Constructing Endomorphism Rings of Large Finite Global Dimension by Ali Mousavidehshikh A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2016 by Ali Mousavidehshikh
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Constructing Endomorphism Rings of Large Finite GlobalDimension
by
Ali Mousavidehshikh
A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics
Notice that if R1 = k[[t5, t8, t11, t12, t14]], then Λ(R) = Λ(R1) but R 6= R1.
2.2 The Construction
Suppose H is a numerical semigroup with generators α1, α2, ..., αs, F (α1, α2, ..., αs) > −1,
and let R1 be the ring of formal power series associated to H. Since R1 6= R̃1 = k[[t]], we
have R1 ( EndR1(m1) ⊆ R̃1 (Theorem 1.1.1). Moreover, m1 contains a non-zero divisor
(Proposition 2.2.1), and EndR1(m1) embeds naturally into R1 (by sending f to f(a)/a,
which is independent of the non-zero divisor a ∈ m1). It is well known that in fact
EndR1(m1) ⊆ R̃1. Furthermore, it is easy to see that EndR1(m1) is itself a ring of formal
power series. Let R2 be any ring of formal power series over k that properly contains
R1 and is contained in EndR1(m1). Notice that R2 is a local Noetherian ring of (Krull)
dimension 1. If R2 = k[[t]], then R2 = EndR1(m1) = k[[t]] in which case we define
M := R1 ⊕R2, E := EndR1(M)
If R2 6= k[[t]], pick R3 such that R2 ( R3 ⊆ EndR1(m2) ⊆ k[[t]] (this is possible by
Theorem 1.1.1). If R3 = k[[t]], define
M := R1 ⊕R2 ⊕R3, E := EndR1(M)
Chapter 2. Main Objects and Tools 16
Notice that R1 ( R2 ( R3 = k[[t]]. If R3 6= k[[t]], repeat the process to obtain R4, and
continue in this fashion. Since R1 is missing only finitely many powers of t there exists
an l such that Rl = R̃1 = k[[t]]. Hence, we have constructed an ascending chain of rings
R1 ( R2 ( R3 ( ... ( Rl−1 ( Rl = k[[t]]
Let
M =l⊕
i=1
Ri, E = EndR1(M)
Notice that k[[t]] = Rl = EndR1(ml−1).
Proposition 2.2.1. Let (R,m, k) be a reduced local Noetherian ring with dim(R) = 1.
Then, m * Z(R) (the set of zero divisors of R).
Proof. Suppose not. Then
m ⊆ Z(R) =⋃
p minimal prime
p (since R is reduced).
By prime avoidance we have m ⊆ p for some minimal prime ideal. In particular, m =
p ⇒ dim(R) = ht(m) = ht(p) = 0, a contradiction (since dim(R) = 1). Therefore,
∃x ∈ m such that x /∈ Z(R).
Proposition 2.2.2. gl. dim(E) ≤ l (see [6] or [7] example 2.2.3(2)).
In one way our construction is more restrictive then the one built in [11]. More specifically,
the rings Ri and EndR1(mi) are always local Noetherian rings of (Krull) dimension 1.
However, it is also less restrictive since we only require Ri+1 ⊆ EndR1(mi). From here
on when we say an ascending chain of rings we mean a chain of rings with the above
restrictions imposed on it.
Given a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
we can represent E as an l × l matrix. More specifically,
Eij = HomR1(Rj, Ri).
The (Jacobian) radical of E denoted by J(E), or rad(E) is the matrix with the following
Chapter 2. Main Objects and Tools 17
entries (see [21]):
(J(E))ij =
Eij if i 6= j
mi if i = j.
Since R1 is a complete local noetherian commutative ring and E is a finitely generated
R-module, Theorem 1.1.4 implies that the right indecomposable projective modules of
E are the matrices Pi = eiE, where ei is the l × l matrix with 1 in the ii-th entry and
zero everywhere else. We identify Pi with its non-zero row. That is, Pi is the i-th row
in E (since all other rows are zero’s). Furthermore, the simple E-modules are Si = eiD,
where D is the l × l diagonal matrix with k as its diagonal entries. We identify Si with
its non-zero row (as we did for the projective modules), that is, Si is the row matrix with
k in its i-th entry and zero everywhere else. Since R1 is in the center of E, to compute
the global dimension of E it suffices to compute the projective dimension of the simple
modules (Theorem 1.1.3).
Lemma 2.2.3. The category of finitely generated projective E-modules is a Krull-Remak-
Schmidt category.
Proof. By Theorem 1.1.3 every simple E-module has a projective cover, and Proposition
1.4.4 completes the proof.
Lemma 2.2.4. Given a simple E-module S, the objects in the projective resolution of S
are isomorphic to a finite direct sum of indecomposable objects (each of which is obviously
projective).
Proof. This follows from example 1.4.5(b) and Theorem 1.1.4.
Example 2.2.5. Let R1 = k[[t3, t4, t5]], R2 = k[[t2.t3]], R3 = k[[t]], then
E =
R1 t3R3 t3R3
R2 R2 t2R3
R3 R3 R3
Notice that Pi = eiA is a 3 × 3 matrix, for i = 1, 2, 3. But as we mentioned, for each i
we identify Pi with its non-zero row. For example,
P1 =
R1 t3R3 t3R3
0 0 0
0 0 0
, S1 =
k 0 0
0 0 0
0 0 0
Chapter 2. Main Objects and Tools 18
In this case we simply write
P1 =(R1 t3R3 t3R3
), S1 =
(k 0 0
)as a row.
Notice that the number of simple and indecomposable projective E-modules is l. We say
that Pi is the projective module, and Si is the simple module associated to Ri.
Notation 2.2.6. Given 1 ≤ i ≤ l, if Pi is a projective E-module associated to the ring
Ri written in row notation with the zero rows taken out (example 2.2.5), we define
Eij := (Pi)j
Recall that
(J(E))ij =
Eij if j 6= i
mi if j = i.
The Jacobian radical of Pi, written in row notation with the zero rows taken out is given
by (see [21])
(J(Pi))j =
(Pi)j if j 6= i
mi if j = i.
Given a ring Ri in a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
Chapter 2. Main Objects and Tools 19
with Γ(Ri) = {β1, β2, β3, ..., βr} (where β1 < ... < βr = C(Ri)), we define
Ri,0 = Ri = lead{0, β1, β2, ..., βr}
Ri,1 = Ri/k = mi = lead{β1, β2, ..., βr}
Ri,2 = lead{β2, ..., βr}
Ri,3 = lead{β3, ..., βr}
.
.
.
Ri,r = lead{βr} = tC(Ri)Rl
It should be noted that in general, there is no connection between Ri,j and Eij. However,
the notation Ri,j is used extensively in the computation portions of this thesis.
Example 2.2.7. Let R1 = k[[t3, t5, t7]], R2 = k[[t3, t4, t5]], R3 = k[[t]]. Then, C(R1) = 5
and Γ(R1) = {3, 5}. In particular,
R1,0 = R1 = lead{0, 3, 5}
R1,1 = m1 = lead{3, 5}
R1,2 = lead{5} = t5R3
Recall that a finitely generated R-module M is torsion-free provided the natural
map M →M ⊗R R is injective, where R is the total quotient ring of R.
Definition 2.2.8. Suppose R and S are local, Noetherian, commutative, reduced rings,
that are also complete with respect to their Jacobian radicals, respectively, and have
Krull dimension 1. We say that S is a birational extension of R provided R ⊆ S and S
is a finitely generated R-module contained in the total quotient ring R of R.
Notice that if S is a birational extension of R, then every finitely generated torsion-
free S-module is a finitely generated torsion-free R-module, but not vice versa. The
following lemma follows by clearing denominators.
Lemma 2.2.9. Suppose S is a birational extension of R. Let C and D be finitely gen-
erated torsion-free S-modules. Then HomR(C,D) = HomS(C,D). Furthermore, if M is
a finitely generated torsion-free R-module, and f : C →M is an R-linear map, then the
image of f is an S-module.
Chapter 2. Main Objects and Tools 20
If R is a ring of formal power series associated to a numerical semigroup H, then R is
local, commutative, Noetherian, reduced, complete with respect to its Jacobian radical,
and has Krull dimension 1.
Lemma 2.2.10. Given a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
for any 1 ≤ a ≤ i ≤ l we have HomR1(Ra, Ri) = HomRa(Ra, Ri) = Ri. In particular,
Eij = HomR1(Rj, Ri) = Ri for j ≤ i
Proof. Notice that Ra is a birational extension of R1. Furthermore, for 1 ≤ a ≤ i ≤ l,
Ra and Ri are finitely generated torsion-free Ra-modules. The result follows by Lemma
2.2.9 and the fact that HomR(R,N) = N for any R-module N .
Notation 2.2.11. By Theorem 1.1.4, Pi → Si → 0 is a projective cover. We denote the
map from Pi → Si by πi. In particular, (Pi, πi) is a projective cover for Si.
Notice that
(Pi)j =
Ri if 1 ≤ j ≤ i
HomR1(Rj, Ri) if i+ 1 ≤ j ≤ l
(Si)j =
0 if i 6= j
k if i = j
We can give an explicit description of the map πi. Let (πi)j : (Pi)j → (Si)j. Then,
(πi)j =
ξi if i = j
0 if i 6= j
where ξi : Ri → Ri/mi is the quotient map. It follows that ker πi = J(Pi) for 1 ≤ i ≤ l.
