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NONCOMMUTATIVE COMPLETE ISOLATED SINGULARITIES
BY
JUSTIN LYLE
A Thesis Submitted to the Graduate Faculty of
WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES
in Partial Fulfillment of the Requirements
for the Degree of
MASTER OF ARTS
Mathematics
May, 2015
Winston-Salem, North Carolina
Approved By:
Frank Moore, Ph.D., Advisor
Ellen Kirkman, Ph.D., Chair
Jeremy Rouse, Ph.D.
Acknowledgments
There are many people who helped to make this thesis possible. First and foremostis my advisor, Dr. Frank Moore. I would, of course, like to thank him for his time andeffort with this project, but more importantly, I wish to thank him for the camaraderieand encouragement he has shown me over the past year and a half. Half of what Iknow about abstract algebra I owe to him.
I would like to also thank Dr. Andy Conner who is responsible for the other half,as well as Dr. Ron Taylor and Dr. Eric McDowell who are responsible for getting meinterested in math in the first place.
Additionally, I would like to acknowledge the wonderful faculty at Wake Forest.In particular, I want to recognize Dr. Jason Parsley, Dr. Steve Robinson, and Dr.Sarah Raynor, who I had the privilege of taking courses with, as well as Dr. JeremyRouse and Dr. Ellen Kirkman who took the time to appear on my thesis committee.
I would like to extend the deepest of gratitude to my family and friends, especiallymy parents Tracey and Kim Lyle, whose support for me has been steadfast andundying throughout my entire life.
Finally, I would like to say a special thanks to my classmates, especially ElliottHollifield, Andrew Kobin, Elena Palesis, and Amelie Schreiber, for letting me bounceideas off them, often in an unsolicited manner, and for otherwise helping to make thelast two years a wonderful experience.
ii
Table of Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 3 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 The Commutative Case and Kleinian Singularities . . . . . . . . . . . 10
3.2 Graded Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 The Skew Group Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 4 Ascent to the Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1 Henselian Rings and Idempotent Lifting . . . . . . . . . . . . . . . . 22
4.2 The Ascent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 5 AR Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1 The Category of Gradeable Modules . . . . . . . . . . . . . . . . . . 35
5.2 AR Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Duality and The Functorial Isomorphism . . . . . . . . . . . . . . . . 40
5.4 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 41
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Appendix A Graded Rings and Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.1 Graded Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 Quotient Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Appendix B Inverse Systems and Ring Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.1 Inverse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.2 Inverse Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
B.3 Ring Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
iii
Abstract
Justin Lyle
Commutative local isolated singularities are a class of rings that have been studiedextensively. Much work has been devoted in the area of noncommutative algebrain generalizing the notion of an isolated singularity for graded rings. However, onemaintains a desire to directly adapt the notion of a Commutative local isolated singu-larity to the noncommutative case. We present some motivating theory in chapter 2building to the definition and some extended theory of graded isolated singularities inchapter 3. In chapter 4 we build theory around ring completions allowing us to passresults about a connected graded isolated singularity A, to it’s completion R = AmA
,which will necessarily be local. Finally, in chapter 5 we are able to use the ideasof previous chapters to prove the existence of Auslander-Reiten sequences for a wellchosen category of modules over R.
iv
Chapter 1: Introduction
In commutative algebra, one has access to geometric notions for several classes
of rings. In particular, regular, Gorestein and Cohen-Macaulay rings correspond to
interesting geometric phenomena. Since each these properties can be characterized by
homological algebra, it’s possible to generalize these classes to the noncommutative
setting. However, certain other classes of rings, while studied extensively in the
commutative case, are so entwined with their geometry that they have not be studied
with nearly the same intensity in the noncommutative realm. In particular, while
extensive theory has been developed for graded isolated singularities, very little theory
exists for rings that are not graded. Moreover, it is unclear that what theory exists
coincides, in any natural way, with the graded theory. As commutative Henselian
local isolated singularities have interesting properties in a category theoretic sense,
it is natural to desire a noncommutative generalization of this case. We are able
to extend the graded theory of isolated singularities, via the completion functor, to
certain noncommutative complete local rings. We have the following main results:
Theorem 1.1. Let A be a connected graded Noetherian k-algebra. Suppose A is a
graded isolated singularity with gldim(tailsA) = d and suppose there exists a graded
subring B ⊆ Z(A) ⊆ A so that A is module-finite over B. Then for all i > d,
ExtiA(M,N) is finite dimensional over k for any finitely generated graded A-modules
M and N .
We would like to use this result to claim that ExtiA(M,N) is complete with respect
to the A≥1-adic filtration. However, there is an obvious problem with this which is
that ExtiA(M,N) does not generally possess an A-module structure. The assumption
that A is module-finite over B allows us to circumvent this issue. We do this via the
1
following result:
Theorem 1.2. Suppose A is a connected graded Noetherian k-algebra with B ⊆
Z(A) ⊆ A a graded subring such that A is a module-finite B-algebra. Denote mA =
A>0 and mB = B>0. Let (−)mAdenote the completion with respect to the mA-adic
filtration, and let (−)mBdenote completion with respect to the mB-adic filtration. If
these functors are restricted to the category of finitely generated A-modules, then we
have (−)mA
∼= (−)mBas functors.
This result allows us to express our noncommutative completion as a commutative
completion, allowing us to access the well-developed theory of commutative comple-
tions. We derive a number of corollaries from this result to eventually obtain that,
for a connected k-algebra A that is a graded isolated singularity,
ExtiA(M,N) ∼= ExtiAmA
(MmA, NmA
)
as BmB-modules, for i > gldim tailsA and for finitely generated graded modules M
and N .
We are able to use this result to prove the existence of almost-split sequences
for Cohen-Macualay gradeable AmA-modules, which are finitely generated Cohen-
Macualay AmA-modules that lie in the image of the completion functor (−)mA
.
2
Chapter 2: Determinants
In studying linear algebra, one quickly recognizes the vast amount of information
captured by the determinant. One who studies linear algebra on a more abstract level
will further note the convenience, especially as a theoretical tool, of using a coordinate
free definition of the determinant. This definition is usually given in terms of exterior
algebras. If we have a field F, an F-vector space V of dimension d, and a group G
acting on V , then this induces a graded action on the exterior algebra
∧(V ) = ∧1(V )⊕ ∧2(V )⊕ · · · ⊕ ∧d(V ).
Since the action is graded, in particular, we have an action of G on ∧d(V ). But ∧d(V )
has dimension(dd
)= 1, so we may take a generator v of ∧d(V ). It follows that for any
g ∈ G, we have g · v = cv for some c ∈ F. We define c = det(g), and it can be proven
that this definition coincides with that of the usual matrix determinant obtained by
viewing G as a group of automorphisms of V and choosing a basis for V . While
certainly a large quantity of information can be gleaned from the determinant, it is
not always necessarily the “right” information on the level of algebras. An example
of this will be seen later in this chapter.
Given this, it is natural to seek out constructions similar to that of our wedge
product construction of the determinant. In order to pursue this idea, we first need
a few definitions.
Definition 2.1. We denote the graded Ext by Ext so that
ExtiA(M,N) =⊕m
ExtiGrMod(A)(M,N(m))
where Ext∗GrMod is the Ext∗ in the category of GrMod(A). If m = An≥1 is the graded
3
maximal ideal of A, we use write H∗m for the local cohomology functors, so that
H im(M) = lim
−→n
ExtiA(A/A≥n,M).
Definition 2.2. A connected graded algebra A is called AS Gorenstein (resp. AS-
regular) of dimenson d and of Gorenstein parameter l if
1. injdimAA = injdimAop A = d <∞ (resp. gldimA = d <∞), and
2. ExtiA(k,A) ∼= ExtiAop(k,A) ∼={k(l) if i = d,
0 if i 6= d
Definition 2.3. Suppose A is a connected graded k-algebra and let m = A>0. For a
graded left A-module M , we define
depthAM = inf RΓm(M) = inf{i | H im(M) 6= 0},
and
ldimA = supRΓm(M) = sup{i | H im(M) 6= 0}.
We say that M is (graded) Maximal Cohen-Macaulay (MCM) if depthAM =
ldimAM = depthAA <∞. If this statement holds when M = A and when M = Aop,
then we say that A is AS-Cohen-Macaulay. If further A admits a balanced dual-
izing complex D in the notion of [Yek92], then we say that A is Balanced Cohen-
Macaulay if D ∼= ωA[d] for some A-A bimodule ωA. In this case we call ωA the
canonical module for A.
These definitions should be thought of as generalizations of the notions Regular,
Gorenstein, and Cohen Macaulay rings from commutative algebra. The AS definitions
coincide with their AS-less versions in the commutative case, and moreover, one has
the familiar chain of implications
4
AS-Regular ⇒ AS-Gorenstein ⇒ AS-Cohen-Macaulay.
With these notions in hand, we may define what is known as the homological
determinant. The homological determinant defines a group homomorphism from the
group of graded automorphisms of a connected graded k-algebra to the multiplicative
group of the field k×. While there exist more general constructions of the homological
determinant, as long as our k-algebra of interest is Noetherian and AS-Gorenstein, we
may define the homological determinant in a manner reminiscent of our “basis-free”
definition of the determinant. Indeed, let A be a Noetherian AS-Gorenstein algebra
of dimension d and of Gorenstein parameter l, let f be a k-linear homomorphism
from a left A-module M to a left A-module N , and let g be a graded automorphism
of A. We define gN to have the same abelian group structure as N but with A action
given by a · n = g(a)n. We say f is a g-linear map if it gives a homomorphism of
A-modules, F : M → gN , i.e., if f(am) = g(a)f(m) for all a ∈ A, m ∈M . Note that
g : A→ A is itself g linear, and of course if f : M →M is g-linear, then f : M → gN
is A-linear. So if we have M → E• an injective resolution, then f lifts to an A-linear
chain map E• → gE•, i.e., to a g-linear chain map E• → E•. That is, g-linear maps
lift to g-linear chain maps, and so a g-linear map f : M → M induces g-linear maps
on local cohomology,
H im(f) : H i
m(M)→ H im(M).
In particular, g : A→ A induces a map
Hdm(g) : Hd
m(A)→ Hdm(A).
By Lemma 2.1 [JZ00], Hdm∼=A A∗(l) where ∗ is the graded vector space dual. By
Lemma 2.2 [JZ00], Hdm(g) = c(g−1)∗ for some c ∈ k×, and we define the homological
determinant of g, to be hdet g = c−1.
As it turns out the homological determinant, much like the determinant, captures
5
very powerful, albeit at times very different, information. However, unlike the deter-
minant, the homological determinant is generally very difficult to compute. However,
it can be computed in a nice way using the trace function, provided we impose ad-
ditional conditions on A. It follows from Lemma 2.6 [JZ00] and Theorem 4.2 [JZ00],
that these conditions are satisfied when A is AS-regular, and in fact this is the only
case with which we will concern ourselves.
Definition 2.4. We define the trace series of g ∈ GrAut(A) to be the series
TrA(g, t) =∞∑i=0
tr(g|Ai)ti ∈ k[[t]].
By Lemma 2.6 [JZ00], if A is an AS-regular algebra of dimension d, then for all
g ∈ Aut(A), we have TrA(g, t) is a rational function in t, and hence can be written as
a Laurent series in t−1. Moreover we have
TrA(g, t) = (−1)d(hdet g)−1t−l + lower terms
where l is the Gorenstein Parameter.
If g is the identity map, it’s not hard to see that TrA(g, t) is the Hilbert series of
A. As a more interesting example, we define the following.
Example 2.5. Let S = k−1[x, y], the skew polynomial ring in 2 variables, where k
is an algebraically closed field. It is classically known that S is AS-regular with
global dimension 2 and Gorenstein parameter 2. Given any diagonal or off-diagonal
g =
(a bc d
)∈ GL(2, k), we can define a graded action on S by first defining the
action on the variables x, y, identified with
(10
)and
(01
)respectively, via the linear
change of coordinates g
(st
)=
(a bc d
)(st
)when
(st
)is
(10
)or
(01
). We then
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extend this action multiplicatively to monomials, and then further extend to all of S
by linearity.
Suppose ζn is a primitive nth root of unity. Consider the following subgroups of
GL(2, k), Cn =
⟨(ζn 00 ζ−1
n
)⟩the cyclic group of order n, Dn =
⟨(ζn 00 ζ−1
n
),
(0 11 0
)⟩the dihedral group of order 2n, andDn =
⟨(ζ2n 00 ζ−1
2n
),
(0 1−1 0
)⟩the dicyclic group
of order 4n.
