1 .
Note that the singularities of ∆ are generically isolated if and only if Li+1 = K i+1 . On
the other hand, K i+1 is precisely the image of Ext iA (k, R)−1 under ϕiR , completing the
proof.
The intersection Li+1 above is also central to the proof of Theorem 1.1, which we now
present as well.
Proof of Theorem 1.1. Once more, since Hmi (R2 ) may only have non-zero components in
the graded degrees 0, 1, and 2, we only need to check that m acts trivially on the degree 0
and degree 1 parts. We begin in degree 1. Let αv , βv , and γv all denote the map induced
by multiplication by xv in the diagram below.
0

Coker θ2i,1
αv

0

0

Hmi (R2 )1
βv

Hmi (R2 )2

Kerθ2i+1,1

0

γv

Hmi+1(R1 )1

0.

The snake lemma provides the exact sequence
0 → Coker θ2i,1 → Kerβv → Kerγv → 0.
Comparing this to the top row of the previous diagram, if we can show that Kerγv is all
of Kerθ2i+1,1 , then we may conclude that Kerβv is all of Hmi (R2 )1 . Note that Kerθ2i+1,1 is
a submodule of Hmi+1(R1 )0 ; to study this submodule, consider the following diagram with
exact rows.
0

Coker θ1i+1,0

Hmi+1 (R1 )0
θ2i+1,1

θ2i+1,1

0

0

Hmi+1 (R1 )1
14

Kerθ1i+2,0

0

τ

Hmi+2 (R)0

0.

As in the previous proof, the rightmost vertical map τ is the restriction of θ2i+2,0 to
Kerθ1i+2,0 . Once more, note that
Kerτ = (Kerθ1i+2,0 ) ∩ (Kerθ2i+2,2 ) := Li+2 .
Through another application of the snake lemma, we get a short exact sequence fitting
into the top row of the following diagram
Coker θ1i+1,0

0

Kerθ2i+1,1
Hmi+1 (R1 )1

0

0

·xv

·xv

·xv

0

Li+2
Hmi+2 (R)0

0.

So, it now remains to show that the rightmost map is zero. But xv acts trivially on Li+2
for all i and for all v if and only if
Li+2 = K i+2 ,
i.e., if and only if the singularities of ∆ are generically isolated.
Now we show that m · Hmi (R2 )0 = 0, independent of the type of isolation of the singularities of ∆. Consider the diagram below:
Hmi (R1 )0

0

Hmi (R2 )0

·xv

Hmi (R1 )0

θ2i,1

0

·xv

Hmi (R1 )1

Hmi (R2 )1

Hmi+1 (R1 )0 .

From the exactness of the rows of this diagram, we can conclude that xv · Hmi (R2 )0 = 0 if
xv · Hmi (R1 )0 ⊆ θ2 · Hmi (R1 )0 .
Generically, all coefficients of θ2 are non-zero. Combining this with the structure of Hmi (R)
outlined in Theorem 2.3, it is immediate that
xv · Hmi (R)−1 ⊆ θ2 · Hmi (R)−1 .
The following diagram now completes the proof.
0

Hmi (R1 )1

δ

Hmi+1 (R)0

·xv

Hmi (R)0

·xv

Hmi (R1 )0

δ

Hmi+1 (R)−1

·θ2

0

0

Hmi (R1 )1

Hmi+1 (R)0

·θ2
δ

Hmi+1 (R)0

15

0

Example 4.8. Combining Theorem 1.1 with Corollary 4.7 provides an infinite family of
interesting examples of rings with some prescribed properties that are quasi-Buchsbaum
but not Buchsbaum. In particular, let M be a d-dimensional manifold that is not a
homology sphere and let Γ be an arbitrary triangulation of M. Suppose further that
max{i : Γ(i) is Cohen-Macaulay} = r + 1.
Equivalently, the depth of k[Γ] is r +1. If we set Γ′ to be the join of Γ with two points, then
Γ′ is a triangulation of the suspension of M and the depth of k[Γ′ ] is r + 2. Since the depth
of the Stanley–Reisner ring is a topological invariant ([Mun84, Theorem 3.1]), if ∆ is an
arbitrary triangulation of the suspension of M then R2 has depth r. Since the singularities
of ∆ are generically isolated, R2 is a quasi-Buchsbaum ring of Krull dimension d and depth
r that is not Buchsbaum; furthermore, the canonical maps ϕiR2 : Ext iA (k, R2 ) → Hmi (R2 )
are surjections in all degrees except 0.