Lemma 2.2.12. Given a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
if 1 ≤ a ≤ i < b ≤ l, then HomR1(Ra, Ri) ) HomR1(Rb, Ri).
Chapter 2. Main Objects and Tools 21
Proof. Given 1 ≤ a < b ≤ l we have
Ra ( Rb =⇒ HomR1(Ra, Ri) ⊇ HomR1(Rb, Ri) for any i
Making the additional assumption 1 ≤ a ≤ i < b ≤ l, we have k ∩HomR1(Rb, Ri) = {0},where k is the base field of R1 (in fact, of all the Ri’s in our chain) and is identified with
the set consisting of scalar multiplication. Lemma 2.2.10 yields
HomR1(Ra, Ri) = Ri ⊇ k
Hence, HomR1(Ra, Ri) ) HomR1(Rb, Ri) for 1 ≤ a ≤ i < b ≤ l.
Lemma 2.2.13. Given a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
if 1 ≤ i < j ≤ l, then HomR1(Rj, Ri) = HomR1(Rj,mi).
Proof. Since mi ( Ri we have HomR1(Rj,mi) ⊆ HomR1(Rj, Ri) for all 1 ≤ j ≤ l. When
i < j ≤ l, then any non-zero map from Rj to Ri cannot send anything to non-zero scalars
(since k ∩HomR1(Rj, Ri) = {0} by Lemma 2.2.12). In particular, every non-zero map in
HomR1(Rj, Ri) is actually a map from Rj to mi. Since the zero map is also a map from
Rj to mi the result follows.
Proposition 2.2.14. Given a chain of ascending rings
R1 ( R2 ( ... ( Rl = k[[t]]
fix an i with 1 ≤ i ≤ l. If mi = tαRj for some α ≥ 0 and 1 ≤ j ≤ l, then
(a) α = e(Ri).
(b) Rj = EndR1(mi).
(c) i ≤ j ≤ l.
(d) If i 6= l, then i < j ≤ l.
(e) If i 6= l, then mi = HomR1(Rj, Ri).
(f) If i 6= l, then for all a with i < a ≤ j, we have HomR1(Ra, Ri) = mi.
Proof. (a) α = e(tαRj) = e(mi) = e(Ri).
(b)
EndR1(mi) = HomR1(mi,mi) = HomR1(te(Ri)Rj, t
e(Ri)Rj) = HomR1(Rj, Rj) = Rj.
Chapter 2. Main Objects and Tools 22
(c) Since Rj = EndR1(mi) ⊇ Ri, we have i ≤ j ≤ l.
(d) If i 6= l, then Rj = EndR1(mi) ) Ri (by construction of the chain), that is i < j ≤ l.
(e) Since i 6= l, part (d) yields i < j ≤ l. By Lemmas 2.2.10 and 2.2.13 we have
HomR1(Rj, Ri) = HomR1(Rj,mi)
= HomR1(Rj, te(Ri)Rj)
= te(Ri) HomR1(Rj, Rj)
= te(Ri)Rj
= mi
(f) If i 6= l, then i < j ≤ l by part (d). For any i < a ≤ j we have Ri ( Ra ⊆ Rj. Lemma
2.2.12 yields
HomR1(Ri, Ri) ) HomR1(Ra, Ri) ⊇ HomR1(Rj, Ri)
In particular, the above chain of inclusions, part (e), and Lemma 2.2.10 yield
mi = HomR1(Rj, Ri) ⊆ HomR1(Ra, Ri) ( HomR1(Ri, Ri) = Ri
Maximality of mi implies that mi = HomR1(Ra, Ri) for all a = i+ 1, . . . , j.
Example 2.2.15. Let
R1 = lead {0, 3, 4, 6}
R2 = EndR1(m1) = lead {0, 3}
R3 = lead {0, 2}
R4 = EndR1(m3) = k[[t]]
Notice that m1 6= t3R2. That is, the converse of part (b) in Proposition 2.2.14 is false.
Lemma 2.2.16. Given an ascending chain of rings
R1 ( R2 ( · · · ( Rl = k[[t]]
then
(a) e(Rl) = C(Rl) = 1, and e(Rl−1) = C(Rl−1).
(b) dimk(Rl/R1) = g(R1)
Chapter 2. Main Objects and Tools 23
Proof. (a) Since Rl = k[[t]] we have e(R1) = 1 = C(R1). Moreover,
Rl = EndR1(ml−1) ⇒ ml−1 = lead {e(Rl−1)}
⇒ Rl−1 = lead {0, e(Rl−1)}
⇒ e(Rl−1) = C(Rl−1)
(b) Let H be the numerical semigroup associated to R1. Suppose b1, b2, ..., br are the
natural numbers missing from H, then Rl/R1 has the set {tbi + R1 : i = 1, 2, . . . , r} as
its basis over k. Hence, dimk(Rl/R1) = g(R1).
Lemma 2.2.17. Let
R1 ( R2 ( ... ( Rl = k[[t]]
be an ascending chain of rings. Then
(a) a1(R1) = e(R1)− 1
(b) 1 ≤ l ≤ g(R1) + 1
(c) e(Ri) = 1⇔ C(Ri) = 0⇔ Ri = k[[t]]⇔ mi = tRi
Proof. (a) a1(R1) = β1 − 1 = e(R1)− 1.
(b) Notice that l = 1 if and only if R1 = k[[t]]. If R1 6= k[[t]], then l ≥ 2. Moreover,
g(R1) is the number of powers of t which are missing from R1, and we atleast put one
power of t in at each stage in our construction, thus, l ≤ g(R1) + 1.
(c) e(Ri) = 1⇔ t ∈ Ri ⇔ C(Ri) = 1⇔ Ri = k[[t]]⇔ mi = tRi = lead {0, 1}.
2.3 A Presentation of a Ring, an Image and Kernel
of a Map
Our aim in this section is to give an elegant way of determining the image and kernel of
a map. We begin with a useful method for describing a ring associated to a numerical
semigroup.
Definition 2.3.1. A presentation of a ring of formal power series R associated to a
numerical semigroup H is a table with the top row consisting of the non-negative integers
upto C(R), and in the bottom row we place a x under a natural number n if tn ∈ R and
a zero if tn /∈ R. We will sometimes shorten the top row when necessary by omitting
the non-negative integers n for which tn /∈ R. There is also an alternate way to shorten
Chapter 2. Main Objects and Tools 24
the table and it is as follows: given integers a, b with a ≤ b, we will use the shorthand
notation a . . . b in the top row to mean all the integers between a and b. We put an x in
the second row if those powers are in R and a zero if they are not.
Since every ring of formal power series R associated to a numerical semigroup H is
uniquely determined by Γ(R), each such ring gives rise to a unique presentation and vice
versa.
Example 2.3.2. Let R = k[[t4, t11, t13, t14]]. Then, R has the following presentation:
Powers of t 0 1 2 3 4 5 6 7 8 9 10 11
R x 0 0 0 x 0 0 0 x 0 0 x
The two shorter versions are
Powers of t 0 4 8 11
R x x x x
Powers of t 0 . . . 4 . . . 8 . . . 11
R x 0 x 0 x 0 x
The power of these presentations is that they enable us to find the image and kernel
is a projective resolution for S1, and it is minimal by Proposition 2.3.6, completing the
proof.
In example 2.2.15, m2 = t3R4 , however, pdE(S2) ≥ 2 by Proposition 2.3.6. In
particular, Proposition 2.3.7 fails to hold if we replace S1 by Si for some 1 < i ≤ l.
2.4 Structure of E
We now turn our attention to the entries of the matrix E and how the entries in a given
row are related to the entries in the rows preceding it and succeeding it.
Lemma 2.4.1. Let
R1 ( R2 ( ... ( Rl = k[[t]]
be an ascending chain of rings. The entries of the matrix E satisfy the following proper-
ties;
(a) If 1 ≤ j ≤ i ≤ l,, then Eij = Ri.
(b) Eij ⊇ Ei(j+1) (that is, there is a descending chain of rings or modules as we go across
a given row. Note that Eij could equal Ei(j+1), for example, E21 = E22 = R2 always).
Furthermore, Eij ⊆ E(i+1)j (there is an ascending chain of rings or modules as we go
down a given column).
(c) If 1 ≤ i ≤ l − 1, then Eii ) Ei(i+1).
(d) HomR1(EndR1(ml), Rl) = Rl. If i 6= l, then HomR1(EndR1(mi), Ri) = mi.
(e) If 1 ≤ i ≤ l − 1, then Ei(i+1) = mi.
(f) Eil = tC(Ri)Rl for 1 ≤ i ≤ l − 1 and Ell = Rl
Proof. (a) This follows from Lemma 2.2.10.
(b) This follows from the fact that HomR1(�, Ri) is a contravariant functor and HomR1(Ri,�)
is a covariant functor.
Chapter 2. Main Objects and Tools 33
(c) This is a consequence of part (b) and Lemma 2.2.12.
(d) The first part follows from part (a) and the fact that EndR1(ml) = Rl. If i 6= l then
by Theorem 1.1.1, Ri ( EndR1(mi). That is, k ∩ HomR1(EndR1(mi), Ri) = {0} (where
k is the base field of R1, in fact, of all the Ri, and it is identified with scalar multiplica-
tion) and Ri = HomR1(Ri, Ri) ) HomR1(EndR1(mi), Ri). Given a non-negative integer
b, tb ∈ EndR1(mi) if and only if tbtx = tb+x ∈ mi for any tx ∈ mi, in particular,
HomR1(EndR1(mi), Ri) ⊇ mi.