Since the homological determinant, as previously mentioned and as proven in
Proposition 2.5 [JZ00], is a group homomorphism, it suffices to determine the ho-
mological determinant of the generators of each of these groups. In total, it suffices
to consider only three elements, a =
(ζn 00 ζ−1
n
), b =
(0 11 0
), and g =
(0 1−1 0
).
Moreover, since the action is extended by linearity, it suffices to consider monomials.
So, take a monomial of degree d, xiyd−i. Observe that a · xiyd−i = ζ inxiζ−(d−i)n yd−i =
ζ2i−dn xiyd−1. Hence we see that tr(a|Sd) =
∑di=0 ζ
2i−dn and thus, Tr(a, t) =
∑∞d=0(
∑di=0 ζ
2i−dn )td.
With some work, it can be seen that this series represents the rational function
1
(1− ζnt)(1− ζ−1n t)
. We can rewrite this as1
t21
( ζnt− 1)( ζ
−1n
t− 1)
=1
t21
(1− ζnt
)
1
(1− ζ−1n
t).
We can then represent this expression as a Laurent series in t−1. We have
Tr(a, t) =1
t2(∞∑i=1
(ζnt
)i)(∞∑j=1
(ζ−1n
t)j).
Observe that the coefficient of t−2 in this expression is 1. Hence we have 1 =
(−1)2(hdet a)−1 = (hdet a)−1 so that hdet a = 1.
Observe that b · xiyd−i = yixj = (−1)i(d−i)xd−iyi. We see that the coefficient of
xiyd−i in a linear combination for (−1)i(d−i)xd−iyi is 0 unless i = d−i. But then 2i = d
7
and hence tr(gSd) = 0 unless d is even, and moreover, when d is even there is only
one monomial giving a nonzero term, so that tr(bSd) =
{0 if d is odd
(−1)i(d−i) if d is even. We
have tr(bSd) =
0 if d is odd
1 if d is even with d2
even
−1 if d is even with d2
odd
. Hence we see that
TrS(b, t) =∞∑i=0
(−1)it2i.
As this is a geometric series, we see that
∞∑i=0
(−1)it2i =1
1 + t2=
1
t2(1 + 1t2
)=
1
t21
(1 + 1t2
)
Expanding this as a Laurent Series gives,
1
t2
∞∑i=0
(−1)it−2i =∞∑i=0
(−1)it−2i−2
The coefficient of the t−2 term in this series is 1 so that (−1)2 hdet(b)−1 = 1 giving
that hdet b = 1.
Observe that g · xiyd−i = (−1)iyixd−i = (−1)i+ijxd−iyi = (−1)i(1+d−i)xd−iyi. We
see that the coefficient of xiyd−i in a linear combination for (−1)i(1+d−i)xd−iyi is 0
unless i = d− i. Just as before we have 2i = d and hence tr(gSd) = 0 unless d is even,
and moreover, when d is even there is only one monomial with giving a nonzero term,
so that tr(gSd) =
{0 if d is odd
(−1)d2
(1+ d2
) if d is even. Since one of d
2or 1 + d
2is even, we have
tr(gSd) =
{0 if d is odd
1 if d is even. Hence we see that
TrS(g, t) =∞∑i=0
t2i.
8
As this is a geometric series, we that
∞∑i=0
t2i =1
1− t2 = − 1
t2(1− 1t2
)= − 1
t21
(1− 1t2
)
Expanding this as a Laurent Series gives,
− 1
t2
∞∑i=0
t−2i =∞∑i=0
−t−2i−2
The coefficient of the t−2 term in this series is −1. Hence, (−1)2 hdet(g)−1 =
hdet(g)−1 = −1 and thus hdet g = −1.
One important takeaway from these examples is that the homological determinant
need not coincide with the usual determinant. For b defined in this example we have
hdet b = 1 6= −1 = det b and for g we have hdet g = −1 6= 1 = det g. However, it turns
out that if S is a commutative AS-regular algebra, then S is a polynomial ring,and if
this polynomial ring happens to be generated in degree 1, then GrAut(S) = GL(d, k)
and in fact the homological determinant does coincide with the usual determinant!
In particular, this implies that the subgroup HSL(S) = {g ∈ GrAut(S) | hdet g = 1}
of GrAut(S), coincides with SL(S) in the case where S is commutative AS-regular,
but that this does not necessarily hold in the noncommutative case.
9
Chapter 3: Isolated Singularities
3.1 The Commutative Case and Kleinian Singularities
The action defined in Example 1 bears a much deeper significance than might be
immediately apparent. In the commutative case, the finite subgroups of SL(2, k), for
an algebraically closed field k, act on the polynomial or power series ring in 2 variables
in such a way that their invariant subrings have “nice” properties in an algebraic
geometry sense. In particular these invariant subrings are isolated singularities, or
are described as having only an isolated singularity. In the commutative case we
define this condition by saying that a commutative local ring (R,m) is an isolated
singularity (or R has only an isolated singularity) if Rp are regular local rings for prime
ideals p of R which are distinct from m. The astute reader may already see where
this definition breaks down in the noncommutative case. In the noncommutative case
one cannot localize willy nilly. In the noncommutative case, we require that the given
multiplicatively closed set at which we wish to localize be a dominator set. As long as
your ring is a Noetherian domain, as will largely be the case for our purposes, it suffices
to consider Ore sets. However, this, very unfortunately, does very little to alleviate
our problem. If we do indeed wish to generalize the notion of isolated singularity to
the noncommutative case, we must find another definition; one preferably equivalent
to the usual commutative definition that doesn’t require localization. In fact, this is
quite doable, and there is a good bit of theory about this provided that your ring is
graded. However, as we will later see, this poses an issue all of its own. For the time
being we focus on motivating these ideas from the commutative case. In particular,
when our ring has dimension 2, we get a special class of rings known as the Kleinian
Singularities.
10
Our goal will be to consider singularities similar to the Kleinian singularities that
arise in the case of the usual polynomial or power series rings k[u, v] or k[[u, v]] over
an algebraically closed field k, but instead over the skew polynomial or skew power
series rings k−1[u, v] or k−1[[u, v]]. Let ζr denote a primitive rth root of unity in k.
In the commutative case, the Kleinian singularities can be completely described as
the invariant subrings of the following groups acting in the linear fashion described
in Example 1. Note that they are all subgroups of SL(2, k):
Cm: The cyclic group of order m for m ≥ 2, generated by
(ζm 00 ζ−1
m
)
Dm: The binary dihedral (dicylic) group of order 4m for m ≥ 1, generated by C2m
and (0 ii 0
)T : The binary tetrahedral group of order 48, generated by D2 and
1√2
(ζ8 ζ3
8
ζ8 ζ78
)
O: The binary octahedral group of order 48, generated by T and
(ζ3
8 00 ζ5
8
)
I: The binary icosahedral group of order 120, generated by
1√5
(ζ4
5 − ζ5 ζ25 − ζ3
5
ζ25 − ζ3
5 ζ5 − ζ45
)and
1√5
(ζ2
5 − ζ45 ζ4
5 − 11− ζ5 ζ3
5 − ζ5
).
So, we want to examine similar singularities arising from the action of these groups
on k−1[u, v] or k−1[[u, v]], for k algebraically closed with char(k) not dividing their
11
order. It is straightforward to see that Cn and Dn act on k−1[u, v] and k−1[[u, v]] via
the action described above. In fact in a similar manner from the fact that Cn and
Dn act on k[u, v] and k[[u, v]] in this manner. However, it can be seen that none of
the other groups listed above, nor any of their conjugates in SL(2, k), act on k−1[u, v]
or k−1[[u, v]], and so of these we focus our attentions solely on Cn and Dn. However,
it turns out that Dn as defined in Example 1 also acts. It is not considered in the
commutative case because it does not lie in SL(2, k), however, as well will see, this
is irrelevant in the noncommutative case.
As an example, we will compute SG in the cyclic case. Note that the following
computations are valid in both k−1[u, v] and k−1[[u, v]]. We begin with Cn. So let
G = Cn for n ≥ 2 and consider S = k[u, v] with k algebraically closed and char(k)
not dividing n. We seek a minimal set of generators for SG. Since Cn acts linearly on
S and since the only nontrivial relation in S is uv = −vu, it is clear that if a, b ∈ S
are such that g · (a + b) = a + b then g · a = a and g · b = b. Hence it suffices to
consider monomials uavb.
Since G is cyclic it suffices to consider the generator, call it σ. Suppose σ · uavb =
uavb. Then we have σ · uavb = ζanuaζ−bn vb = ζa−buavb = uavb. Hence ζa−b = 1 and
thus we see that a − b ≡ 0 mod n. These monomials are the monomials of the
form ur, usvs, vt where r, t ≡ 0 mod n. Hence, by definition of polynomial ring, we
see that SG = k[un, uv, vn]. Now we compute the relations in SG. So, let x = un,
y = uv, and z = vn. Observe that xy = unuv = un+1v = (−1)nuvun = (−1)nyx,
and so xy − (−1)nyx = 0. Similarly we see that yz − (−1)nzy = 0. Moreover,
xz = unvn = (−1)n2vnun = (−1)n
2zx, and so xz − (−1)n
2zx = 0. Finally, we have
yn = (uv)n = (−1)n(n−1)
2 unvn = (−1)n(n−1)
2 xz. Hence we see that xz − (−1)n(n−1)
2 yn =
12
0. All of this information is captured by the statement
SG = k[u, v]G = k[un, uv, vn] ∼= k(−1)n [x, y, z]
(xz − (−1)n(n−1)
2 yn).
These computations give us an explicit description of the the invariant subrings
SG, as quotients of skew polynomial or skew power series rings. We omit the com-
putations for Dn for reasons discussed later. Understanding the behavior of these
invariant subrings, and the MCM modules over them, will be the primary motivation
for the theory presented in this paper. In the commutative case, the Kleinian singu-
larities are all isolated singularities and it is this fact which allows us to understand
them, and so, in order to proceed we need a notion of isolated singularity. Since
k−1[x1, . . . , xn] is graded, we can use the notion of a graded isolated singularity.
3.2 Graded Isolated Singularities
Definition 3.1. Let A be a connected graded k-algebra. We denote by GrA the
category of graded right A-modules with degree zero A-module homomorphisms, and
by grA the full subcategory consisting of finitely generated graded right A-modules.
We denote by torsA the full subcategory of grA consisting of torsion modules, and
we let tailsA := grA/ torsA in notion of Appendix A. We usually denote by M the
image of M in tails(A). Now we can define the global dimension of tailsA to be
gldim(tailsA) := sup{i | ExtitailsA(M,N ) 6= 0 for some M,N ∈ tailsA}.
You should interpret tailsA as a noncommutative projective scheme associated
to A, a connection explored much more thoroughly in [AZ94]. Then we can define
a notion of graded isolated singularity by the smoothness of this noncommutative
projective scheme.
13
Definition 3.2. A Noetherian connected graded k-algebra A is called a graded isolated
singularity if tailsA has finite global dimension.
It’s not hard to see that if A has finite global dimension, then so does tails(A), in
particular, if A is AS-regular, then A is a graded isolated singularity. It can be proven
that this condition is equivalent to the usual notion of isolated singularity, provided
of course that A is commutative.
In the commutative case one obtains a nice result about isolated singularities.
Theorem 3.3. Let R be a commutative isolated singularity. Then there exists J ∈ N
such that for all R-modules M ,N we have ExtiR(M,N) is finite dimensional over kR
for all i ≥ J .
We are able to directly generalize this under some additional hypotheses. We
begin with some lemmas.
Lemma 3.4. Suppose B is a connected graded noetherian ring. If M is a graded
finitely generated torsion B module, then M is finite dimensional over kB.
Proof. Since M is graded we have M =⊕
j∈NMj. Since M is finitely generated, we
may take m1,m2, . . . ,ml generators for M . Set di = degmi. Since M is torsion, for all
mi, there exists ni such that B≥nimi = 0. Let d = max{di} and n = max{ni}. Then,
for all i we have B≥nmi = 0. Now, take m ∈M . Then m = b1m1 + b2m2 + · · ·+ blml
where bi ∈ A. Since B is N-graded, it follows that M is left bounded. In particular,
M =⊕∞
j=min diMj. Since d is the maximum degree of a generator of M , it follows
that the maximum degree of any element of M is n+d−1, so that M =⊕n+d−1
min diMj.
By Artin-Zhang Proposition 2.1 (cite), A is locally finite and thus, since M is finitely
generated, we have that M is locally finite. Hence M is the direct sum of finitely
14
many finite dimensional vector spaces over kB, and is therefore a finite dimensional
vector space over kB.
Lemma 3.5. Suppose B ⊆ Z(A) ⊆ A is a graded subring. Suppose M is a finitely
generated left A-module and that N is a torsion left A-module. Then HomA(M,N)
is a torsion B-module.