4.3

Another surjection

By [SV86, Proposition I.3.4], the quasi-Buchsbaum property of some R2 is equivalent to
the fact that every homogeneous system of parameters of R2 contained in m2 is a weakly
regular sequence. In light of the typical definition of the Buchsbaum property in terms
of l.s.o.p.’s being weakly regular sequences (see [SV86])) along with the characterization
appearing in Theorem 4.1 by surjectivity of the maps ϕiM : Ext iA (k, M) → Hmi (M), what
happens when we consider the natural maps Ext iA (A/m2 , R2 ) instead? Our next result
establishes another measure of the gap between the structure of R2 and the Buchsbaum
property:
Proposition 4.9. Suppose ∆ has isolated singularities. Then the canonical maps ψRi 2 :
Ext iA (A/m2 , R2 ) → Hmi (R2 ) are surjective.
Proof. Once more, surjectivity needs only to be demonstrated in degrees 0, 1, and 2. We
will begin with the degree 2 piece. The exact sequence (3.1) with j = 1 and l = 2 gives
rise to the commutative diagram below, where the horizontal maps are isomorphisms.
Ext iA (A/m2 , R1 )1

δ

ψi

i+1
ψR

R1

Hmi (R1 )1

2
Ext i+1
A (A/m , R)0

δ

Hmi+1 (R)0

2
i+1
By [Mi91, Corollary 4.5], the map ψRi+1 : Ext i+1
A (A/m , R)0 → Hm (R)0 is equivalent to
i+1
i
the identity map on H∅ (∆). Hence, ψR1 is an isomorphism in degree 1. By the same
argument, the exact sequence (3.5) and the corresponding commutative diagram show
that ψRi 2 : Ext iA (A/m2 , R2 )2 → Hmi (R2 )2 is an isomorphism.

16

The arguments for further graded pieces are similar. First consider the following diagram with exact rows again induced by (3.1).
0

Ext iA (A/m2 , R)0

2
Ext i+1
A (A/m , R)−1

Ext iA (A/m2 , R1 )0
ψi

i
ψR

i+1
ψR

R1

Hmi (R1 )0

Hmi (R)0

0

Hmi+1 (R)−1

By [Mi89, Corollary 2], the left and right vertical maps are isomorphisms. Exactness then
implies that the middle vertical map is surjective. So, ψRi 1 is an isomorphism in degree 1
and a surjection in degree 0. For the surjectivity of ψRi 2 in degree 1, consider the following
commutative diagram induced by (3.5) with exact rows.
0

Ext iA (A/m2 , R1 )1

2
1
Ext i+1
A (A/m , R )0

Ext iA (A/m2 , R2 )1

ψi

ψi+1
1

ψi

R1

R2

Hmi (R1 )1

0

R

Hmi (R2 )1

Hmi+1 (R1 )0

We have just demonstrated that the left and right vertical maps are at least surjections.
Again by exactness, we now have that ψRi 2 is a surjection in degree 1. Now consider one
final commutative diagram.
Ext iA (A/m2 , R1 )0

Ext iA (A/m2 , R2 )0

ψi

ψi

R1

R2

Hmi (R1 )0

Hmi (R2 )0

From Section 3, we know that the bottom map is an isomorphism. Since ψRi 1 is a surjection
in degree zero, ψRi 2 must be as well.

5

An enumerative theorem

Although R2 is not Buchsbaum, the quasi-Buchsbaum property does allow for a computation of the Hilbert series of the generic Artinian reduction of R by a h.s.o.p. of a particular
type. Let ∆ have generically isolated singularities and say Θ = θ1 , . . . , θd is a homogeneous
system of parameters for ∆ such that θ1 , θ2 is a linear regular sequence, while θ3 , . . . , θd
are quadratic forms. For 2 ≤ i ≤ d − 1, there are exact sequences
·θi+1

i
−−−
→ Rji → Ri+1 → 0.
0 → (0 :Ri θi+1 )j−2 → Rj−2

Since R2 is quasi-Buchsbaum, the sequence θ3 , . . . , θd is a weakly regular sequence by
[SV86, Proposition I.2.1(ii)]. Furthermore, the proof of the proposition shows that (0 :R2
17

θ3 ) = Hm0 (R2 ). On the other hand, [Suz87, Theorem 3.6] states that Ri is quasi-Buchsbaum
for 2 ≤ i ≤ d − 1. Hence, the sequence above can be re-written as
i
0 → Hm0 (Ri )j−2 → Rj−2
→ Rji → Ri+1 → 0.

for 2 ≤ i ≤ d − 1. If Hilb(M; t) denotes the Hilbert series of a Z-graded A-module M,
then these exact sequences imply
Hilb(Ri+1 ; t) = (1 − t2 ) Hilb(Ri ; t) + t2 Hilb(Hm0 (Ri ); t).
A standard calculation then shows
d

Hilb(R ; t) = (1 + t)

d−2

d

(1 − t) Hilb(R; t) +

d−1
X

i=2


t2 (1 − t2 )d−1−i Hilb(Hm0 (Ri ); t) . (5.1)

P
The first term reduces to (1 + t)d−2 di=0 hi (∆)ti , following [Sta75]. To analyze the sum,
[Suz87, Lemma 3.5] provides the exact sequence
0 → Hmj (Ri )k → Hmj (Ri+1 )k → Hmj+1(Ri )k−2 → 0
for 2 ≤ i ≤ d − 2 and 0 ≤ j ≤ d − i − 2. So, as vectors spaces over k, for 2 ≤ i ≤ d − 1
there are isomorphisms


i−2
M

M j 2
Hm (R )−2j  .
Hm0 (Ri )j ∼
=

j=0
(i−2
j )
That is,

Hilb(Hm0 (Ri ); t)


i−2 
X
i − 2 2j
t Hilb(Hmj (R2 ); t).
=
j
j=0

(5.2)