Thus,
Ri ) HomR1(EndR1(mi), Ri) ⊇ mi,
the maximality of mi gives the desired result.
(e) Given 1 ≤ i ≤ l − 1, we have Ri ( Ri+1 ⊆ EndR1(mi). Parts (a), (c) and (d) imply
that
Ri = Eii ) Ei(i+1) ⊇ HomR1(EndR1(mi), Ri) = mi.
Maximality of mi gives Ei(i+1) = mi.
(f) Given 1 ≤ i ≤ l − 1, since Rl = k[[t]]
Eil = HomR1(Rl, Ri) = tC(Ri)Rl
The second part follows form part (a).
The preceding proposition gives us a very nice description of the entries of E on its
main diagonal, below it, the entries right above the main diagonal (Ei(i+1)), and the
entries in column l. In particular,
E =
R1 m1 ∗ ∗ ∗ ∗ ∗ ∗ tC(R1)Rl
R2 R2 m2 ∗ ∗ ∗ ∗ ∗ tC(R2)Rl
R3 R3 R3 m3 ∗ ∗ ∗ ∗ tC(R3)Rl
R4 R4 R4 R4 m4 ∗ ∗ ∗ tC(R4)Rl
......
......
. . . . . . ∗ ∗ ...
Rl−1 Rl−1 Rl−1 Rl−1 . . . tC(Rl−1)Rl
Rl Rl Rl Rl . . . Rl
Chapter 2. Main Objects and Tools 34
The ∗ entries are unknown and must be computed on a base by base cases. Notice that
ml−1 = tC(Rl−1)Rl by Lemma 2.2.16(a). We conclude this section by looking at the entries
of E when the ring R has the property e(R) = C(R).
Lemma 2.4.2. Let
R1 ( R2 ( ... ( Rl = k[[t]]
be an ascending chain of rings, and suppose e(Ri) = C(Ri) for some 1 ≤ i ≤ l. Then
Eij =
Ri if 1 ≤ j ≤ i
te(Ri)Rl if i+ 1 ≤ j ≤ l
Proof. By Proposition 2.4.1 Eij = Ri for 1 ≤ j ≤ i. Moreover, e(Ri) = C(Ri) implies
that
Ri = lead {0, e(Ri)}
Then, for any j with i+ 1 ≤ j ≤ l
HomR1(Rj, Ri) = lead {e(Ri)} = te(Ri)Rl
Proposition 2.4.3. Let
R1 ( R2 ( ... ( Rl = k[[t]]
be an ascending chain of rings. Then, pdE(Sl) = 2.
Proof. By Lemma 2.2.16(a) we have Rl−1 = lead {e(Rl−1)} with e(Rl−1) = C(Rl−1). By
Lemma 2.4.2
E(l−1)j =
Ri if 1 ≤ j ≤ l − 1
te(Rl−1)Rl if j = l
Let
∆ = (1, t)
Pl−1⊕Pl
Chapter 2. Main Objects and Tools 35
The image of (1, t) has the following presentation:
Value of j ∆j 0 1 . . . e(Rl−1) . . .
1 ≤ j ≤ l − 1 Rl−1 x 0 0 x x
⊕tRl 0 x x x x
j = l Rl−1 0 0 0 x x
⊕tRl 0 x x x x
That is, Im(1, t) = ker πl = J(Pl). The kernel of (1, t) has the following presentation:
Value of j e(Rl−1)− 1 e(Rl−1) . . .
1 ≤ j ≤ l 0 x x
x x x
Thus,
ker(1, t) =
(te(Rl−1)
−te(Rl−1)−1
)Pl
Hence,
0 Sl Pl
Pl−1
⊕Pl
Pl 0πl (1, t)
(te(Rl−1)
−te(Rl−1)−1
)
is a projective resolution for Sl. Furthermore, it is minimal by Proposition 2.3.6, com-
pleting the proof.
2.5 Family of Starting Rings
In this section we construct a family of starting rings recursively and state some of the
properties of these starting rings. These rings will play a fundamental role in constructing
endomorphism rings of large global dimension.
Chapter 2. Main Objects and Tools 36
Definition 2.5.1. Let n ≥ 6 be an even integer. Define
R11 := lead
{0, n,
3n
2
}and
R21 := lead
{0, n,
3n
2, C(R2
1)
}
where3n
2+ 2 ≤ C(R2
1) ≤ 2n. For each i ≥ 3, let
Ri1 := lead
{0,jn
2, C(Ri
1) : j = 2, 3, ..., i+ 1
}
where C(Ri1) = C(Ri−1
1 ) +n
2for i ≥ 3.
The following results are a direct consequence of our construction, we record them
here for future reference.
Lemma 2.5.2. G(Ri1) = i+ 1, e(Ri
1) = n for i = 1, 2, 3, ...
Lemma 2.5.3. C(Ri1) = C(R2
1) + (i− 2)n
2for all i ≥ 2.
Lemma 2.5.4. Λ(R11) = {n}, and Λ(Ri
1) =
{n,
3n
2
}for all i ≥ 2.
Lemma 2.5.5.
a1(R11) = n− 1, a2(R
11) =
n
2− 1
and for i ≥ 2,
aj(Ri1) =
n− 1 if j = 1n
2− 1 for 2 ≤ j ≤ i
C(Ri1)− (i+ 1)
n
2− 1 if j = i+ 1
Notice that C(Ri1) = C(Ri−1
1 ) +n
2for i ≥ 3. This implies that
C(Ri1)− (i+ 1)
n
2− 1 = C(R2
1)−3n
2− 1 for i ≥ 3
Chapter 2. Main Objects and Tools 37
Notice the above equality is also true when i = 2. Hence, for i ≥ 2 we have
aj(Ri1) =
n− 1 if j = 1n
2− 1 for 2 ≤ j ≤ i
C(R21)−
3n
2− 1 if j = i+ 1
Lemma 2.5.6. Γ(R11) =
{n,
3n
2
}, and for i ≥ 2 we have Γ(Ri
1) ={βi1, β
i2, ..., β
ii+1
}where
βij =
n+ (j − 1)n
2if 1 ≤ j ≤ i
C(Ri1) if j = i+ 1
Definition 2.5.7. Suppose
R1 ( R2 ( . . . ( Rl = k[[t]]
is a chain of ascending rings. Fix i, and suppose Γ(Ri) = {β1, β2, ..., βr} with β1 < β2 <
... < βr = C(Ri). Let γ1 be the number of positive powers of t between tC(Ri)−β1 and
tC(Ri) (inclusive) which are missing from Ri. Let γ2 be the number of positive powers of
t between tC(Ri)−β2 and tC(Ri)−β1−1 (inclusive) which are missing from Ri, continuing this
process, this stops at γr, the number of positive powers of t between tC(Ri)−βr = t0 and
tC(Ri)−βr−1−1 (inclusive) which are missing from Ri. Define
Φ(Ri) = {γj | j = 1, 2, ..., r}
Lemma 2.5.8. Φ(R11) = {γ11 , γ12} =
{n− 1,
n
2− 1}
, and for i ≥ 2 we have Φ(Ri1) ={
γi1, γi2, ..., γ
ii+1
}where
γij =
n− 2 if j = 1n
2− 1 if 2 ≤ j ≤ i− 1
n
2if j = i
C(Ri1)− βii − 1 if j = i+ 1
Notice that
C(Ri1)− βii − 1 = C(R2
1) + (i− 2)n
2−(n+ (i− 1)
n
2
)− 1 = C(R2
1)−3n
2− 1
Chapter 2. Main Objects and Tools 38
That is, for i ≥ 2 we have
γij =
n− 2 if j = 1n
2− 1 if 2 ≤ j ≤ i− 1
n
2if j = i
C(R21)−
3n
2− 1 if j = i+ 1
Suppose
R1 ( R2 ( . . . ( Rl = k[[t]]
is a chain of ascending rings. Given 1 ≤ i ≤ l, if e(Ri) < C(Ri) then Λ(Ri) is not empty.
Let
Λ(Ri) = {α1, ..., αs}, Γ(Ri) = {β1, ..., βr},
where the elements are listed in ascending order. Since Λ(Ri) ⊆ Γ(Ri), for each αa ∈Λ(Ri) there exists a βja ∈ Γ(Ri) such that αa = βja . Since α1 = β1 we have a = 1 = j1.
However, for a ≥ 2 this need not be the case. This leads us to the following useful
definition.
Definition 2.5.9. Given 1 ≤ i ≤ l, let
Λ(Ri) = {α1, ..., αs}
Γ(Ri) = {β1, ..., βr}
Φ(Ri) = {γ1, ..., γr}
For each a ∈ {1, 2, ..., s}, define
λa = i+
ja∑h=1
γh
Since j1 = 1 we have λ1 = i+ γ1. We define
χ(Ri) = {λ1, ..., λs}
If e(Ri) = C(Ri), we define χ(Ri) = ∅. If follows that |χ(Ri)| = |Λ(Ri)|.
Chapter 2. Main Objects and Tools 39
Example 2.5.10. Let R1 = k[[t5, t11, t14, t17, t18]], then
C(R1) = 14
Λ(R1) = {5, 11}
Γ(R1) = {5, 10, 11, 14}
Φ(R1) = {γ1, γ2, γ3, γ4} = {3, 4, 1, 2}
Using the notation above we have α1 = 5 = β1, α2 = β3, that is, a = 1 = j1 and j2 = 3.