Proof. To begin, take m1, . . . ,ml a generating set for M as a left A-module. Since
N is torsion as an A-module, and since B is a graded subring of A, it follows that N
is a torsion B module. Take f ∈ HomA(M,N). Consider f(m1), f(m2), . . . , f(ml).
Since N is torsion as a B-module we have, for any i, that there exists ni such that
B≥nif(mi) = 0. Let n = max{ni}, let b ∈ B≥n, and take m ∈ M . Consider bf . We
have bf(m) = bf(a1m1 + a2m2 + · · ·+ alml) for some a1, . . . , al ∈ A. Then,
bf(a1m1 + a2m2 + · · ·+ alml) = b(a1f(m1) + a2f(m2) + · · ·+ alf(ml))= ba1f(m1) + ba2f(m2) + · · ·+ balf(ml)= a1bf(m1) + a2bf(m2) + · · ·+ albf(ml)= 0 + 0 + · · ·+ 0= 0.
Hence bf = 0 so that B≥nf = 0. Ergo, HomA(M,N) is a torsion B-module.
Lemma 3.6. Suppose B ⊆ Z(A) ⊆ A is a graded subring with A module finite over
B. Let M be a left A-module. Then M is a finitely generated as a B-module if and
only if it is finitely generated as an A-module.
Proof. [⇒] It’s clear.
[⇐] Let m1, . . . ,ml be a generating set for M as an A-module. Let a1, . . . , ak be a
generating set for A as a B-module. Take m ∈M . Then there exist α1m1, . . . , αl ∈ A
15
such that m = α1 + · · · + αlml. But then for each i, there exists bi1 , . . . , bik ∈ B
such that αi = bi1a1 + · · · + bikak. Hence m =∑l
i=1 αimi =∑l
i=1(∑k
j=1 bijaj)mi =∑li=1
∑kj=1 bijajmi. Ergo, {ajmi}i=1,...,l,j=1,...,k is finite generating set for M as a B-
module.
Theorem 3.7. Suppose A is a connected graded Noetherian k-algebra such that
there exists a graded subring B ⊆ Z(A) ⊆ A with A module-finite over B. Then
ExtitailsA(M,N ) = 0 implies ExtiA(M,N) is finite dimensional over k for all i.
Proof. For this proof, we will denote by kB, the residue field of B, for clarity.
To begin, take a minimal injective resolution (of A modules) of N → E•,
0→ Nη−→ E0 d0−→ E1 d1−→ · · ·
Take i ∈ N. Consider τ(Ei), the set of all torsion elements of Ei. Then τ(Ei) is
a submodule of Ei since A is a Noetherian. Denote I i = E(τ(Ei)), the injective hull
of τ(Ei). Then I i is an essential extension of τ(Ei), and is thus torsion, as seen in
Appendix A. Since Ei is injective, we have I i ⊆ Ei and moreover, there exists Qi such
that Ei ∼= Qi ⊕ I i. It follows that Qi is torsion-free since Qi ∼= Ei/I i, and injective
as it is a direct summand of Ei. We examine the following diagram.
0 N I0 I1 · · ·
0 N Q0 ⊕ I0 Q1 ⊕ I1 · · ·
0 N Q0 Q1 · · ·
idN
η|N
i0
d0|I0
i1
d1|I1
idN
η
p0
d0
p1
d1
η d1 d2
16
Where ij and pj are the natural embedding and projection maps, and where di is
the natural induced map of quotients. It follows that I• and Q• are complexes and
it’s easy to see that this diagram commutes. This gives rise to a short exact sequence
of complexes,
0 I•(N) E•(N) Q•(N) 0
Now we apply HomA(M,−), noting that we maintain exactness of the columns of
this diagram since they are all split exact, and we consider the long exact sequence
in homology obtained by applying the zig-zag lemma,
0 H0(HomA(M, I•(N))) H0(HomA(M,E•(N))) H0(HomA(M,Q•(N)))
H1(HomA(M, I•(N))) H1(HomA(M,E•(N))) H1(HomA(M,Q•(N)))
H2(HomA(M, I•(N))) H2(HomA(M,E•(N))) H2(HomA(M,Q•(N)))
H i(HomA(M, I•(N))) H i(HomA(M,E•(N))) H i(HomA(M,Q•(N)))
α0
α1
Now, from Proposition 7.15 [AZ94], we have that for every i, ExtitailsA(M,N ) ∼=H i(HomA(M,Q•(N))). Thus, if ExtitailsA(M,N ) = 0, we have
im{H i(HomA(M, I•(N))))→ H i(HomA(M,E•(N)))} = H i(HomA(M,E•(N)))
by exactness. Hence by the first isomorphism theorem, we have
H i(HomA(M,E•(N))) ∼= H i(HomA(M, I•(N))))
ker{H i(HomA(M, I•(N))))→ H i(HomA(M,E•(N)))} ,
17
as B-modules. We have from Lemma 3.5 that HomA(M, I i) is a torsion B-module,
and hence H i(HomA(M, I•(N)))) is torsion since it is a subquotient of HomA(M, I i)
and subquotients of torsion modules are torsion. Moreover we have that
ker{H i(HomA(M, I•(N))))→ H i(HomA(M,E•(N)))}
is torsion, since submodules of torsion modules are torsion. Since quotients of tor-
sion modules are torsion, it follows that H i(HomA(M,E•(N))) is torsion. But,
ExtiA(M,N) = H i(HomA(M,E•(N))) by definition. Hence ExtiA(M,N) is a torsion
B-module. Since M and N are finitely generated with B commutative Noetherian
ExtiA(M,N) is finitely generated as a B module. Hence, from Lemma 3.4 we have
that ExtiA(M,N) is finite dimensional over kB. But, since B is a graded subring of
A, we have that kB ⊆ kA (in fact since A is module finite over B, kA is a finite field
extension of kB). Hence ExtiA(M,N) is finite dimensional over kA.
Corollary 3.8. If A is a graded isolated singularity with d = gldim(tailsA), then for
all i > d, ExtiA(M,N) is finite dimensional over k.
3.3 The Skew Group Ring
In keeping with our goal in studying noncommutative Kleinian singularities, before
we proceed in building theory for them around our definition of graded isolated sin-
gularity, we need to prove that the rings we have defined are in fact graded isolated
singularities. Otherwise our motivating example would be rather ill formed. In order
to prove that these rings are graded isolated singularities we need a definition.
Definition 3.9. Let S be a ring and G ⊆ Aut(S) a finite group of automorphisms
of S. let S#G denote the skew group ring of S and G. As an S module, S#G =
18
⊕σ∈G S#σ is free on the elements of G; the product of two elements s#σ and t#τ is
(s#σ)(t#τ) = sσ(t)#στ.
Thus moving σ past t “twists” the ring element.
We have the following result from Mori and Ueyama.
Theorem 3.10. (Mori and Ueyama) Let S be a noetherian AS-regular algebra over
k of dimension d ≥ 2 and G ≤ HSL(S) be a finite subgroup such that char k does not
divide |G|. Then the following are equivalent.
1. SG is a graded isolated singularity, and
Φ : S#G→ EndSG(S); s#g 7→ [t 7→ sg(t)]
is an isomorphism of graded algebras.
2. S#G/(e) is finite dimensional over k where e = 1|G|∑
g∈G 1#g ∈ S#G.
Example 3.11. If S = k−1[x, y] and if G = 〈A〉, where A =
(ζn 00 ζ−1
n
), acts linearly
on S as before, then S#G/(e) is finite dimensional over k, where e =1
|G|∑
g∈G 1#g ∈
S#G, and where ζn is a primitive nth root of unity.
Proof. Since G is cyclic, it’s irreducible representations are one dimensional and are
defined by mapping A to ζ in for some 0 ≤ i < n. Let χi be the character of the
representation defined by A 7→ ζ in. Now define ei =∑n−1
j=0 χi(A−j)#Aj noting that
e0 = e. Observe that for any i
(x#1)ei = (x#1)n−1∑j=0
χi(A−j)#Aj =
n−1∑j=0
(x#1)(χi(A−j)#Aj)
19
=n−1∑j=0
(x#1)((ζ in)−j#Aj) =n−1∑j=0
(x#1)(ζ−ijn #Aj) =n−1∑j=0
ζ−ijn x#Aj
=n−1∑j=0
ζ−ij−jn ζjnx#Aj =n−1∑j=0
ζ−ij−jn Ajx#Aj
=n−1∑j=0
((ζ i+1n )−j#Aj)(x#1) =
n−1∑j=0
(χi+1(A−j)#Aj)(x#1)
= (n−1∑j=0
χi+1(A−j)#Aj)(1#x) = ei+1(x#1).
Moreover, we also have, for any i,
ei(y#1) = (n−1∑j=0
χi(A−j)#Aj)(y#1) =
n−1∑j=0
(χi(A−j)#Aj)(y#1) =
n−1∑j=0
((ζ in)−j#Aj)(y#1)
n−1∑j=0
(ζ−ijn #Aj)(y#1) =n−1∑j=0
(ζ−ijn Aj(y)#Aj) =n−1∑j=0
(ζ−ijn ζ−jn y#Aj) =n−1∑j=0
(ζ−ij−jn y#Aj)
n−1∑j=0
(y#1)(ζ−ij−jn #Aj) = (y#1)n−1∑j=0
(ζ−ij−jn #Aj) = (y#1)n−1∑j=0
((ζ i+1n )−j#Aj)
= (y#1)n−1∑j=0
χi+1(A−j)#Aj) = (y#1)ei+1.
One may use these relations to write down a finite basis for S#G/(e) over k.
While this approach works well in the case when G is cyclic, it fails to be as
insightful for the dihedral or the dicyclic case. However, we can gain our desired
result by simply applying a theorem from Ueyama [Uey13].
20
Theorem 3.12 (Ueyama). Let A be an AS-regular algebra of dimension 2, and let
G be a finite subgroup of GrAutA such that hdetσ = 1 for all σ ∈ G. Then AG is a
graded isolated singularity.
From our computations in Example 1, this immediately gives us that SG is a
graded isolated singularity when S = k−1[x, y] and when G = Cn or Dn (with |G|
invertible in k). However, since there exist elements in Dn with homological deter-
minant −1, this theorem does not apply to Dn. In fact, it turns out that SG is not
a graded isolated singularity when G = Dn and so we now cease to consider it. This
is despite the fact that we get an isolated singularity for SG with G = Dn in the
case where S = k[x, y], as in this case we have SL(S) = HSL(S). Moreover for the
same reasons, Dn gives an isolated singularity in the noncommutative case, whereas
it does not in the commutative case. The examination of the noncommutative case
reveals to us that being in SL(S) is not so important with regard to the property of
being an isolated singularity. Here it is the homological determinant, rather than the
determinant, that captures the appropriate information.
21
Chapter 4: Ascent to the Completion
4.1 Henselian Rings and Idempotent Lifting
In the commutative case the property of S being an isolated singularity is related to
deep information about the category of graded MCM-modules over S, provided that
S satisfies another key property, that S be Henselian.
Definition 4.1. Let A be a local ring with maximal ideal m. We say that A is
Henselian if for every monic polynomial
F (x) = xn + an−1xn−1 + · · ·+ a1x+ a0 ∈ A[x]
such that F (x) = f1(x)f2(x) for some relatively prime monic polynomials f1(x), f2(x) ∈
k[x], there are monic polynomials F1(x), F2(x) ∈ A[x] such that F (x) = F1(x)F2(x)
with F1(x) = f1(x) and F2(x) = f2(x)
This property essentially says that a factorization in k[x] of the image of a poly-
nomial F ∈ A[x] (under the canonical quotient map) can be lifted to a factorization
in A[x]. Quite unfortunately, k−1[x1, x2, . . . , xn] does not satisfy this property and
neither do any of our desired quotients. However, there is one useful, albeit very
technical, tool we can use to get around this, ring completion. A detailed treatment
of ring completions can be found in Appendix B.
In the commutative case, complete local rings are Henselain, but it turns out that
this does not hold in general. However, as Aryapoor shows in [Ary09], if A is local
and almost commutative, i.e., if A has a commutative associated graded ring, then
complete does imply Henselian. Unfortunately, k−1[[x, y]] has k−1[x, y] as its associated
graded ring, so that this result does not apply. However, we are able to expand on
his result to cover our case.
22
Theorem 4.2. Let A be a local ring such that there exists a subring R ⊆ Z(A) with
R ∼= k. Then A complete (and separated) implies A Henselian.