Now define

and


µi = dimk Hmi (R2 )0 = dimk Coker θ1i,0 ⊕ Kerθ1i+1,0 ,

ν i = dimk Hmi (R2 )1 = dimk Coker θ2i,1 ⊕ Kerθ2i+1,1 ,
β∅i (∆) = dimk H∅i (∆),

so
Hilb(Hmi (R2 ); t) = µi + ν i t + β∅i+1(∆)t2 .
Combining this equality with equations (5.1) and (5.2) implies the following theorem describing Hilb(Rd ; t).
18

Theorem 5.1. If q = 2p is even, then
dimk (Rqd )



X


p−1 
q 
X
pβ∅k (∆)
d−2
k
k
p−1 d − 2
.
(−1) µ +
hi (∆) + (−1)
=
d−1−p
p
q−i
i=0
k=0

If q = 2p + 1 is odd, then
dimk (Rqd )

X


p−1
q 
X
d−2
p−1 d − 2
(−1)k ν k .
hi (∆) + (−1)
=
p
q
−
i
i=0
k=0

Remark 5.2. Note that µi is actually a topological invariant of ∆. If k∆k is the geometric
realization of ∆ and Σ is the set of isolated singularities of ∆, then µi = dimk H∅i−1 (k∆krΣ)
(see [NS12, Theorem 4.7]). At present there is no similar description for ν i , as it is not
clear how to trace the geometry of ∆ all the way through to Hmi (R2 )1 in such a precise
manner.

6

Comments

There are many possible abstractions of the results that have been presented. Perhaps the
most immediate consideration is in introducing singularities of dimension greater than 0.
In this case, the structure of Hmi (k[∆]) outlined in Theorem 2.3 becomes more involved
and hinders the calculations of Section 3. For instance, if ∆ contains singular faces even
of dimension 1, then θ1i,j will not, in general, be an isomorphism in degrees j < 0. This
implies that there is some i such that Hmi (k[∆]/θ1 k[∆])j is non-zero for infinitely many
values of j, i.e., k[∆]/θ1 k[∆] does not have finite local cohomology. In fact, Miller, Novik,
and Swartz classified when this is the case for quotients of k[∆] by arbitrarily many generic
linear forms:
Theorem 6.1. ([MNS11, Theorem 2.4]) A simplicial complex ∆ is of singularity dimension at most m − 1 if and only if k[∆]/(θ1 , . . . , θm )k[∆] has finite local cohomology.
In the case that m = 1, we know that k[∆]/θ1 k[∆] not only has finite local cohomology,
but it is also Buchsbaum if and only if the singularities of ∆ are homologically isolated.
So, one may pose the following question.
Question 6.2. If ∆ is of singularity dimension m−1, is there an analog of the homological
isolation property for singularities of arbitrary dimension classifying when k[∆]/(θ1 , . . . , θm )k[∆]
is Buchsbaum?
A possible property could be that all pairs of images of maps of the form H i (∆, cost∆ (F ∪
{u})) → H i (∆, cost∆ F ) and H i (∆, cost∆ (F ∪ {v})) → H i (∆, cost∆ F ) occupy linearly independent subspaces of H i (∆, cost∆ F ) for all faces F and all vertices u and v in the
appropriate dimensions.
19

On the other hand, when m = 1 we know that k[∆]/θ1 k[∆] has finite local cohomology
and that k[∆]/(θ1 , θ2 )k[∆] is quasi-Buchsbaum if and only if the singularities of ∆ are
generically isolated. The leads to our next question.
Question 6.3. If ∆ is a simplicial complex of singularity dimension m − 2 and k[∆] is
of depth at least m with θ1 , . . . , θm a regular sequence on k[∆], is there an analog of the
generic isolation property classifying when k[∆]/(θ1 , . . . , θm )k[∆] is quasi-Buchsbaum?
Again, one candidate property would be that given m + 1 generic linear forms, the
pairwise intersections
Kerθil,0 ∩ Kerθjl,0
are all trivially equal to K l .
Lastly, we have been able to provide many examples of complexes ∆ with isolated
singularities in which k[∆]/(θ1 , θ2 )k[∆] is quasi-Buchsbaum but not Buchsbaum. In light
of these examples, we present the following conjecture.
Conjecture 6.4. In the setting of Theorem 1.1, if the singularities of ∆ are not homologically isolated then k[∆]/(θ1 , θ2 )k[∆] is never Buchsbaum.

Acknowledgements
The author would like to thank Isabella Novik for proposing a problem that led to the
results in this paper. The author is also grateful to Satoshi Murai for pointing out an error
in an earlier version, leading to the notion of generic isolation of singularities.

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