Moreover,
λ1 = 1 + γ1 = 4
λ2 = 1 +3∑
h=1
γh = 1 + 3 + 4 + 1 = 9
which yields χ(R1) = {4, 9}.
Lemma 2.5.11. χ(R11) = {λ11} = {γ11 + 1} = {(n− 1) + 1 = n}, and for i ≥ 2 we have
χ(Ri1) = {λi1, λi2} where
λij =
1 + γi1 = n− 1 if j = 1
1 + γi1 + γi2 if j = 2
Moreover,
λ2j =
1 + γ21 = n− 1 if j = 1
1 + γ21 + γ22 =3n
2− 1 if j = 2
and for i ≥ 3
λij =
1 + γi1 = n− 1 if j = 1
1 + γi1 + γi2 =3n
2− 2 if j = 2
2.6 The symbol d e
In this section we focus on the symbol d e and its properties.
Definition 2.6.1. Let X be a 1 × l row matrix, and Xj be its j-th entry. Given an
Chapter 2. Main Objects and Tools 40
integer a ≥ 0, we define Xdae to be a 1× (l + a) row matrix with the following entries:
(Xdae)j =
X1 if 1 ≤ j ≤ a
Xj−a if a+ 1 ≤ j ≤ l + a
That is, we are putting a string of X1’s (in fact, a of them) at the beginning of X to
obtain Xdae. Notice that
Xd0e = X
Given integers a, b ≥ 0,
(Xdae)dbe = Xda+ be = (Xdbe)dae
Notation 2.6.2. Let p ∈ N, and X i be a 1× l row matrix for i = 1, 2, ..., p. We define
X =
p⊕i=1
X i =
X1
X2
...
Xp
It follows that X is a p× l matrix.
Definition 2.6.3. Given
X =
p⊕i=1
X i
where X i are 1× l row matrices, we define
Xdae =
p⊕i=1
X idae =
X1daeX2dae
...
Xpdae
Given an integer a ≥ 0, if f : X → Y is a map given in matrix form, where
X =
p⊕i=1
X i, Y =
p⊕i=1
Y i
Chapter 2. Main Objects and Tools 41
and X i, Y i are 1 × l row matrices, then f has p columns. Since the number of rows of
Xdae, Y dae is also p, we define fdae : Xdae → Y dae by setting fdae = f (the only thing
we have done is change the domain and co-domain). We abuse notation and we use f in
place of fdae.
Let
R1 ( R2 ( R3 ( ... ( Rl−1 ( Rl = k[[t]]
be an ascending chain constructed in section 2.2. The maps πi : Pi → Si are not given
by a matrix . We define
πidae : Pidae → Sidae
as follows;
(π1dae)j =
ξi if 1 ≤ j ≤ a+ 1
0 if a+ 2 ≤ j ≤ l + a
and for 2 ≤ i ≤ l,
(πidae)j =
ξi if j = i+ a
0 if j 6= i+ a
where ξi : Ri → Ri/mi is the quotient map, Pi, Si are 1× l row matrices, and
(S1dae)j =
k if 1 ≤ j ≤ a+ 1
0 if a+ 2 ≤ j ≤ l + a
(P1dae)j =
R1 if 1 ≤ j ≤ a+ 1
(P1)j−a if a+ 2 ≤ j ≤ l + a,
for 2 ≤ i ≤ l,
(Sidae)j =
k if j = i+ a
0 if j 6= i+ a
(Pidae)j =
Ri if 1 ≤ j ≤ a+ 1
(Pi)j−a if a+ 2 ≤ j ≤ l + a
Chapter 2. Main Objects and Tools 42
Example 2.6.4. Suppose R1 = k[[t3, t4, t5]], R2 = k[[t2, t3]], R3 = k[[t]]. Then
E =
R1 t3R3 t3R3
R2 R2 t2R3
R3 R3 R3
and
S1d2e =(k k k 0 0
)S2d2e =
(0 0 0 k 0
)P1d2e =
(R1 R1 R1 t3R3 t3R3
)= (P1d1e)d1e
P2d2e =(R2 R2 R2 R2 t2R3)
)P1
⊕P2
d2e =
(P1d2eP2d2e
)=
(R1 R1 R1 t3R3 t3R3
R2 R2 R2 R2 t2R3
)=
P1d2e⊕
P2d2e
Moreover,
π1d2e : P1d2e → S1d2e and (π1d2e)j =
ξ1 if 1 ≤ j ≤ 3
0 if 4 ≤ j ≤ 5
π2d2e : P2d2e → S2d2e and (π2d2e)j =
ξ2 if j = 4
0 if j 6= 4
The following two results are an immediate consequence of our definitions above and
we record them here for future reference.
Lemma 2.6.5. Let p ∈ N. Suppose
X =
p⊕i=1
X i, Y =
p⊕i=1
Y i, Z =
p⊕i=1
Zi
where X i, Y i, Zi are 1× l rows. If
Xdae f→ Y dae g→ Zdae
Chapter 2. Main Objects and Tools 43
is exact at Y dae (i.e. ker g = Im(f)) for some a ∈ N0, then
Xdbe f→ Y dbe g→ Zdbe
is exact at Y dbe for every b ∈ N0.
Lemma 2.6.6. Using the notation in Proposition 2.6.5,
Im(X
f→ Y)⊆ J(Y )
if and only if
Im(Xdae f−→ Y dae
)⊆ J(Y dae) for any a ∈ N0
Chapter 3
“Lazy” Construction
In this chapter we concentrate on a construction of our chain which maximizes its length,
called the “lazy” construction. More specifically, in section 3.1 we give the precise defi-
nition of this construction and introduce some of the necessary notation. In section 3.2
we compute the global dimension of endomorphism rings for specific starting rings. In
Section 3.3 we give some of the results which are a consequence of this construction.
Section 3.4 focuses on computing the global dimension of endomorphism rings when the
length of the chain is small. In section 3.5 we combine this construction with the family
of starting rings constructed in section 2.5 to obtain a set of endomorphism rings whose
global dimensions are arbitrarily large (but finite).
3.1 The Construction
Given a numerical semigroup H, let R be the ring of formal power series associated to
H. Then, H has a minimal generating set, say {α1, α2, ..., αs} written in ascending order.
That is,
H = 〈α1, α2, ..., αs〉 ⇔ R = k[[tα1 , tα2 , ..., tαs ]]
Given a non-negative integer b with b 6= αi, we define
H[[b]] = 〈α1, α2, ..., αs, b〉
Since gcd(α1, α2, ..., αs) = 1 implies that gcd(α1, α2, ..., αs, b) = 1, the set H[[b]] is a
numerical semigroup. We define R[[tb]] to be the ring of formal power series associated
to H[[b]]. It should be noted that H ⊆ H[[b]], and equality holds if and only if b ∈ H.
44
Chapter 3. “Lazy” Construction 45
We are now in position to describe the “lazy” construction.
Let H be a numerical semigroup with minimal generating set {α1, α2, ..., αs}, and
F (α1, α2, ..., αs) ≥ 1 (i.e. 1 /∈ H). Let R1 be the ring of formal power series associated
to H. We define
Ri = Ri−1[[tC(Ri−1)−1]] for i ≥ 2
Since only finitely many powers of t are missing from R1, there exists an l ≥ 2 such that
Rl = k[[t]]. In particular, we have constructed the following ascending chain of rings:
R1 ( R2 ( · · · ( Rl = k[[t]] (3.1)
Let
M :=
(l⊕
i=1
Ri
), E := EndR1(M)
We say the ascending chain in (3.1), M , and E are constructed via the “lazy” construc-
tion.
For 1 ≤ i ≤ l − 1 we have Ri 6= k[[t]]. Lemma 2.1.3(e) yields tC(Ri)−1 /∈ Ri and
tx ∈ Ri for all x ≥ C(Ri). In particular, Ri ( Ri+1 ⊆ EndR1(mi). Hence, gl. dim(E) ≤ l
(Proposition 2.2.2).
Lemma 3.1.1. Let
R1 ( R2 ( ... ( Rl
be an ascending chain of ring constructed via the ”lazy” construction. Then,
(a) 1 ≤ l = g(R1) + 1, e(Rl) = C(Rl) = 1. If l ≥ 2, then e(Rl−1) = C(Rl−1) = 2.
(b) e(Ri) ≤ e(Ri−1) for i = 2, 3, ..., l. Moreover, if e(Ri−1) = C(Ri−1) then e(Ri) =
e(Ri−1)− 1.
(c) C(Ri) ≤ C(Ri−1)− 1 for i = 2, 3, ..., l. Moreover, if e(Ri−1) = C(Ri−1), then C(Ri) =
C(Ri−1)− 1.
(d) If e(Ri) = C(Ri) for some i = 1, 2, ..., l, then e(Rj) = C(Rj) for all i ≤ j ≤ l.
Proof. (a) Notice that l = 1⇔ R1 = k[[t]]. If R1 6= k[[t]], then l ≥ 2. Since g(R1) is the
number of powers of t which are missing from R1 and we put them in one at a time to
construct our chain, we have l = g(R1) + 1. Also, Rl = k[[t]] ⇒ e(Rl) = C(Rl) = 1. If
l ≥ 2, then Rl−1 = k[[t2, t3]]⇒ e(Rl−1) = C(Rl−1) = 2.