Proof. Let F (x) = xn +an−1xn−1 + · · ·+a1x+a0 ∈ A[x] such that F (x) = f1(x)f2(x)
for relatively prime monic polynomials f1, f2 ∈ k[x]. We will inductively construct a
sequence of monic polynomials {F1,r} and {F2,1} in A[x] such that
F1,r(x) = f1(x), F2,r(x) = f2(x),
F1,r+1(x)− F1,r(x) ∈ mr[x], F2,r+1(x)− F2,r(x) ∈ mr[x],
and
F (x)− F1,r(x)F2,r(x) ∈ mr[x].
Since the canonical quotient map is surjective, there exist F1,1(x), F2,1(x) ∈ A[x]
such that F1,1(x) = f1(x) and F2,1(x) = f2(x). Now suppose we have defined F1,r(x)
and F2,r(x). Write
F1,r+1(x) = F1,r(x) +G1(x), F2,r+1(x) = F2,r +G2(x).
We see that finding our desired F1,r+1(x) and F2,r+1(x) is equivalent to findingG1(x), G2(x) ∈
mr[x] with deg(G1(x)) < deg(f1(x)), deg(G2(x)) < deg(f2(x)) and
F (x)− F1,rF2,r −G1(x)F2,r(x)− F1,r(x)G2(x) ∈ mr+1[x].
But this is the same as finding G1(x), G2(x) ∈ mr[x] with
deg(G1(x)) < deg(f1(x)), deg(G2(x)) < deg(f2(x))
and
[F (x)− F1,rF2,r −G1(x)F2,r(x)− F1,r(x)G2(x) = 0
23
in mr/mr+1. If we consider mr/mr+1 as a vector space over k, then, using the fact
that k is central, we can see that this is the same as finding G1(x) and G2(x) in mr[x]
such that deg(G1(x)) < deg(f1(x)), deg(G2(x)) < deg(f2(x)) and
(F (x)− F1,rF2,r)− f2(x)G1(x)− f1(x)G2(x) = 0
in mr+1[x]. However, this is possible, as f1(x) and f2(x) are relatively prime. Hence
we have our construction. Now since A is separated in the m-adic topology, we have⋂rm
r = {0} so that⋂rm
r[x] = {0}. Since A is m-adically complete, it follows
that our sequence converges giving us an F1(x), F2(x) such that F (x)−F1(x)F2(x) ∈⋂rm
r[x] = {0} with F1(x) = f1(x) and F2(x) = f2(x). Thus, F (x) − F1F2 = 0
implying F (x) = F1(x)F2(x) with F1(x) = f1(x) and F2(x) = f2(x). Therefore, A is
Henselian.
There are many equivalent definitions of Henselian rings in the commutative case;
Leuschke and Wiegand [LW12] alone present 5! Of particular interest to us, a com-
mutative ring B is Henselian if and only if for every module-finite B-algebra A, each
idempotent of A/J(A) lifts to an idempotent of A.
Given this characterization of commutative Henselian rings, we can prove the
following proposition, beginning, of course, with a lemma.
Lemma 4.3. Let B be a commutative ring and A a module-finite B-algebra. Then
AJ(B) ⊆ J(A).
Proof. Let f ∈ AJ(B). We want to show that for any λ ∈ A, we have 1 − λf is a
unit in A. That is, that A(1 − λf) = A. Clearly we have A(1 − λf) + AJ(B) = A,
and thus, we have A(1− λf) = A by Nakayama’s Lemma. Ergo, AJ(B) ⊆ J(A).
24
Proposition 4.4. Suppose (B,mB, kB) is a Henselian local Noetherian ring and that
(A,mA, kA) is a module-finite extension of B. Then for any finitely generated inde-
composable A-module M , we have that EndA(M) is local.
Proof. Note that E = EndA(M) is a module-finite B algebra. Since B is local, we
have that J(B) = mB. Hence, by the lemma, EmB ⊆ J(E). Hence we have a
projection map p : E/EmB → E/J(E). But E/EmB is clearly finite dimensional
over k so that im p = E/J(E) is finite dimensional over k. Hence E/J(E) is a
finitely generated Artinian B-algebra, hence semisimple. Since M is indecomposable,
E has no nontrivial idempotents. Since B is commutative Henselian, idempotents
in E/J(E) lift to idempotents in E, so that E/J(E) has no nontrivial idempotents.
By the Artin-Wedderburn theorem, we have that E/J(E) ∼=∏k
i=1Mni(Di) where
the Di are division rings. But then, the element (0, . . . , 0, Ini, 0, . . . , 0) is a nonzero
idempotent in∏k
i=1Mni(Di). Thus, (0, . . . , 0, Ini
, 0, . . . , 0) must be 1 in∏k
i=1Mni(Di).
Ergo, k = 1. Moreover, the matrix in Mn1(D1) consisting of a one in the top left corner
and 0’s elsewhere is a nonzero idempotent in Mn1(D1). Thus, this matrix must be
the identity, and so we must have n1 = 1. Ergo, E/J(E) is a division ring. Hence,
J(E) is maximal in E, so that E is local.
Corollary 4.5. Same hypothesis. Then KRS holds in the category of finitely gener-
ated A-modules.
4.2 The Ascent
We have seen that ring completion gives us a process by which we can naturally extend
a given ring, thereby forming a new ring which always satisfies properties we desire.
Since our graded isolated singularities don’t satisfy many of these key properties:
25
Henselian, Local etc., it is natural to attempt to extend these singularities via ring
completion. In some way, we hope to be able to pass along information about our
graded singularities via the completion functor. In this section, we prove that this is
in fact possible.
Theorem 4.6. Suppose A is a connected graded locally-finite k algebra so that A =⊕∞k=0Ak. Then let m denote the homogeneous maximal two-sided ideal A≥1. Let A
denote the completion of A with respect to the m-adic filtration. Then A ∼=∏∞
k=0Ak
as k-algebras (the ring structure is the convolution product coming from⊕∞
k=0Ak).
In particular, if A = k−1[x1, x2, . . . , xn], then A = k−1[[x1, x2, . . . , xn]].
Proof. As seen in Appendix B,
A ∼= {~a ∈∏k
A/mk | ai + mi = aj + mi for i ≥ j}.
We define maps φi :∏∞
k=0Ak → A/mi by φi(f) = f +mi. By the universal property
of inverse limits, there is a unique ring homomorphism θ :∏∞
k=0Ak → A defined
by θ(f) = (f + m, f + m2, . . . ). We define a map π : A → ∏∞k=0Ak by sending
(f1 + m, f2 + m2, . . . ) to f1 + (f2 − f1) + (f3 − f2) + · · · . This map lands in∏∞
k=0Ak
as fi+1 − fi ∈ mi+1, hence having degree at least i + 1, and one may check that it is
independent of choice of representatives fi. Moreover, it is obviously an inverse for θ
so that θ is an isomorphism.
It is a fact that the theory of commutative ring completions is somewhat more de-
veloped than that of noncommutative ring completions. Certain theorems that hold
in the commutative case simply, but not surprisingly, don’t hold in the noncommu-
tative setting. Of particular importance is the Artin-Rees Lemma. However, we are
able to circumvent this issue in a nice way.
26
Theorem 4.7. Suppose A is a connected graded Noetherian kA-algebra with B ⊆
Z(A) ⊆ A a graded subring such that A is a module-finite B-algebra. Denote mA =
A>0 and mB = B>0. Let (−)mAdenote the completion with respect to the mA-adic
filtration, and let (−)mBdenote completion with respect to the mB-adic filtration.
Then if M is a finitely generated A-module (resp. M is a module finite A-algebra)
then MmA∼= MmB
as AmA-modules (resp. as AmA
-algebras).
Proof. We first show that {mnA} and {A≥n}, and {mn
B} and {A≥n} are cofinal. Of
course we have, for any n, that An>0 ⊆ A≥n by definition of graded. Since A is
Noetherian it follows, as in Appendix A, that A is a finitely generated k algebra, and
moreover, the generators may be taken to be homogeneous. Let g1, . . . , gl be such a
generating set and let d be the largest degree amongst the gi. Pick k ∈ N and pick
a ∈ A≥kd. Let W denote the set of words on g1, . . . , gl, and write a =∑
w∈W aww
with each aw ∈ kA. Since a ∈ A≥kd, the degree of each word in this expression must
be at least kd. However, the maximum degree among the gi’s is d, and so each word
must be the product of at least k of the gi. Hence, a ∈ Ak>0 so that A≥kd ⊆ Ak>0, and
we have that {mnA} is cofinal with {A≥n}. From [FJ74] we have that B is Noetherian.
Hence B is a finitely generated kB algebra. As for A, let g′1, . . . , g′m be a homogeneous
generating set for B as a kB-algebra, let d′ be the largest degree among the g′i, and
let W ∗ denote the set of words on g′1, . . . , g′l. Finally, let a1, . . . , as be a homogeneous
generating set for A as a B-module, and let dA/B be the maximum degree among the
ai.
Suppose M is a finitely generated A-module. Let m1, . . . ,ml be a homogeneous
generating set for M as an A-module. Let dM be the maximum degree among
the mi. Pick k ∈ N. Clearly we have that (mBM)k ⊆ A≥kM . Now pick a′m ∈
27
A≥kd′+dA/B+dMM . We have
am = a(l∑
i=1
bimi) (Where each bi ∈ B)
=l∑
i=1
abimi
=l∑
i=1
(m∑j=1
b′jaj)bimi (Where each b′j ∈ B)
=l∑
i=1
m∑j=1
b′jbiajmi (Since B ⊆ Z(A))
=l∑
i=1
m∑j=1
(∑
wj∈W ∗kwj
wj)(∑
wi∈W ∗k∗wi
wi)biajmi (Where each kwj, k∗wj
∈ kB)
=l∑
i=1
m∑j=1
∑wj∈W ∗
∑wi∈W ∗
kwjk∗wi
wjwibiajmi.
Take any term, kwjk∗wi
wjwibiajmi in this sum. Note that ajmi ∈M , since M is a
left A-module. The maximal degree of ajmi is, by construction, dA/B+dM . Since B is
connected, it follows that wjwi has degree at least kd′+dA/B +dM−dA/B−dM = kd′.
But each generator g′i has degree at most d′ and thus the word wjwi must have at
least k terms of g′1, . . . , g′l, giving us that am ∈ mk
BM . Ergo, {A≥nM} and {mnBM}
are cofinal filtrations. Since {mnA} and {A≥n} are cofinal, it follows that {mn
AM}
and {A≥nM} are cofinal. Since the cofinal relation is an equivalence relation, we
have that {mnAM} and {mn
BM} are cofinal. As seen in Appendix B, this implies that
MmA∼= MmB
as AmA-modules. If further M is an A-algebra, then these completions
occur in the category of A-algebras, so that this is an isomorphism of AmA-algebras.
Corollary 4.8. AmA∼= AmB
as rings.
Corollary 4.9. If (−)mAand (−)mB
are restricted to the category of finitely generated
A-modules, then (−)mA
∼= (−)mBas functors, and moreover (−)mA
is exact.
28
Proof. The first claim is clear from the definitions of fmA, fmB
and the isomorphisms
induced by cofinality. The second claim follows since, from the commutative theory,
(−)mBis exact on finitely generated B-modules.
What we have effectively done is express our noncommutative completion functor
as a commutative completion functor. As a result, we obtain access to the very
developed theory of commutative ring completions.
One may show that the subspace topology of the mA-adic topology on A is home-
omorphic to the mB-adic topology on any subring B of A, provided that B is Noethe-
rian, hence inducing isomorphic completions. Moreover, as a topological fact, we have
that the completion of B with the subspace topology is contained in the completion of
A in the mA-adic topology. That is, BmB⊆ AmA
is a subring. Moreover, if B ⊆ Z(A)
then BmB⊆ Z(AmA
), as each component of a sequence of elements of B is in Z(A).
Additionally, we obtain the following
Corollary 4.10. AmAis a module-finite BmB
algebra.
Proof. Since B is Noetherian with A a finitely generated B-module, we have that A
is finitely presented as a B-module. Let Bp → Bq → A → 0 be a presentation of A.
Since BmAis exact on finitely generated B-modules, we have that
BpmB→ Bq
mB→ AmB
→ 0
is exact. Since completion (more generally inverse limits) commute with direct sums
and since (−)mB
∼= (−)mA, we have that
(BmB)p → (BmB
)q → AmA→ 0
is exact. Hence AmAis a finitely presented BmB
-module, ergo finitely generated.
29
Corollary 4.11. AmAis Noetherian.
Proof. We know that BmBis Noetherian from the commutative theory since B is.
Since AmAis module-finite over BmB
we have that AmAis Noetherian as a left and
right BmB-module, hence as a left and right AmA
-module.