Chapter 3. “Lazy” Construction 46
(b) Since Ri−1 ( Ri we have e(Ri) ≤ e(Ri−1). If e(Ri−1) = C(Ri−1), then Ri−1 =
lead {0, e(Ri−1)} ⇒ Ri = lead {0, e(Ri−1)− 1} ⇒ e(Ri) = e(Ri−1)− 1.
(c) Since Ri−1 ( Ri we have C(Ri) ≤ C(Ri−1) − 1. If e(Ri−1) = C(Ri−1), then Ri−1 =
lead {0, C(Ri−1)} ⇒ Ri = lead {0, C(Ri−1)− 1} ⇒ C(Ri) = C(Ri−1)− 1.
(d) By part (b) e(Ri+1) = e(Ri)− 1, and by part (c) C(Ri+1) = C(Ri)− 1. In particular,
e(Ri+1) = e(Ri)− 1 = C(Ri)− 1 = C(Ri+1). A similar proof shows the result is true for
i+ 2, i+ 3, ..., l.
The projective modules under the lazy construction have a very nice description. We
state this as a lemma for future reference.
Lemma 3.1.2. Suppose
R1 ( R2 ( . . . ( Rl = k[[t]]
is a chain of ascending rings constructed via the “lazy” construction. Fix i, and let
Φ(Ri) = {γj | j = 1, 2, ..., r}, then the i-th projective module Pi has the following entries;
(Pi)j = Eij =
Ri,0 = Ri if 1 ≤ j ≤ i
Ri,1 = mi if i+ 1 ≤ j ≤ i+ γ1
Ri,2 if i+ γ1 + 1 ≤ j ≤ i+ γ1 + γ2·...
Ri,r = tC(Ri)Rl = c(Ri) if l − γr + 1 ≤ j ≤ l
Moreover, |Φ(Ri)| = |Γ(Ri)| = G(Ri) and
l = i+r∑j=1
γj
In particular,
l − γr + 1 = i+ 1 +r−1∑h=1
γh
If e(Ri) = C(Ri), then Φ(Ri) = {γ1 = C(Ri)− 1 = e(Ri)− 1}. More specifically, the
above formula for Pi coincides with Lemma 2.4.2.
Chapter 3. “Lazy” Construction 47
Example 3.1.3. Let R1 = k[[t5, t8, t17, t19]] = lead{0, 5, 8, 13, 15}, then
l = 11
C(R1) = 15
Γ(R1) = {5, 8, 10, 13, 15}
G(R1) = 5
γ1 = 3, γ2 = 2, γ3 = 1, γ4 = 3, γ5 = 1
The first row of E is
(P1)j = E1j =
R1,0 = R1 if j = 1
R1,1 = m1 if 2 ≤ j ≤ 4
R1,2 if 5 ≤ j ≤ 6
R1,3 if j = 7
R1,4 if 8 ≤ j ≤ 10
R1,5 = t15R11 if j = 11
3.2 Special Rings I
In this section we compute the global dimension for some special starting rings. All the
constructions in this section are via the lazy construction.
Lemma 3.2.1. Suppose
R1 = lead {0, n}
with n > 1. Then, gl. dim(E) = 2.
Proof. Let mi be the maximal ideal of Ri. Notice that l = n by Lemma 3.1.1(a), and
the rings in our chain are
Ri = lead {0, n− i+ 1} where 1 ≤ i ≤ n
Chapter 3. “Lazy” Construction 48
For a fixed i, where 1 ≤ i ≤ n− 1, we have
(Pi)j = Eij =
Ri = Ri,0 if 1 ≤ j ≤ i
tn−i+1Rn = Ri,1 if i+ 1 ≤ j ≤ n,
(Pn)j = Enj = Rn = Rn,0 for 1 ≤ j ≤ n
Moreover, Ri,1 = mi = tn−i+1Rn for i = 1, 2, ..., n. We compute the minimal projective
resolutions of Si. For i = 1 we have
(kerπ1)j =
tnRn if j = 1
(P1)j if j 6= 1= tnRn if 1 ≤ j ≤ n
In particular,
0 S1 P1 Pn 0π1 tn
is a projective resolution for S1, and it is minimal by Proposition 2.3.6. For i = 2, 3, ..., n,
let
∆i = (1, tn−i+1)
Pi−1⊕Pn
Then
Value of j (∆i)j 0 n− i+ 1 n− i+ 2 . . .
1 ≤ j ≤ i− 1 Ri−1,0 x 0 x x
⊕tn−i+1Rn,0 0 x x x
i ≤ j ≤ n Ri−1,1 0 0 x x
⊕tn−i+1Rn,0 0 x x x
That is,
∆i = (1, tn−i+1)
Pi−1⊕Pn
= J(Pi) = ker πi
Chapter 3. “Lazy” Construction 49
Moreover, ker(1, tn−i+1) has the following presentation:
Value of j 0 1 . . . n− i+ 1 n− i+ 2 . . .
1 ≤ j ≤ n 0 0 0 0 x x
0 x x x x x
Therefore,
0 Si Pi
Pi−1
⊕Pn
Pn 0πi (1, tn−i+1)
(tn−i+2
−t
)
is a projective resolutions for Si, and it is minimal by Proposition 2.3.6. The result
follows by Theorem 1.1.3.
An immediate consequence of Lemma 3.2.1 is: if e(R1) = C(R1), then gl. dim(E) =
2. Moreover, combining Lemma 3.1.1(d) with the proof in Lemma 3.2.1 shows that if
Ri−1 = lead {0, e(Ri−1)} for some 2 ≤ i ≤ l, then the minimal projective resolution of Si
is given by
0 Si Pi
Pi−1
⊕Pl
Pl 0πi (1, te(Ri))
(te(Ri−1)
−t
)
That is, pdE(Sj) = 2 for all j with i ≤ j ≤ l. This leads us to the following which we
present as a lemma for future reference.
Lemma 3.2.2. Suppose 2 ≤ i ≤ l with C(Ri−1) = e(Ri−1). Then,
(a) If e(R1) = C(R1) then gl. dim(E) = 2
(b) pdE(Sj) = 2 for all i ≤ j ≤ l.
(c) pdE(Sj) = 2 for z(R1) + 2 ≤ j ≤ l.
Proof. The only parts we need to prove is parts (c). This follows from the fact that
e(Rz(R1)+1)) = C(Rz(R1)+1)).
Lemma 3.2.3. Suppose b ∈ N, n > 1 with
R1 = lead {0, xn : x = 1, 2, ..., b}
Chapter 3. “Lazy” Construction 50
Then gl. dim(E) = 2.
Lemma 3.2.1 is a special case of this lemma (by setting b = 1).
Proof. The rings in our chain are
R2 = lead{0, xn, bn− 1 : x = 1, 2, ..., b− 1}
R3 = lead{0, xn, bn− 2 : x = 1, 2, ..., b− 1}...
Rb(n−1)+1 = lead {0, 1} = k[[t]]
with l = b(n− 1) + 1 by Lemma 3.1.1(a). For a fixed i with 1 ≤ i ≤ (b− 1)(n− 1) + 1,
The fifth case is when R1 = k[[t4, t5, t6, t8]]. In this case
E =
R1 R1,1 R1,2 R1,3 R1,4
R2 R2 R2,1 R2,1 R2,1
R3 R3 R3 R3,1 R3,1
R4 R4 R4 R4 R4
R5 R5 R5 R5 R5
Similar calculations to the ones done in section 3.2 show that the minimal projective
resolutions of S1 and S2 are as follows;
0 S1 P1
P2
⊕P3
⊕P4
0
P5
⊕P5
π1 (t4, t5, t6)
−t4 −t4
t3 0
0 t2
Chapter 3. “Lazy” Construction 69
0 S2 P2
P1
⊕P5
P2
⊕P3
⊕P4
0
P5
⊕P5
π1 (1, t4)
(t4 t5 t6
−1 −t −t2
)
−t4 −t4
t3 0
0 t2
Moreover, since z(R1) = 1 we have pdE(Si) = 2 for 3 ≤ i ≤ 5 (Lemma 3.2.2). Hence,
gl. dim(E) = 3 by Theorem 1.1.3.
3.5 Constructing Endomorphism Rings of Large Global
Dimension
Throughout this section {Ri1 : i ∈ N} is a set of starting rings constructed in section 2.5
and for each natural number i, the ascending chain
Ri1 ( Ri
2 ( . . . ( Rili
= k[[t]]
is constructed via the lazy construction. We define
M i =
li⊕j=1
Rij and Ei = EndRi
1(M i)
In section 3.5.1 we investigate the lengths of the chains, projective modules, and the
projective dimension of the first simple modules for i = 1, 2. Once this is done, this
forms the backbone of the proofs of the main results in this chapter. In section 3.5.2
the first main result of this thesis is proved: we establish a lower bound for the global
dimension of Ei for each i ∈ N. Section 3.5.3 focuses on the module we get when we
remove the starting ring from our chain and instead start our chain from the second ring.
We conclude this section by proving the second main result of this thesis in section 3.5.4.
More precisely, we compute the global dimensions of Ei for a given set of starting rings.
Chapter 3. “Lazy” Construction 70
3.5.1 Minor Results II
This section is devoted to building the machinery needed in the next three sections. The
following proposition gives us a formula for li, the length of the chain with starting ring
Ri1.
Lemma 3.5.1. Given an even integer n ≥ 6, then l1 =3n
2− 1 and for i ≥ 2
li = i(n
2− 1)
+ C(R21)− n = C(R2
1) + (i− 2)n
2− i
Moreover,
li+1 = li +n
2− 1 for i ≥ 2
Proof.
l1 = 1 + g(R11) (Lemma 3.1.1)
= 1 + a1(R11) + a2(R
11)
= 1 + (n− 1) +(n
2− 1)
(Lemma 2.5.5)
=3n
2− 1
and for i ≥ 2,
li = 1 + g(Ri1) (Lemma 3.1.1)
= 1 +
G(Ri1)∑
j=1
aj(Ri1)
= 1 +i+1∑j=1
aj(Ri1) (Lemma 2.5.2)
= 1 + a1(Ri1) +
i∑j=2
aj(Ri1) + ai+1(R
i1)
= 1 + (n− 1) + (i− 1)(n
2− 1)
+ C(R21)−
3n
2− 1 (Lemma 2.5.5)
= i(n
2− 1)
+ C(R21)− n
= C(R21) + (i− 2)
n
2− i
Chapter 3. “Lazy” Construction 71
In particular, for i ≥ 2
li+1 = (i+ 1)(n
2− 1)
+ C(R21)− n
= i(n
2− 1)
+ C(R21)− n+
n
2− 1
= li +n
2− 1
Notice that l2 is not necessarily equal to l1 + n2− 1. To see this, the preceding
proposition yields
l2 = C(R21)− 2 and l1 +
n
2− 1 =
3n
2− 1 +
n
2− 1 = 2n− 2.
That is, the two are equal if and only if C(R21) = 2n (which is not true in general). The
following proposition allows us to go back and forth between the projective Ei modules
and projective Ei+1 modules. It plays a fundamental role in the main theorems we prove
later on in this thesis.
Proposition 3.5.2. Given an even integer n ≥ 6, let {Ri1 | i ∈ N} be a set of starting
rings constructed in section 2.5. Then
(a) For a given i ≥ 2 and j = 1, 2, ..., li,
(P i+1n2+j−1)a = (P i
j )a−n2+1 if n
2≤ a ≤ li+1
In particular, for i ≥ 2,
(P i+1n2
)a = (P i1)a−n
2+1 if n
2≤ a ≤ li+1
(b) For a given i ≥ 2 and j = 1, 2, ..., li,
(P i+1n2+j−1)a =
Rij if 1 ≤ a ≤ n
2− 1
(P ij )a−n
2+1 if n
2≤ a ≤ li+1
=((P ij
⌈n2− 1⌉))
a
In particular, for i ≥ 2,
(P i+1n/2 )a =
Ri1 if 1 ≤ a ≤ (n/2)− 1
(P i1)a−(n/2)+1 if (n/2) ≤ a ≤ li+1
=((P i1
⌈n2− 1⌉))
a
Chapter 3. “Lazy” Construction 72
Proof. (a) Fix an i ≥ 2 and j = 1, 2, ..., li. Given a with n2≤ a ≤ li+1,
(P ij )a−n
2+1 = Ei
j(a−n2+1) = HomRi
1(Ri
a−n2+1, R
ij)
= HomRi+1n2
(Ri+1n2+a−n
2+1−1, R
i+1n2+j−1)
= HomRi+1n2
(Ri+1a , Ri+1
n2+j−1)
= HomRi+11
(Ri+1a , Ri+1
n2+j−1)
= Ei+1(n2+j−1)a
= (P i+1n2+j−1)a
For i ≥ 2, setting j = 1 yields
(P i+1n2
)a = (P i1)a−n
2+1 if n
2≤ a ≤ li+1
(b) Given i ≥ 2 and j = 1, 2, ..., li,
(P i+1n2+j−1)a =
Ri+1n2+j−1 if 1 ≤ a ≤ n
2+ j − 1
(P i+1n2+j−1)a if n
2+ j ≤ a ≤ li+1
=
Ri+1n2+j−1 if 1 ≤ a ≤ n
2− 1
(P in2+j−1)a if n
2≤ a ≤ li+1
=
Rij if 1 ≤ a ≤ n
2− 1
(P ij )a−n
2+1 if n
2≤ a ≤ li+1
=((P ij
⌈n2− 1⌉))
a
For i ≥ 2, setting j = 1 gives
(P i+1n2
)a =
Ri1 if 1 ≤ a ≤ n
2− 1
(P i1)a−n
2+1 if n
2≤ a ≤ li+1
=((P i1
⌈n2− 1⌉))
a
Remark 3.5.3. Since (Sij)1 = 0 for 2 ≤ j ≤ li and li+1 = li +n
2− 1 for i ≥ 2, we have
Sij
⌈n2− 1⌉
= Si+1j+n
2−1 for i ≥ 2, 2 ≤ j ≤ li
Chapter 3. “Lazy” Construction 73
A consequence of Proposition 3.5.2 is
P 3q+2n−1 = P 3q+1
n2
⌈n2− 1⌉
= P 3q1 [n− 2]
P 3q+23n2−2 = P 3q+1
n−1
⌈n2− 1⌉
= P 3qn2dn− 2e = P 3q−1
1
⌈3n
2− 3
⌉for q ≥ 1. We conclude this section by computing the projective dimension of S1
1 as an
E1-module and S21 as an E2-module.
Lemma 3.5.4. pdE1(S11) = 1 and gl. dim(E1) = 2.
Proof. By Lemma 3.2.4
0 S11 P 1
1 P 1n 0
π11 tn
is the minimal projective resolution of S11 . That is, pdE1(S1
1) = 1. The second part
follows by Proposition 3.3.2.
The following notation will be very useful throughout this thesis.
Notation 3.5.5. Let
ε = C(R21)−
3n
2, ε1 = C(R2
1)− n, ε2 = C(R21)−
n
2
ζ = (tn, t3n2 ), τ =
(t3n2 t2n
−tn −t 3n2
):=
(τ1
τ2
), φ =
(tε1
−tε
)
Notice that
ζτ = 0, τφ = 0, ζφ = 0, τ1φ = 0, τ2φ = 0
Lemma 3.5.6. The minimal projective resolution of S21 is as follows;
0 S21 P 2
1
P 2n−1
⊕P 2
3n2−1
P 2l2
0π21 ζ φ
(3.9)
In particular, pdE2(S21) = 2.
Chapter 3. “Lazy” Construction 74
Proof. Since
R21 = lead
{0, n,
3n
2, C(R2
1)
}R2n−1 = lead {0, ε1}
R23n2−1 = lead {0, ε}
R2l2
= k[[t]]
Lemma 3.1.2 yields
(P 21 )j =
R21,0 = R2
1 if j = 1
R21,1 = m2
1 if 2 ≤ j ≤ n− 1
R21,2 if n ≤ j ≤ 3n
2− 1
R21,3 if 3n
2≤ j ≤ l2
(P 2n−1)j =
R2n−1,0 = R2
n−1 if 1 ≤ j ≤ n− 1
R2n−1,1 = m2
n−1 = tε1R2l2
if n ≤ j ≤ l2
(P 23n2−1)j =
R23n2−1,0 = R2
3n2−1 if 1 ≤ j ≤ 3n
2− 1
R23n2−1,1 = m2
3n2−1 = tεR2
l2if 3n
2≤ j ≤ l2
and (P 2l2
)j = R2l2
= k[[t]] for 1 ≤ j ≤ l2. Let
∆ = (tn, t3n2 )
P 2n−1
⊕P 2
3n2−1
Chapter 3. “Lazy” Construction 75
This gives us the following table;
Value of j ∆j n 3n2C(R2
1) . . .
1 ≤ j ≤ n− 1 tnR2n−1 x 0 x x
⊕t3n2 R2
3n2−1 0 x x x
n ≤ j ≤ 3n2− 1 tnm2
n−1 0 0 x x
⊕t3n2 R2
3n1−1 0 x x x
3n2≤ j ≤ l2 tnm2
n−1 0 0 x x
⊕t3n2 m2
3n2−1 0 0 x x
Therefore, ker ζ has the following representation;
Value of j ε ε+ 1 . . . ε1 . . .
1 ≤ j ≤ l2 0 0 0 x x
x x x x x
That is,
(tn, t3n2 )
P 2n−1
⊕P 2
3n2−1
= J(P 21 ) = ker π2
1
and
φ(P 2l2
) = ker
P 2n−1
⊕P 2(3n/2)−1
ζ−→ P 21
(
J(P 2n−1)
⊕J(P 2
(3n/2)−1)
= J
P 2n−1
⊕P 2(3n/2)−1
Hence, (3.9) is a projective resolution for S2
1 (it is minimal by Proposition 2.3.6), com-
pleting the proof.
3.5.2 Lower Bound for gl. dim(Ei)
In this section we prove the first main result of this thesis. More precisely, we obtain a
lower bound for the global dimension of Ei. We begin with a useful definition.
Chapter 3. “Lazy” Construction 76
Definition 3.5.7. Let n be a positive even integer and a ≤ b be non-negative integers,
we define
Aba(n) ={xn
2: x = a, a+ 1, ..., b
}
Now we are in position to prove the first main result of this thesis.
Theorem 3.5.8. Let {Ri1|i ∈ N} be a set of starting rings constructed in section 2.5.