Corollary 4.12. If M is a finitely generated A-module, then MmA∼= M ⊗B BmB
as
left AmA-modules.
Proof. From the previous corollary we know that (−)mBis exact on the category of
finitely generated A-modules. Let M be a finitely generated left A-module. Hence A
is a finitely generated B module. Since B is Noetherian, we have that M is a finitely
presented B-module. Let Bp a−→ Bq b−→ M → 0 be a presentation of M . We note
that as a general categorical fact, inverse limits commute with direct sums, so that
completion does. Consider the following diagram
BpmB
BqmB
MmB0
Bp ⊗B BmBBq ⊗B BmB
M ⊗B BmB0
f
a⊗1
g
b⊗1
h
a b
The rows of this diagram are exact by exactness of (−)mBon finitely generated
B-modules, and by right exactness of ⊗B. There are natural B-module isomorphisms
a3, a2, and a1, respectively, giving that
Bp ⊗B BmB∼= (B ⊗B BmB
)p ∼= (BmB)p ∼= Bp
mB.
We define f = a1 ◦ a2 ◦ a3 so that f is a B-module isomorphism. We have that
f((bi)pi=1 ⊗ (bn)∞i=1) = a1(a2(a3((bi)
pi=1 ⊗ (bn)∞i=1) = a1(a2((bi ⊗ (bn)∞i=1)pi=1)
30
= a1(((bibn)∞i=1)pi=1) = ((bibn)pi=1)∞n=1
We define the map g similarly, and see that g is a B-module isomorphism. We
now define h : M ⊗B BmB→ MmB
by first defining h∗ : M × BmB→ MmB
by
h∗((m, (bn)) = (mbn), noting that h∗ is bilinear, and letting h be the induced map.
In particular, h∗ is A-linear in the first slot and B-linear in the second, so that
h is a homomorphism of A − B bimodules. In particular, h is a map of left A-
modules. By constuction of the maps f , g, and h, we see that the above diagram
commutes. Since f and g are isomorphisms, it follows that h is an isomorphism, so
that M ⊗B BmB∼= MmB
as left A modules. It’s not hard to see that this map is also
preserves the right BmBstructure. In fact, this same argument shows that any struc-
ture inherited by M ⊗B BmBfrom either M or BmB
is preserved by this isomorphism.
Unfortunately, for our purposes we must dig a bit deeper.
Quite counter-intuitively, we may actually define an AmAaction on M × BmB
in a natural way. However, to be safe about issues regarding well-definedness, we
induce this action through the action on MmBby defining it through the inverse
isomorphism h−1. To begin, let a1, . . . , ak be a generating set for A as a B-module,
and let m1, . . . ,ml be a generating set for M as a B-module. We have that
(an)(mn) = (anmn) = ((k∑i=1
bniai)(l∑
j=1
b′njmj)) =k∑i=1
l∑j=1
aimj(bnib′nj).
But then,
h−1((anmn)) =k∑i=1
l∑j=1
aimj ⊗ (bnib′nj),
since h−1 is abelian group homomorphism. But then we have that
31
k∑i=1
l∑j=1
aimj ⊗ (bnib′nj) =
k∑i=1
ai(l∑
j=1
mj ⊗ (b′nj))(bni) =k∑i=1
ai(bni)(l∑
j=1
mj ⊗ (b′nj)).
Note that, h−1((mn)) = h−1(∑l
j=1(b′njmj)) = h−1(∑l
j=1(mjb′nj)) =
∑lj=1 mj ⊗ (b′nj).
Thus we see that this induced action is given, on elementary tensors, by (an)(m ⊗
(bn) = (∑k
i=1 bniai)(m ⊗ (bn)) =∑k
i=1 aim ⊗ (bnibn). This is perhaps the action one
would want to define naturally, but it would not be clear that this action does not
depend on how we write the terms ai of (an). By defining the action through the
action on MmB, we circumvent this issue altogether. Moreover, by construction, we
see that h is AmB-linear so that M ⊗B BmB
∼= MmBas left AmB
-modules as well. One
potential issue this action could give is if M itself has an AmBstructure, thus giving
one to M ⊗B BmB. However, as mentioned previously, M ⊗B BmB
∼= MmBas AmB
-
modules under this action as well, so that this action and the one defined previously
coincide.
Corollary 4.13. AmAis a flat A-algebra.
Proof. Similar to the previous corollary, we obtain that, for a left A-module M ,
AmA⊗A M ∼= MmA
as left AmA-modules. Now, it suffices to show that, for every
left ideal I ⊆ A, that the map f : AmA⊗A I → AmA
, defined on elementary tensors
by f((an) ⊗ a) = (ana) is injective. But, we have that f is an isomorphism when
restricted to its image ImA. Moreover, ImA
i−→ AmAis injective since (−)mA
is exact on
finitely generated A-modules. Hence AmAis a flat A-module.
32
Corollary 4.14. If M and N are A-modules, then
HomA(M,N)⊗B BmB∼= HomAmA
(MmA, NmA
)
as BmB-modules. From the previous corollary, this implies that
ExtiA(M,N)⊗B BmB∼= Exti
AmA
(MmA, NmA
)
as BmB-modules.
Proof. Let N be finitely generated A-modules and let Apa−→ Aq
b−→ M → 0 be a
presentation of N as a left A-module. We define contravariant functors F and G of a
finitely generated left A-module M by
F (M) = HomA(M,N)⊗B BmBand G(M) = HomAmA
(AmA⊗AM, AmA
⊗A N).
Note that both F and G are left exact. We define a morphism of functors λ : F → G
by defining λ : F (M)→ G(M) on objects by λ(f ⊗ b) = b(1AmA⊗ f).
We have the following commutative diagram with exact rows
0 G(M) G(Aq) G(Ap)
0 F (M) F (Aq) F (Ap).
λ
b∗⊗1
λ
a∗⊗1
λ
(b⊗1)∗ (a⊗1)∗
There are canonical BmB-module isomorphisms giving that
F (Ap) = HomA(Ap, N)⊗B BmB∼= (HomA(A,N))p ⊗B BmB
∼= Np ⊗B BmB∼= (N ⊗B BmB
)p ∼= (NmA)p.
Likewise we have canonical BmB-isomorphisms giving that
33
(NmA)p ∼= (AmA
⊗A N)p ∼= (HomAmA(AmA
, AmA⊗A N))p
∼= HomAmA((AmA
)p, AmA⊗A N) ∼= HomAmA
(AmA⊗A Ap, AmA
⊗A N) = G(Ap).
One may check that the composition of all these isomorphisms is λ : F (Ap) →
G(Ap) so that λ is a B-module isomorphism. Similarly λ : F (Aq) → G(Aq) is
a B-module isomoprhism. It follows that λ : F (M) → G(M) is a BmB-module
isomorphism.
Finally, we have the following:
Corollary 4.15. If A is a graded isolated singularity, so that gldim tailsA = d, then
for all i > d, ExtiA(M,N) ∼= ExtiAmA
(MmA, NmA
) as BmB-modules for any finitely
generated graded modules M and N .
Proof. Let i > d. Since M and N are finitely generated graded modules over a
graded isolated singularity we have, by Theorem 3.7, that ExtiA(M,N) is Artinian,
hence complete. Since M and N are finitely generated, the previous corollary then
gives the result.
34
Chapter 5: AR Theory
5.1 The Category of Gradeable Modules
Using the ideas of the previous chapter, we are now able to pass information from a
graded isolated singularity A, to its completion, provided we assume there is a graded
subring B ⊆ Z(A) ⊆ A so that A is module-finite over B. More generally, we will
pass information from the category grA, of finitely generated graded A modules, into
an appropriate category of modules over AmA. We define this category in a natural
way. Let A be a connected graded Noetherian kA-algebra. As in previous chapters,
we let mA = A>0. We let CM(AmA) be the category of MCM AmA
-modules with
AmA-homomorphisms, and let CMgr(A) denote the category of graded MCM modules
over A with degree zero graded homomorphisms. Likewise, we denote by M(AmA)
the category of finitely generated AmAmodules and we maintain the convention of
writing grA for the category of finitely generated graded A-modules with degree zero
homomorphisms.
Definition 5.1. Suppose (R,m) is a complete local ring. Similar to the graded case,
for a left R-module M , we define depthRM = inf RΓm(M) = inf{i | H im(M) 6= 0},
and ldimR = supRΓm(M) = sup{i | H im(M) 6= 0}. We say that M is Maximal
Cohen-Macaulay (MCM) if depthRM = ldimRM = depthRR. If this holds
when M = R and M = Rop we say that R is AS Cohen-Macaulay. If additionally,
R has a balanced dualizing complex D, then we say that R is balanced Cohen-
Macaulay (CM) if D is isomorphic to a shift of an A-A bimodule, which we call
the canonical module for A, denoted ωR.
We need the following facts.
35
Proposition 5.2. Let M be a gradeable AmA-module so that X is a finitely generated
graded A-module with XmA= M . Then M is MCM if and only if X is.
Proof. Since, as seen in the previous chapter, mnA and A≥n are cofinal filtrations, it
follows that
H im(X)mB
= lim−→n
ExtiA(A/mnA, X)
mB
∼= lim−→n
ExtiA(A/mnA, X)⊗B BmB
∼= lim−→n
(ExtiA(A/mnA, X)⊗B BmB
) ∼= lim−→n
ExtiAmA
( (A/mnA)mA
, XmA)
∼= lim−→n
ExtiAmA
(AmA/(mn
A)mA,M) ∼= lim
−→n
ExtiAmA
(AmA/mn
AmA
,M) ∼= H im
AmA
(M).
Since ExtiA(A/mnA, X) is a torsion B-module, it follows that H i
mA(X) ∼= H i
mAmA
(M).
The result follows a fortiori.
Proposition 5.3. Let A be a graded AS-Gorenstein ring. Then AmAis AS-Gorenstein.
Proof. Suppose A is AS-Gorenstein of dimension d and Gorenstein parameter `. Let
id0X = max{i | ExtiA(k,X) 6= 0}. Since A is AS-Gorenstein we see that id0A =
idA. But then from the previous chapter, we have, since ExtiA(k,X) and k are
torsion (hence Artinian) A-modules for any i, that ExtiA(k,X) ∼= ExtiAmA
(k, XmA) ∼=
ExtiAmA
(k,M). In particular, for i 6= d, we have ExtiAmA
(k,M) = 0, and we have
ExtdAmA
(k,M) ∼= k(l) ∼= k. Given the previous chapter, it’s not difficult to see that
AmA[d] is a balanced dualizing complex for AmA
. From, [WZ01] Lemma 5.6 (2), we
have that id0 AmA= id AmA
so AmAis AS-Gorenstein.
36
In what follows, we will introduce the assumption A is AS-Gorenstein so that AmA
is. While we believe the following results hold if one introduces the slightly weaker
hypothesis that A is balanced Cohen-Macaulay, the theory is greatly simplified for
AS-Gorenstein rings, and there are a couple of results that we have yet to generalize
to this setting. We attempt to make this explicit by only assuming balanced Cohen-
Macaulay where possible.
Definition 5.4. Let M be a finitely generated left AmA-module. We say that M is
gradeable if there is a finitely generated graded left A-module X over R such that
M ∼= XmA. Similarly an AmA
-homomorphism f : M → N of gradeable modules is said
to be a gradeable homomorphism if there is a graded homomorphism of graded
modules g : X → Y so that the following diagram commutes in CM(AmA).
M XmA
N YmA
f
∼=
∼=
g
We define the category of gradeable left AmA-modules, denoted CMgr(AmA
) to
be the category with objects gradeable AmA-modules and morphisms gradeable AmA
-
homomorphisms.
The gradeable category is defined the way it is precisely to allow us to bounce back
and forth with ease between the graded side and the complete side via the completion
functor. In this sense, we can “pull up” results from the graded side to the complete
side.
37
5.2 AR Sequences
We are now finally equipped with the tools we need to coax out the aforementioned
deep implications between isolated singularities and the categories of MCM modules
over them. With begin by formulating a notion introduced by Maurice Auslander
and Idun Reiten, which they called almost-split sequences. We refer to them by their
now more common name, Auslander-Reiten sequences.
Definition 5.5. Suppose (A,m, k) is a Henselian CM local ring with canonical module
ω. Let M and N be non-zero indecomposable MCM A-modules, and let
0→ Ni−→ E
p−→M → 0
be an exact sequence of left (or right) A-modules.
(i) We say this sequence is an AR sequence ending in M if it is non-split, but
for every MCM module X and every homomorphism f : X → M which is not
a split surjection, f factors through p.
(ii) We say that this sequence is an AR sequence starting from N if it is non-
split, but for every MCM module Y and every homomorphism g : N → Y which
is not a split injection, g lifts through i.
We will generally be concerned with AR sequences ending in M , however, one
may show that the two conditions are in fact equivalent, and so we will often refer to
such a sequence simply as an AR sequence for M .
Definition 5.6. Let R be a Noetherian ring and M a finitely generated R-module
with projective presentation
P1ϕ−→ P0 →M → 0.
38
The Auslander Transpose TrM of M is defined by
TrM = cok(ϕ∗ : P ∗0 → P ∗1 ).
In other words, TrM is defined by the exactness of the sequence
0→M∗ → P ∗1ϕ∗−→ P ∗0 → TrM → 0.
Here (−)∗ = HomR(−, R) or (−)∗ = HomRop(−, R), as appropriate.
Definition 5.7. If A is Henselian, we define redsyzRn (M) to be the reduced nth
syzygy module. That is, redsyzRn (M) is the module obtained by deleting any non-
trivial free direct summands from the nth syzygy module syzRn (M). In particular
redsyzR0 (M) is obtained by deleting any free direct summands from M .
For later purposes, it is necessary to know the following
Proposition 5.8. If A is AS-Gorenstein of dimension d and Gorenstein parameter
`, so that R = AmAis AS-Gorenstein of dimension d, and if M is an indecomposable
left A-module, then redsyzAop
j (TrM) is indecomposable for any 0 ≤ j ≤ d.
Proof. Note that since R is AS-Gorenstein we have, by canonical duality, that
ExtiR(M,R) ∼= ExtiRop(R,M∗) ∼={M∗ if i = 0
0 otherwise
since R is a projective R-module. The result follows from Proposition 3.8 in [AB69]
and following the proof of Proposition 13.4 in [LW12].
Syzygy and Auslander Transpose are but a couple examples where things are well-
defined only up to projective direct summands; to circumvent such issues, it is often
helpful to work in a category where all projective modules are isomorphic to 0.
39
Definition 5.9. We define the stable category of A-modules, denoted A-Mod to
be the quotient category of AMod formed by taking morphisms to be elements of the
quotient HomA(M,N) = HomA(M,N)/P(M,N), where P(M,N) is the subgroup of
HomA(M,N) of all A-homomorphisms from M to N that factor through a projective
A-module.
One may see, fairly easily, that every projective object in this category is isomor-
phic to the zero object. As with the usual Hom, the stable Hom group HomA(M,N)
is naturally a left EndA(N)-module and a right EndA(M)-module. Further, one may
see that B(M,M) is a two-sided ideal of EndA(M) so that EndA(M) is in fact a ring.
5.3 Duality and The Functorial Isomorphism
For a local ringA, we define (−)∨ = HomA(−, EA(A/J)) or (−)∨ = HomAop(−, EA(A/J))
as appropriate, where J is the Jacobson radical of A (so that A/J is a division ring),
and call M∨ the Matlis Dual of M . Note that EA(A/J) is an A − A bimodule so
that this definition makes sense. A detailed treatment of Matlis duality can be found
in both [WZ01] and [Cha00]. They prove that the definition we have given for Matlis
Duality, which is the standard definition used for commutative local rings, is indeed
the appropriate definition to use, that is, it satisfies the desired properties one attains
in the commutative case. The facts about Matlis duality that are most important to
us, are that Matlis duality is actually a true duality in that M∨∨ ∼= M , that Matlis
duality takes Noetherian modules to Artinian modules and vice versa, and that the
number of generators of a finitely generated A-module M is equal to dimA/J soc(M∨).
We pose the following conjecture:
Conjecture 5.10. Let A be a connected graded Noetherian k-algebra that is a graded
isolated singularity and B ⊆ Z(A) ⊆ A a graded subring so that A is module-
40
finite over B. Let R = AmAand let M and N be left R-modules. Denote τ(M) =
HomRop(redsyzRop
d TrM,ωR). Then,
HomS(HomR(M,N), ES(S/J(S)) ∼= Ext1R(N, τ(M))
as functors in both variables M and N for suitable choices of S. In particular, if
1. S = EndR(M), then the isomorphism is as left EndR(N)-modules.
2. S = EndR(N), then the isomorphism is as right EndR(M)-modules.
3. S = EndR(M) with M = N , then the isomorphism is as EndR(M)-bimodules.
Moreover, one can easily see that if this holds in this case, then we in fact have
ES(S/J(S) ∼= Ext1R(M, τ(M))
We know that an appropriate analogue to this conjecture is true in both the
commutative and the graded case, and so it seems at least reasonable that this be
true in our setting.
5.4 Existence and Uniqueness
Theorem 5.11. As before, let A be a connected graded Noetherian AS-Gorenstein
isolated singularity with B ⊆ Z(A) ⊆ A a graded subring so that A is module-finite
over B. Let R = AmA. Let M be a indecomposable gradeable CM left R-module. Then
there exists a unique AR sequence for M
α : 0→ τ(M)→ E →M → 0.
More precisely, Ext1R(M, τ(M)) has a one-dimensional socle, and any representative
for a generator of that socle is an AR-sequence for M .
41
The uniqueness proof is verbatim the proof from the commutative case. The
existence proof is the main goal of this paper, and naturally requires a lemma.
Lemma 5.12. Suppose X and M are left R-modules with M non-projective so that
EndR(M) is local, and suppose we have an R-homomorphism f : X → M . If the
induced map f ∗ : HomR(M,X)→ EndR(M) is surjective, then f is a split surjection.
Proof. Since f ∗ is surjective, we have that idM +P(M,M) ∈ im(f ∗). Hence there
exists some h ∈ P(M,M) so that idM +h ∈ im(f ∗). Pick g ∈ Hom(M,X), so that
f ∗(g) = idM +h. But then, by definition of f ∗, f ∗(g) = fg, so that fg = idM +h.
Since EndR(M) is local, it follows that either h is a unit or idM +h is a unit. We
claim that h is a not a unit. So suppose that it is. Let q be such that hq = idM . Since
h ∈ P(M,M), there exists a projective module P , and maps a, b giving the following
commutative diagram
M M
P
h
a b
We have the following exact sequence
0 M P coker a 0a
But then qh = qba = idM so that qb gives a splitting for a. Hence the above
exact sequence is left split, hence split by the splitting lemma. So, P ∼= M ⊕ coker a.
But then M is the direct summand of a projective, hence projective. However, we
have assumed that M is non-projective, so that h is not a unit. So, it must be that
idM +h is a unit. Let l be so that fgl = (idM +h)l = idM . Since gl is a right inverse
homomorphism of f , it follows that f is a split surjection.
42
Now we finally have all the tools required to prove this theorem.
Proof. First, we have that BmB⊆ Z(R) ⊆ R is a subring of R so that R is module-
finite over BmB. Since BmB
is complete and commutative, we have by Hensel’s Lemma,
that it is Henselian. By Proposition 4.4, it follows that EndR(M) is local. Since
EndR(M) is a quotient of EndR(M), it follows that EndR(M) is local. For brevity,
denote S = EndR(M) and let J be the Jacobson radical of S. Since S is trivially a one-
generated S-module, it follows that dimS/J soc(S∨) = 1. But, from the conjecture,
S∨ = HomS(S,ES(S/J)) ∼= ES(S/J) ∼= Ext1R(M, τ(M)) as S − S bimodules so that
the socle of Ext1R(M, τ(M)) is also one-dimensonal over S/J . Let
α : 0→ τ(M)→ E →M → 0
be an extension generating the socle of Ext1R(M, τ(M)). By Proposition 5.7, we
know that redsyzRop
d (Tr(M)) is indecomposable so that its canonical dual τ(M) is
indecomposable. Now let f : X → M be a homomorphism of MCM R-modules.
Pullback along f then induces a map f : Ext1R(M, τ(M)) → Ext1
R(X, τ(M)). If f
does not factor through the map (E →M), then f(α) is non-split, that is, non-zero in
Ext1R(X, τ(M)). Since α generates soc(Ext1
R(M, τ(M)), it follows that f is injective.
Since the isomorphism from the conjecture is functorial, f must be the same map as
the map
HomS(S,ES(S/J))→ HomS(Hom(M,X), ES(S/J))
induced by f . Since f is injective, it follows from Matlis duality, that the induced
map f ∗ : HomR(M,X)→ EndR(M) is surjective. But then, by the previous lemma,
it follows that f is a split surjection so that α is an AR-sequence for M .
It remains to show that this sequence lies in the gradeable category. One easily
sees that if M is gradeable with M ∼= XmAfor a graded finitely generated A-module
43
X, then τ(M) ∼= τgr(X)mA. Now the unique AR-sequence for M is a generator for
the socle of Ext1R(M, τ(M)). However, as X and τgr(X) are MCM, it follows from
Lemma 5.7 [Uey13] that Ext1A(X, τgr(X)) is finite dimensional over k so that
Ext1A(X, τgr(X)) ∼= Ext1
A(X, τgr(X))mB
∼= Ext1AmA
(XmA, τgr(X)mA
) ∼= Ext1R(M, τ(M)).
It follows that the socle of Ext1A(X, τgr(X)), as a graded-module over EndA(X) (the
stable endomorphism ring ofM), must be in part of maximal degree in Ext1A(X, τgr(X)).
Hence we can take a socle element as a homogeneous element in Ext1A(X, τgr(X)). Let
ρ : 0 → τgr(X) → Y → X → 0 be such a socle element. Then we obtain an AR-
sequence for M by completing ρ, and so, by uniqueness of AR-sequences, it follows
that every module and homomorphism appearing in the AR-sequence for M is grade-
able.
44
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48
Appendix A: Graded Rings and Categories
Here we present some of the elementary facts regarding graded rings and modules
and some categorical facts.
A.1 Graded Rings and Modules
Definition A.1. If G is a semigroup, then a ring R is called G-graded if there exists
a family of subgroups {Rg}g∈G of R such that
1. R =⊕
g Rg as abelian groups.
2. RaRb ⊆ Rab for all a, b ∈ G.
We most often focus on the case when G = Z, and in this case we simply say
that R is graded rather than Z-graded. If R is a graded ring with Rn = 0 for all
n < 0, then we say R is nonnegatively graded in addition to referring to R as
N-graded.
Proposition A.2. If R is a graded ring, then R0 is a subring of R, and Rn is an
R0-R0 bimodule.
Proof. Since R is graded, we have R0 is an abelian group and R0R0 ⊆ R0 so that R0
is closed under multiplication. Since R =⊕
n∈ZRn, we may write 1 =∑
n xn where
xn ∈ Rn and only finitely many of the xn’s are nonzero. Then for any i, we have
xi = 1 · xi =∑
n xnxi. Comparing degrees gives that xi = x0xi. Hence,
x0 = 1 · x0 =∑n
xnx0 =∑
xn = 1.
Ergo, R0 is a subring of R. We have that Rn is a R0-R0 bimodule as R0Rn ⊆ Rn and
RnR0 ⊆ Rn.
49
In some sense graded is not a restrictive condition in the least, as every ring pos-
sesses the trivial grading. That is, we may view a ring R as a graded ring by defining
R0 = R and Rn = 0 for any n 6= 0. Of course the examples we will consider are
generally much more interesting, a class of which is given by the following definition.
Definition A.3. Suppose k is a field and let A be a k-algebra. Then A is a graded
algebra if it is graded as a ring. Furthermore, we say that A is connected if A0∼= k.
Two natural, and important, definitions are the following:
Definition A.4. Let S =⊕
n Sn be a graded ring. A subring R of S is called a
graded subring S if R =∑
n(Sn ∩R).
Definition A.5. Let S =⊕
n Sn be a graded ring. A left S-module M is called a
graded module if there is a collection {Mn} of abelian groups so that M =⊕
nMn,
and for any i, j, we have SiMj ⊆Mi+j. We say that a submodule N of M is a graded
submodule if N =⊕
n(Mn ∩N).
Definition A.6. Suppose A is a graded k-algebra. We say that A is locally finite
if An is a finite dimensional k vector space for all n.
The following is well known.
Proposition A.7. A graded ring A is right (resp. left) Noetherian if and only if it
is graded right (resp. left) Noetherian, that is, if and only if every graded right(resp.
left) ideal is finitely generated.
The following is a list of highly useful facts which appear in [AZ94].
Proposition A.8. Suppose A is a Noetherian connected N-graded ring. Then,
50
1. A is locally finite.
2. If M is a finitely generated graded A-module, M is locally finite.
The following is well known in the commutative case, but it is difficult to track
down in the noncommutative case, so we provide a proof.
Proposition A.9. Suppose A is an N-graded ring. If A is Noetherian then A0 is
Noetherian and A is a finitely generated algebra over A0.