(a) If q ≥ 1, then
0 S3q+21 P 3q+2
1
P 3q+2n−1
⊕P 3q+2
3n2−2
(J(P 3q−11 ))
⌈3n2− 3⌉
0π3q+21 ζ µ
(3.10)
is an exact sequence, where
µ =
(tn2
−1
), Im(ζ) = J(P 3q+2
1 )
(b) If q ≥ 0, then
0 S3q+21 W0 W1 W2 · · · Wq+1 Wq+2 0
d0 d1 d2 d3 dq+1 dq+2
(3.11)
Chapter 3. “Lazy” Construction 77
is a minimal projective resolution for S3q+21 , where
Wj =
P 3q+21 if j = 0
P 3q+2(n−1)+3(j−1)(n
2−1)
⊕
P 3q+2(n−1)+3(j−1)(n
2−1)+(n
2−1)
if j = 1, 2, ..., q
P 3q+2(n−1)+3q(n
2−1)
⊕
P 3q+2(n−1)+3q(n
2−1)+n
2
if j = q + 1
P 3q+2l3q+2
if j = q + 2
and
dj =
π3q+21 if j = 0
ζ if j = 1
τ for j = 2, ..., q + 1
φ if j = q + 2
In particular, pdE3q+2(S3q+21 ) = q + 2 for q ∈ N0 ⇒ gl. dim(E3q+2) ≥ q + 2 for q ∈ N0.
Notice that if q = 0 the second row for Wj above is omitted.
Proof. (a) Fix q ≥ 1, definition 2.5.1 and the lazy construction yield
R3q+21 = lead
{0,jn
2, C(R3q+2
1 ) : j = 2, 3, ..., 3q + 3
}R3q+2n−1 = lead
{0,jn
2, C(R3q+2
n−1 ) : j = 2, 3, ..., 3q + 1
}R3q+2
3n2−2 = lead
{0,jn
2, C(R3q+2
3n2−2
): j = 2, 3, ..., 3q
}
Chapter 3. “Lazy” Construction 78
where
C(R3q+2n−1 ) = C(R3q+2
1 )− n
C(R3q+2
3n2−2
)= C(R3q+2
1 )− 3n
2
and
G(R3q+21 ) = 3q + 3
G(R3q+2n−1 ) = G(R3q+2
1 )− 2 = 3q + 1
G(R3q+2
3n2−2
)= G(R3q+2
1 )− 3 = 3q
Lemmas 3.1.2 and 2.5.8 yield
(P 3q+21 )j =
R3q+21,0 if j = 1
R3q+21,1 if 2 ≤ j ≤ n− 1
R3q+21,2 if n ≤ j ≤ 3n
2− 2
R3q+21,3 if 3n
2− 1 ≤ j ≤ 2n− 3
R3q+21,a if an
2− a+ 2 ≤ j ≤ (a+1)n
2− a for 2 ≤ a ≤ 3q + 1
R3q+21,3q+2 if (3q+2)n
2− 3q ≤ j ≤ (3q+3)n
2− (3q + 1)
R3q+21,3q+3 if (3q+3)n
2− 3q ≤ j ≤ l3q+2
(P 3q+2n−1 )j =
R3q+2n−1,0 if 1 ≤ j ≤ n− 1
R3q+2n−1,1 if n ≤ j ≤ 2n− 3
R3q+2n−1,2 if 2n− 2 ≤ j ≤ 5n
2− 4
R3q+2n−1,3 if 5n
2− 3 ≤ j ≤ 3n− 5
R3q+2n−1,a if an
2+ n− a ≤ j ≤ (a+3)n−2(a+2)
2for 2 ≤ a ≤ 3q − 1
R3q+2n−1,3q if (3q+2)n
2− 3q ≤ j ≤ (3q+3)n
2− (3q + 1)
R3q+2n−1,3q+1 if (3q+3)n
2− 3q ≤ j ≤ l3q+2
Chapter 3. “Lazy” Construction 79
(P 3q+2
3n2−2
)j
=
R3q+23n2−2,0 if 1 ≤ j ≤ 3n
2− 2
R3q+23n2−2,1 if 3n
2− 1 ≤ j ≤ 5n
2− 4
R3q+23n3−2,2 if 5n
2− 3 ≤ j ≤ 3n− 5
R3q+23n3−2,3 if 3n− 4 ≤ j ≤ 7n
2− 6
R3q+23n2−2,a if (a+3)n
2− (a+ 1) ≤ j ≤ (a+4)n−2(a+3)
2for 2 ≤ a ≤ 3q − 2
R3q+23n2−2,3q−1 if (3q+2)n
2− 3q ≤ j ≤ (3q+3)n
2− (3q + 1)
R3q+23n2−2,3q if (3q+3)n
2− 3q ≤ j ≤ l3q+2
Using Lemma 3.5.1 we get
l3q+2 −(3q + 3)n
2− 3q + 1 = C(R2
1)−3n
2− 1
Let
∆ = (tn, t3n2 )
P 3q+2n−1
⊕P 3q+2
3n2−2
f(a) =
(a+ 3)n
2− (a+ 1)
h(a) =(a+ 4)n
2− (a+ 3)
ρ(a) = C(R3q+21 )− an
2
Chapter 3. “Lazy” Construction 80
The image of (tn, t3n2 ) has the following presentation:
Value of j ∆j n 3n2
2n A3q+35 (n) C(R3q+2
1 ) · · ·1 ≤ j ≤ n− 1 tnR3q+2
n−1,0 x 0 x x x x
⊕t3n2 R3q+2
3n2−2,0 0 x 0 x x x
n ≤ j ≤ 3n2− 2 tnR3q+2
n−1,1 0 0 x x x x
⊕t3n2 R3q+2
3n2−2,0 0 x 0 x x x
3n2− 1 ≤ j ≤ 2n− 3 tnR3q+2
n−1,1 0 0 x x x x
⊕t3n2 R3q+2
3n2−2,1 0 0 0 x x x
2n− 2 ≤ j ≤ 5n2− 4 tnR3q+2
n−1,2 0 0 0 x x x
⊕t3n2 R3q+2
3n2−2,1 0 0 0 x x x
For 2 ≤ a ≤ 3q − 2 we have
Value of j ∆j A3q+3a+4 (n) C(R3q+2
1 ) · · ·f(a) ≤ j ≤ h(a) tnR3q+2
n−1,a+1 x x x
⊕t3n2 R3q+2
3n2−2,a x x x
and (note that h(3q − 2) + 1 = f(3q − 1))
Value of j ∆j(3q+3)n
2C(R3q+2
1 ) · · ·f(3q − 1) ≤ a ≤ h(3q − 2) + n
2tnR3q+2
n−1,3q x x x
⊕t3n2 R3q+2
3n2−2,3q−1 x x x
h(3q − 2) + n2
+ 1 ≤ a ≤ l3q+2 tnR3q+2n−1,3q+1 0 x x
⊕t3n2 R3q+2
3n2−2,3q 0 x x
In particular, Im(ζ) = J(P 3q+21 ) = ker π3q+2
1 . The kernel of ζ has the following presenta-
Chapter 3. “Lazy” Construction 81
tion:
Value of j n A3q3 (n) ρ(3) · · · (3q+1)n
2· · · ρ(2) · · ·
1 ≤ j ≤ 5n2− 4 0 x 0 0 x 0 x x
x x x x x x x x
For 2 ≤ a ≤ 3q − 2 the kernel of ζ has the following presentation:
Value of a (a+1)n2
A3qa+2(n) ρ(3) · · · (3q+1)n
2· · · ρ(2) · · ·
f(a) ≤ j ≤ h(a) 0 x 0 0 x 0 x x
x x x x x x x x
and
Value of a 3qn2
ρ(3) · · · (3q+1)n2
· · · ρ(2) · · ·f(3q − 1) ≤ a ≤ h(3q − 2) + n
20 0 0 x 0 x x
x x x x x x x
h(3q − 2) + n2
+ 1 ≤ a ≤ l3q+2 0 0 0 0 0 x x
0 x x x x x x
Since
R3q−11 = lead
{0,xn
2, C(R3q−1
1 ) : x = 2, 3, ..., 3q}
with G(R3q−11 ) = 3q (Lemma 2.5.2) and C(R3q−1
1 ) = C(R3q+21 )− 3n
2(by construction of the
starting rings). By Lemma 3.1.2 the projective module P 3q−11 has the following entries;
(P 3q−11 )j =
R3q−11,0 if j = 1
R3q−11,1 if 2 ≤ j ≤ n− 1
R3q−11,a if an
2− a+ 2 ≤ j ≤ (a+1)n
2− a for 2 ≤ a ≤ 3q − 2
R3q−11,3q−1 if (3q−1)n
2− (3q − 3) ≤ j ≤ 3qn
2− (3q − 2)
R3q−11,3q if 3qn
2− (3q − 3) ≤ j ≤ l3q−1
By Lemma 3.5.1 we have
l3q+2 = l3q−1 +3n
2− 3
Chapter 3. “Lazy” Construction 82
and
an
2− a+ 2 +
3n
2− 3 = f(a)
(a+ 1)n
2− a+
3n
2− 3 = h(a)
In particular,
((J(P 3q−1
1 ))
⌈3n
2− 3
⌉)j
=
R3q−11,1 if 1 ≤ j ≤ 5n
2− 4
R3q−11,a if f(a) ≤ j ≤ h(a) for 2 ≤ a ≤ 3q − 2
R3q−11,3q−1 if f(3q − 1) ≤ j ≤ h(3q − 2) + n
2
R3q−11,3q if h(3q − 2) + n
2+ 1 ≤ j ≤ l3q+2
which yields ker ζ = µ((J(P 3q−11 ))
⌈3n2− 3⌉). Therefore, the sequence in (3.10) is an exact
sequence, as desired.