Proof. Suppose A is Noetherian. Hence A is graded Noetherian.
It follows that A>0 and A/A>0∼= A0 are Noetherian. Since A is graded Noetherian,
we may take g1, g2, . . . , gm to be a finite homogeneous generating set for A>0 as an
A-module. Let R be the A0 algebra generated by g1, g2, . . . , gm. We claim A = R.
Clearly R ⊆ A. If A * R, then there exists an a ∈ A>0 of least degree so that a /∈ R.
But since a ∈ A>0, we may express a as a linear combination, a1gi1 + · · ·+algil where
deg(gik) < deg a and deg ak ≤ deg a with ak homogeneous for all k. But then each
ak ∈ R and hence a ∈ R.
Definition A.10. If A is a graded Noetherian ring, then we say an element x ∈ A
is torsion if there is an s so that xA≥s = 0. The torsion elements in M form a
graded A-submodule which we denote by τ(M) and call it the torsion submodule
of M . A module is said to be torsion-free if τ(M) = 0 and is said to be torsion
if τ(M) = M . If A is further a connected k-algebra, one may show that τ defines
a left exact functor and that the right derived functors of τ are the local cohomology
functors with respect to the graded maximal ideal A>0. One may further check that
τ(M) is the smallest submodule of M so that M/τ(M) is torsion-free.
The following also appears in [AZ94].
51
Proposition A.11. Suppose A is graded Noetherian ring and let M → E be an
essential extension of graded A-modules. Then M torsion implies that E is torsion.
A.2 Quotient Categories
One fundamental construction we invoke is that of the Quotient Category. Through
this construction we are able to define tailsA for a connected graded k-algebra A,
and through this define our notion of graded isolated singularity.
Definition A.12. Let C be a category. A congruence relation ∼ on C is given
by: for any X, Y ∈ obj C, an equivalence relation ∼X,Y on HomC(X, Y ) such that
if f1, f2 ∈ HomC(X, Y ) such that f1 ∼X,Y f2 and g1, g2 ∈ HomC(Y, Z) such that
g1 ∼Y,Z g2, then g1 ∼X,Z f1, g1 ∼X,Z f2, g2 ∼X,Z f1, and g2 ∼X,Z f2.
Definition A.13. Given a category C and congruence relation ∼, we define the quo-
tient category C/ ∼ to be the category whose objects are the objects of C, and whose
morphisms are defined by HomC/∼(X, Y ) = HomC(X, Y )/ ∼X,Y , the set of equivalence
classes of the relation ∼X,Y . Composition of morphisms is defined via representatives,
and this composition is well-defined since ∼X,Y is a congruence relation.
One particularly important construction arises from the notion of a Serre subcat-
egory.
Definition A.14. Let C be an abelian category. We say that a subcategory S of C
is a Serre Subcategory if it is closed under the formation of subobjects, quotients,
and extensions.
Given an abelian category C and a Serre subcategory S, we may form a the
quotient category C/S by taking obj(C/S) = obj(C) and
HomC/S(X, Y ) = lim−→U,S
Hom(U, Y/S)
52
where U runs through all subobjects of X with X/U ∈ obj(S) and S runs through
all subobjects of Y belonging to S. One may define a canonical quotient functor
T : C → C/S and show that TX = 0 for X ∈ obj(C) if and only if X ∈ obj(S).
53
Appendix B: Inverse Systems and Ring Completion
Here we present the theory behind inverse systems, inverse limits, and ring com-
pletions in all its beautiful gory detail.
B.1 Inverse Systems
Definition B.1. Let I be a partially ordered set and let C be a category. An inverse
system in C is an ordered pair ((Mi)i∈I , (ϕji )j�i), abbreviated {Mi, ϕ
ji}, where (Mi)i∈I
is an indexed family of objects in C and (ϕji : Mj → Mi)j�i is an indexed family of
morphisms for which ϕii = 1Mifor all i and such that the following diagram commutes
whenever k � j � i:
Mk Mi
Mj
ϕki
ϕkj ϕji
We could also describe an inverse system as a contravariant functor M : I → C.
We may view I as a category whose objects are the elements of I and with morphisms
exactly one morphism κij : i → j whenever i � j. In the notation above, M(i) = Mi
and M(κij) = ϕji . The conditions above then merely state that M is a contravariant
functor.
Example B.2. Let R be a ring and let I be an ideal. Then each In is an ideal and
there is a descending filtration R ⊃ I ⊃ I2 ⊃ I3 ⊃ · · · . If M is a left R-module,
then there is a descending filtration M ⊃ IM ⊃ I2M ⊃ I3M ⊃ · · · . If m ≥ n, then
54
ImM ⊂ InM and thus we can define ϕmn : M/ImM → M/InM to be the induced
R-module homomorphism on quotients of the natural inclusion map ImM ↪→ InM .
That is, ϕmn (a+ ImM) = a+ InM . Then of course for any i, ϕii(a+ I iM) = a+ I iM
so that ϕii = idIiM , and moreover if k ≥ j ≥ i, we have IkM ⊂ IjM ⊂ I iM . Further,
ϕji (ϕkj (a + IkM) = ϕji (a + IjM) = a + I iM = ϕki (a + IkM) so that ϕki = ϕji ◦ ϕkj
and the appropriate diagram commutes. Hence {M/InM,ϕmn } is an inverse system
in RMod over the partially ordered set N. Of course, this argument also works for
right R-modules mutatis mutandis.
This is our motivating example for considering inverse systems, and it is through
this particular inverse system that we will define the notion of ring completion. We
proceed with yet another definition.
B.2 Inverse Limits
Definition B.3. Let I be a partially ordered set, let C be a category, and let {Mi, ϕji}
be an inverse system in C over I. We define the inverse limit of {Mi, ϕji} to be an
object lim←−Mi and a family of projections (αi : lim←−Mi →Mi)i∈I such that
1. ϕjiαj = αi if i � j.
2. For every X ∈ obj(C) and all morphisms fi : X → Mi satisfying ϕjifj = fi for
all i � j, there exists a unique morphism θ : X → lim←−Mi making the following
diagram commute:
55
lim←−Mi X
Mi
Mj
αi
θ
fi
αj fjϕji
We generally denote the inverse limit by lim←−Mi. This notation is perhaps a bit
defective as the inverse limit depends strongly on the morphisms of the inverse system,
however, aside from being standard practice, we generally only discuss the inverse
limit in cases where the morphisms are understood. It turns out, as one might expect
for the solution to a universal mapping problem, that the object of the inverse limit
is unique up isomorphism in C when it exists. In fact, a stronger result is true. The
inverse limit is unique up to a unique isomorphism in C commuting with the projection
maps. However, the inverse limit may not exist at all. In fact, as seen in chapter 4,
it does not generally exist in GrA for a connected graded k-algebra A. However, one
can prove that they do exist in many familiar categories, and further one can even
write down a nice expression for the object of the inverse limit in these conditions.
We begin in an obvious place.
Proposition B.4. Inverse limits exist for any inverse system {Mi, ϕji} in RMod over
any partially ordered index set I, and moreover,
lim←−Mi∼= {~m ∈
∏k
Mk | mi = ϕji (mj) when i � j}.
Proof. Let L =∏
kMk | mi = ϕji (mj) when i � j}. We first need to check that L
is a submodule of∏
kMk. Clearly ~0 ∈ L. Take ~a,~b ∈ L. Then for all i � j we
have ai = ϕji (aj) and bi = ϕji (bj). Hence ai − bi = ϕji (aj) − ϕji (bj) = ϕji (aj − bj) by
56
R-linearity of ϕji . But by definition of∏
kMk, we have (~a −~b)k = ~ak −~bk for any k
so that L is an abelian subgroup of∏
kMk. Moreover, if r ∈ R, then rai = rϕji (aj) =
ϕji (raj) by R-linearity of ϕji . By definition of∏
kMk, we have (r~a)k = r~ak and
therefore r~a ∈ L. Ergo, L is a submodule of∏
kMk. Let pi be the natural projection
of∏
kMk to Mi and define αi : L → Mi by αi = pi|L. Then, by construction,
ϕji (αj(~a)) = ϕji (aj) = ai = αi(~a) when i � j. Now let X be a left R-module and
suppose we have R-linear maps fi : X →Mi satisfying ϕjifj = fi for all i � j. Define
θ : X → ∏kMk by θ(x) = (f1(x), f2(x), . . . ). Since ϕjifj = fi for all i � j, it follows
that im θ ⊆ L. Then we have, for any i, αiθ(x) = αi((f1(x), f2(x), . . . )) = fi(x)
so that θ makes the appropriate diagram commute. It remains to show that θ is
unique. Suppose ϕ : X → L makes the diagram commute. Since ϕ maps into L,
we have ϕ = (m1,m2, . . . ) with mi ∈ Mi and ϕji (mj) = mi. But then αiϕ(x) = mi.
But then by commutativity of the diagram we have αiϕ(x) = mi = fi(x). Hence
ϕ(x) = (f1(x), f2(x), . . . ) so that ϕ = θ. Therefore, we have that {lim←−Mi, αi} exists
and lim←−Mi∼= L.
Corollary B.5. Inverse limits exist in the category AbGrps. In particular if {Gi, ϕji}
is an inverse system in the category abelian groups, then
lim←−Gi∼= {~g ∈
∏k
Gk | gi = ϕji (gj) when i � j}.
Exactly the same argument shows that inverse limits exist and are isomorphic
to the analogous subobjects of the direct product in the categories ModR, Rings,
ComRings, RAlg, and AlgR, mutatis mutandis. It is also a fact, that we will not
prove, that inverse limits exist in the categories Groups, Sets, and TopSpaces.
57
B.3 Ring Completion
Now that we have formulated the notion of inverse limit and proved its existence in
a number of useful categories, it’s only natural to apply this notion to our previous
example of an inverse system.
Definition B.6. Let R be a ring, I be an ideal, and M is a left R-module. Consider
the descending filtration M ⊃ IM ⊃ I2M ⊃ I3M ⊃ · · · . For m ≥ n, ImM ⊂ InM
and thus we can define ϕmn : M/ImM → M/InM by ϕmn (a + ImM) = a + InM . We
have that {M/InM,ϕmn } is an inverse system in RMod over N. As we have seen
the inverse limit of this system exists. We define M = lim←−M/InM and call M the
I-adic completion of M .
Of course a similar construction works in rings, however, one must take care,
as R/I is only a ring when I is a two-sided ideal. This, of course, presents no
problems in the case when R is commutative, however, we are principally concerned
with noncommutative rings, and so we must take this into consideration.
There is an alternate construction of the I-adic completion. As before, let R be
a ring, I an ideal, and M a left R-module. Again, we consider the inverse system
{M/I iM,ϕji} with the appropriate induced maps ϕji . We may now put a topology
M . Define open sets about a point x ∈ R, to be the cosets x + InM . The union
of all these opens sets forms a topology on R known as the Krull Topology. Now,
in order to proceed with this construction, we must introduce the assumption that⋂∞n=0 I
nM = {0}. We shall see why this condition is important as we proceed. While
this condition might seem somewhat restrictive, there are a large class of rings and
ideals for which this holds. Indeed, any commutative Noetherian local ring satisfies
this for its maximal ideal m, by the Krull Intersection Theorem, as does any connected
nonnegatively-graded k-algebra A for its homogeneous maximal ideal A≥1. Provided
58
we have this condition, given a nonzero x ∈ M , there exists an j such that x ∈ IjM
but x /∈ IkM for any k > j. We define ||x|| = 2−j, and additionally define ||0|| = 0,
noting that without our condition, || · || would not be defined for every nonzero x. We
then define the function d(x, y) = ||x−y||. It’s not hard to see that d is a metric. It’s
also not difficult to see that the open balls corresponding to this metric are precisely
the open sets of the Krull Topology. Hence, the Krull Topology is metrizable, being
induced by this metric. Now as we want our module to be, in some sense, complete,
we define M to be the analytic completion of (M,d), which is unique up to a unique
invertible isometry φ : M → M . Moreover, φ(M) is dense in M , and now (M, d)
is complete. What is not obvious is that M ∼= lim←−M/InM . It is perhaps not even
obvious that M has a compatible R-module structure. To see this we identify M
with something we know is an R-module. Let C(M)I be the set of Cauchy sequences
in (M,d), and let C(M)0 be the set of sequences in (M,d) that converge to 0. That
is, those sequences such that for all k ∈ N, there exists an N ∈ N such that j ≥ N
implies ri ∈ IkM . Any such convergent sequence is automatically Cauchy. Now we
have a natural R-module structure on the direct product∏
n∈NM . It’s not difficult
to see that C(M)I is a submodule of∏
n∈NM . Similarly one can see that C(M)0
is a submodule of C(M)I . Then we can consider C(M)I/C(M)0 which of course
has a natural R-module structure. We may now show that C(M)I/C(M)0 = M as
sets. Observe that C(M)I/C(M)0 = {E(an) | (an) ∈ C(M)I} where E(an) denotes the
equivalence class of (an) under the equivalence relation ∼ defined by (an) ∼ (bn) if
(an)−(bn) = (an−bn) ∈ C(M)0. However, this is true if and only if for all k ∈ N, there
exists N ∈ N such that for all j ≥ N we have aj−bj ∈ IkM which is true if and only if
d(aj, bj) = ||aj− bj|| ≤ 2−k. However this is true if and only if limj→∞ d(aj, bj) = 0 so
that we have C(M)I/C(M)0 is the set of equivalence classes E ′(an) under the equivalence
relation ∼′ defined by (an) ∼′ (bn) if and only if limj→∞ d(aj, bj) = 0. This is, by
59
definition, M . Hence M has a natural R-module structure, that is, the natural
structure arising from the quotient C(M)I/C(M)0.
Now, given a sequence (rn) ∈ C(M)I we may consider the sequence (rn + IkM)
for any k ∈ N. Since (rn) is Cauchy, it follows that (rn+ IkM) is eventually constant,
having a stable value r+IkM . Hence we may define a map pk : C(M)I →M/IkM by
pk((rn)) = r+ IkM where r+ IkM is the stable value of (rn + IkM). One may check
that pk is an R-module homomorphism. Moreover, pk is clearly surjective, as given
a + IkM ∈ M/IkM , we have that the constant sequence a, a, . . . , which of course is
Cauchy, maps to a + IkM under pk. Further, we see that if (rn) ∈ C(M)0 then the
stable value of (rn + IkM) is IkM so that C(M)0 ⊆ ker pk. Hence we get an induced
surjective map πk : C(M)I/C(M)0 → M/IkM . Moreover, one easily checks that if
i ≥ j, one has ϕjiπj = πi. By the universal property of inverse limit, there exists a
unique R-linear map θ, so that the following diagram commutes:
lim←−Mi C(M)I/C(M)0
M/I iM
M/IjM
αi
θ
πi
αj πj
ϕji
From proposition We have that
lim←−M/InM ∼= {~a ∈∏k
M/IkM | ai = ϕji (aj) when i ≥ j},
and θ is defined by θ(x) = (π1(x), π2(x), . . . ). So, take ~a ∈ {~a ∈ ∏kM/IkM | ai =
ϕji (aj) when i ≥ j}. Then aj = rj + IjM for some rj ∈M , for each j ∈ N. Consider
(rj). The condition ϕji (aj) = ai implies (rj) is Cauchy. Consider π((rn)), where
60
π : C(M)I → C(R)I/C(M)0 is the natural projection. By definition of θ, we have that
θ(π(rj)) = ~a so that θ is surjective.
Suppose we have θ((an)+C(R)0) = (a0+M,a1+IM, a2+I2M, . . . ) = (b0+M, b1+
IM, b2 + I2M, . . . ) = θ((bn) + C(M)0). Hence ai− bi ∈ I iM for all i. But this implies
that (ai)− (bi) = (ai − bi) ∈ C(M)0 so that (ai) + C(M)0 = (bi) + C(M)0. Thus, θ is
an isomorphism, and we have successfully identified M , M , and C(M)I/C(M)0.
Now, as mentioned previously, if⋂∞k=0 I
k 6= {0}, we cannot even define the metric
d, however we may still define C(M)I directly as those sequences (rn) such that for
all k ∈ N there exists N ∈ N such that i, j ≥ N implies ri − rj ∈ IkM . Likewise we
can construct C(M)0 without reference to d. We can define φ : M → C(M)I/C(M)0
by φ(m) = (m,m,m, . . . ) (since (m,m,m, . . . ) is obviously Cauchy for any m ∈M).
But then the kernel of this map is the set {m ∈ M | (m,m,m, . . . ) ∈ C(M)0}. That
is, it is the set of m ∈M , such that for all k ∈ N, there is an N ∈ N, such that j ≥ N
implies r ∈ IkM . In other words, it is the set⋂∞k=0 I
kM . In particular, in the case
above, when⋂∞k=1 I
kM = {0}, we have that φ is injective. This perhaps gives a deeper
indication of the importance of this condition. As we saw before, when⋂∞k=0 I
k = {0},
the I-adic topology is Hausdorff, and this is enough for this topology, to ensure that
it is in fact metrizable. Further, we also have that the map φ : M → M is one-to-
one. It turns out these the converses of these implications are true as well, and so if
any of these equivalent conditions hold we say that M is I-adically separated. If
additionally, we have that φ is surjective, hence an isomorphism, we say that M is
I-adically complete.
A similar construction works for rings. If R is a ring and I is a two-sided ideal, then
we have the inverse system {R/Ik, ϕjk}. We are able to construct R and C(R)I/C(R)0
is a similar manner as we did for R-modules, and likewise identify them as before.
However, if I is not two-sided, then {R/Ik, ϕjk} is not an inverse system in Rings,
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and so we cannot even begin to speak of the inverse limit (in Rings) of such a system,
though we can still consider the completion in RMod or ModR, as appropriate.
The following is a useful fact.
Proposition B.7. Let R be a ring and I a two-sided ideal in R. Then there is a
bijective correspondence between maximal left ideals of R, the I-adic completion of R,
and R/I. In particular, if R/I is local, then so is R.
Proof. We have the natural projection map α1 : R→ R/I. Let J = kerα1. We claim
that for any j ∈ J , we have that 1 + j is a unit in R. To see this fix j ∈ J , and
write j = (rn) + C(R)0 where (rn) ∈ C(R)I . Define (vn) by vk =∑k+1
i=0 (−1)irik. Let
k ≥ 0 be given. Then there is an N ∈ N such that for all i, j ≥ N , ri − rj ∈ Ik. Let
M = max{N, k}. Then suppose, without loss of generality, that we have j ≥ i ≥M .
Observe that
vi − vj = (i+1∑d=0
(−1)drdi )− (
j+1∑d=0
(−1)drdj ) = (i+1∑d=0
(−1)d(rdi − rdj ))− (
j+1∑l=i+2
(−1)lrlj).
Observe that for any t ∈ N, ri−rj ∈ Ik ⇒ ri+Ik = rj+Ik ⇒ rti +I
k = rtj+Ik ⇒ rti−
rtj ∈ Ik. Hence∑i+1
d=0(−1)d(rdi −rdj ) ∈ Ik. Of course we have −(∑j+1
l=i+2(−1)lrlj) ∈ I i+2.
Hence vi − vj ∈ Ik + I i+2. But since i ≥ M ≥ k, it follows that I i+2 ⊆ Ik so that
Ik + I i+2 = Ik. hence vi − vj ∈ Ik and we have that (vn) ∈ C(R)I . Now define
(zn) = (1)− (1 + rn)(vn) = (1)− (1 + rn)(n+1∑i=0
(−1)irin)) = (rn+2n )
It’s easy to see that (rn+2n ) ∈ C(R)0. Thus we have 1 − (1 + rn)(vn) ∈ C(R)0 so
that ((1 + (rn))(vn)) + C(R)0 = ((1 + j) + C(R)0)((vn) + C(R)0) = 1 + C(R)0. Ergo,
(vn) + C(R)0 is an inverse for 1 + j so that 1 + j is a unit. Since J = kerα1, it follows
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that J is a two-sided ideal. Ergo, J ⊆ J(R). Hence J is contained in every maximal
left (and right) ideal of R. The result follows from the Correspondence theorem.
In particular, for a connected graded k algebra A, where k is a field, and for its
maximal homogeneous two-sided ideal A≥n, we have that A/A≥n ∼= k. Since k is a
field, it is local, so have by this proposition that A is local.
Definition B.8. Suppose M is a left R-module and I is an ideal in R. We say that
a filtration of submodules {Ni} of M is cofinal with the filtration {IjM} if for all
k ∈ N there is a j ∈ N so that IjM ⊆ Nk and for all n ∈ N there is an m ∈ N so
that Nm ⊆ InM .
Cofinal filtrations are important as they induce the same topology on M . In
the case when⋂k I
k = {0} they induce equivalent metrics, so that lim←−M/Ni∼=
lim←−M/IkM . In fact, one can show that this is still true even without this condition.
Proposition B.9. Let A be a ring and I a left ideal in A. If M and N are left
A-modules and if f : M → N is an A-linear map, then f induces an A-linear map
f : M → N . Moreover, f is surjective if f is. We may then define a covariant
functor (−) : AMod →A Mod by M 7→ M for all M ∈ obj(AMod) and f 7→ f for
any morphism f .
Proof. For any n, we have the natural map ϕn : M/InM → N/InN . By definition
of inverse limit, we have maps αn : M = lim←−M/InM → M/InM . Composing these
gives maps βn : M → N/InM . One easily sees that these maps make the appropri-
ate diagram commute so that the universal property of inverse limits gives us a map
f : M → N .
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If f is surjective, then the maps βn are all surjective. As seen in Proposition 7, our
map f may be defined by f(x) = (β1(x), β2(x), . . . ). Take any sequence (yn) ∈ N .
For any n we have, since yn ∈ N/InN and by surjectivity of βn, that there is a
bn ∈M/InM so that β(bn) = yn. Hence f((bn)) = (yn) so that f is surjective.
It’s not hard to see that for any M ∈ obj(AMod), we have idM = idM . Suppose we
now have s : S → T and t : T → Q in AMod. Let sn be the composition of the
natural maps αn : S → S/InS with πSn : S/InS → T/InT and tn the composition of
the natural maps γn : T → T/InT with πTn : T/InT → Q/InQ. Likewise for (t ◦ s)
we let (t ◦ s)n be the composition of the natural maps S → S/InS and the induced
maps S/InS → Q/InQ, which are then tn ◦ sn. We then have, as seen before, that
t ◦ s = ((t ◦ s)n) = (tn ◦ sn) = tn ◦ (sn) = t ◦ s
Hence (−) is a covariant functor.
Of course, the same result holds for rings, when I is a two-sided ideal, and algebras
by applying this proof mutatis mutandis. Of particular note, the subcategory AAlg
of AMod is mapped into the subcategory AAlg of AMod by this functor.
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Justin Lyle
Contact Information
5123 Winster Drive Apt. 001 [email protected]
Winston-Salem, NC 27106 770-881-1665
Research InterestsCommutative/Noncommutative Ring Theory, Homological Algebra, Representation The-ory, Algebraic Geometry, and Invariant Theory.
EducationWake Forest University
M.S. in Mathematics, May 2015
Berry College
B.S. in Mathematics, May 2013 (magna cum laude)
• Dean’s List 2010-2013
TalksThe Representation Theory of Finite Groups, Math Club Talk, Wake Forest University.(November 2014)Godel’s Incompleteness Theorems: Exploring the Holes in Mathematical Logic, SeniorSeminar, Berry College. (March 2012)
Teaching ExperienceSpring 2015 Teaching Assistant, Linear AlgebraFall 2014 Private Tutor, Single Variable CalculusFall 2014 Teaching Assistant, Discrete MathematicsSpring 2014 Teaching Assistant, Multivariable CalculusFall 2013 Teaching Assistant, Single Variable Calculus
2013-Present Math Center Tutor, Wake Forest University• Tutored: Single/Multivariable Calculus, Discrete Mathematics, Linear Algebra, Or-
dinary Differential Equations, Tier 1/Tier 2 Graduate Level Abstract Algebra, andGraduate Level Linear Algebra.
2012-2013 Math Tutor, Academic Support Center at Berry College2011-2013 Math Lab Tutor, Berry College Math Department
Honors and Awards2014 Outstanding Graduate Student Award2013 Barton Mathematics Award
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Graduate Coursework
q Group Theoryq Module Theoryq Galois Theoryq Representation Theoryq Homological Algebra
q Real Analysisq Functional Analysisq General Topologyq Probability Theoryq Differential Geometry
Research ExperienceMasters Thesis at Wake Forest University – April 2014 to May 2015
Title: Noncommutative Complete Isolated SingularitiesAdvisor: Dr. Frank MooreWe studied completions of noncommutative graded isolated sin-gularities. In particular, we proved the existence of almost splitsequences in the category gradeable modules over such a comple-tion.
Clubs and Organizations2014-present Pi Mu Epsilon Mathematics Honor Society2013-present Wake Forest Math Club2012 Berry College Putnam Exam Team (Score of 11)2010-2013 Dead Poets Society, Berry College Chapter
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