(b) We proceed by induction on q. The case q = 0 is Lemma 3.5.6. Assume the result
holds for q− 1 (with q ≥ 1). By Theorem 1.1.4 and part (a) of this theorem the minimal
projective resolution of S3q+21 has the following beginning;
0 S3q+21 P 3q+2
1
P 3q+2n−1
⊕P 3q+2
3n2−2
π3q+21 ζ
(3.12)
Moreover, part (a) gives the following exact sequence;
0 S3q+21 P 3q+2
1
P 3q+2n−1
⊕P 3q+2
3n2−2
(J(P 3q−11 ))
⌈3n2− 3⌉
0π3q+21 ζ φ
(3.13)
By induction, pdE3q−1(S3q−11 ) = (q − 1) + 2 = q + 1 (since S
3(q−1)+21 = S3q−1
1 ) and
0 S3q−11 L0 L1 L2 · · · Lq Lq+1 0
f0 f1 f2 f3 fq fq+1
(3.14)
Chapter 3. “Lazy” Construction 83
is a minimal projective resolution for S3q−11 , where
Lj =
P 3q−11 if j = 1
P 3q−1(n−1)+3(j−1)(n
2−1)
⊕
P 3q−1(n−1)+3(j−1)(n
2−1)+(n
2−1)
if j = 1, 2, ..., q − 1
P 3q−1(n−1)+3(q−1)(n
2−1)
⊕
P 3q−1(n−1)+3(q−1)(n
2−1)+n
2
if j = q
P 3q−1l3q−1
if j = q + 1
and
fj =
π3q−11 if j = 0
ζ if j = 1
τ for j = 2, ..., q
φ if j = q + 1
Since Im(f1) = ker(f0) = J(P 3q−11 ), the exact sequence in (3.14) yields the following
exact sequence;
0 J(P 3q−11 ) L1 L2 · · · Lq Lq+1 0
f1 f2 f3 fq fq+1
(3.15)
Lemma 2.6.5 and (3.15) imply that the following sequence is exact;
0 J(P 3q−11 )
⌈3n2− 3⌉
L1
⌈3n2− 3⌉
L2
⌈3n2− 3⌉
0 Lq+1
⌈3n2− 3⌉
Lq⌈3n2− 3⌉
· · ·
f1 f2
f3fqfq+1
(3.16)
Chapter 3. “Lazy” Construction 84
Splicing (3.13) and (3.16) yields the following exact sequence;
0 S3q+21 P 3q+2
1
P 3q+2n−1
⊕P 3q+2
3n2−2
L1
⌈3n2− 3⌉
0 Lq+1
⌈3n2− 3⌉
Lq⌈3n2− 3⌉
· · · L2
⌈3n2− 3⌉
π3q+21 ζ τ = µζ
f2
fqfq+1 f3
(3.17)
Let
Wj =
P 3q+21 if j = 0
P 3q+2n−1 ⊕ P
3q+23n2−2 if j = 1
Lj−1[3n2− 3] if j = 2, ..., q + 2
and
dj =
π3q+21 if j = 0
ζ if j = 1
τ if j = 2
fj−1 if j = 3, ..., q + 2
In particular, (3.17) becomes the following exact sequence;
0 S3q+21 W0 W1 W2 · · · Wq+1 Wq+2 0
d0 d1 d2 d3 dq+1 dq+2
(3.18)
Chapter 3. “Lazy” Construction 85
For j = 2, ..., q we have
Wj = Lj−1
⌈3n
2− 3
⌉
=
P 3q−1n−1+3((j−1)−1)(n
2−1)⌈3n2− 3⌉
⊕P 3q−1n−1+3((j−1)−1)+(n
2−1)⌈3n2− 3⌉
=
P 3q+2n−1+3(j−2)(n
2−1)+3(n
2−1)
⊕P 3q+2n−1+3(j−2)(n
2−1)+(n
2−1)+3(n
2−1)
(Proposition 3.5.2)
=
P 3q+2n−1+3(j−1)(n
2−1)
⊕P 3q+2n−1+3(j−1)(n
2−1)+(n
2−1)
and
Wq+1 = Lq
⌈3n
2− 3
⌉
=
P 3q−1n−1+3(q−1)(n
2−1)⌈3n2− 3⌉
⊕P 3q−1n−1+3(q−1)(n
2−1)+n
2
⌈3n2− 3⌉
=
P 3q+2
n−1+3(q−1)(n2−1)+3( 3n
2−1)
⊕P 3q+2n−1+3(q−1)(n
2−1)+n
2+3(n
2−1)
(Proposition 3.5.2)
=
P 3q+2n−1+3q(n
2−1)
⊕P 3q+2n−1+3q(n
2−1)+n
2
Wq+2 = Lq+1
⌈3n
2− 3
⌉= P 3q−1
l3q−1
⌈3n
2− 3
⌉= P 3q+2
l3q−1+n−1+(n2−2) (Proposition 3.5.2)
= Pl3q−1+3n2−3
= Pl3q+2 (Lemma 3.5.1)
Chapter 3. “Lazy” Construction 86
That is, (3.18) is a projective resolution for S3q+21 . By Theorem 1.1.4
0 S3q+21 W0
d0
is a projective cover for S3q+21 . Since (3.12) is the start of the minimal projective resolution
for S3q+21 and (3.14) is the minimal projective resolution for S3q−1
1 , we have
Im(d1) = Im(ζ) = ker π3q+21 = J(P 3q+2
1 )
Im(d2) = Im(τ) = ker ζ ⊆ J(W1)
and for 3 ≤ j ≤ q + 2,
Im(Lj−1
fj−1−→ Lj−2
)⊆ J(Lj−2) by minimality of (3.14).
This implies that
Im(dj) = Im(fj−1)
⊆ J
(Lj−2
⌈3n
2− 3
⌉)(Lemma 2.6.6)
= J(Wj−1)
Thus, (3.18) is a minimal projective resolution for S3q+21 , as desired. The second part is
a consequence of what we just proved.
Notation 3.5.9. Let
η =
(tC(R
21)−(n/2)
−tC(R21)−n
), σ =
(t3n/2
−tn
), ε = C(R2
1)−3n
2
The following two theorems cover the cases when i is congruent to zero or 1.
Theorem 3.5.10. Let {Ri1|i ∈ N} be a set of starting rings constructed in section 2.5.
(a) The minimal projective resolution of S31 is as follows;
0 S31 P 3
1
P 3n−1
⊕P 3
3n2−2
P 3l3
0π31 ζ η
Chapter 3. “Lazy” Construction 87
In particular, pdE3(S31) = 2.
(b) If q ≥ 2, then
0 S3q1 P 3q
1
P 3qn−1
⊕P 3q
3n2−2
(J(P 3q−31 ))
⌈3n2− 3⌉
0π3q1 ζ µ
is an exact sequence, where
µ =
(tn2
−1
)
(c) If q ≥ 1, then
0 S3q1 W0 W1 W2 · · · Wq Wq+1 0
d0 d1 d2 d3 dq dq+1
is a minimal projective resolution for S3q1 , where
Wj =
P 3q1 if j = 0
P 3q(n−1)+3(j−1)(n
2−1)
⊕
P 3q(n−1)+3(j−1)(n
2−1)+(n
2−1)
if j = 1, 2, . . . , q
P 3ql3q
if j = q + 1
and
dj =
π3q1 if j = 0
ζ if j = 1
τ if j = 2, . . . , q
η if j = q + 1
In particular, pdE3q(S3q1 ) = q + 1 for q ∈ N⇒ gl. dim(E3q) ≥ q + 1 for q ∈ N.
Proof. The proof of part (a) is similar to the proof given in Lemma 3.5.6 and proofs of
parts (b) and (c) is similar to the proof given in Theorem 3.5.8.
Chapter 3. “Lazy” Construction 88
Theorem 3.5.11. Let {Ri1|i ∈ N} be a set of starting rings constructed in section 2.5.
(a) The minimal projective resolution of S11 and S4
1 are as follows;
0 S11 P 1
1 P 1n 0
π11 tn
0 S41 P 4
1
P 4n−1
⊕P 4
3n2−2
P 4l4−(ε−1) 0
π41 ζ σ
In particular, pdE1(S11) = 1 and pdE4(S4
1) = 2.
(b) If q ≥ 2, then
0 S3q+11 P 3q+1
1
P 3q+1n−1
⊕P 3q+1
3n2−2
(J(P 3q−21 ))
⌈3n2− 3⌉
0π3q+11 ζ µ
is an exact sequence, where
µ =
(tn/2
−1
)
(c) If q ≥ 1, then
0 S3q+11 W0 W1 W2 · · · Wq Wq+1 0
d0 d1 d2 d3 dq dq+1
is a minimal projective resolution for S3q+11 , where
Wj =
P 3q+11 if j = 0
P 3q+1(n−1)+3(j−1)(n
2−1)
⊕
P 3q+1(n−1)+3(j−1)(n
2−1)+(n
2−1)
if j = 1, 2, . . . , q
P 3q+1l3q+1−(ε−1) if j = q + 1
Chapter 3. “Lazy” Construction 89
and
dj =
π3q+11 if j = 0
ζ if j = 1
τ if j = 2, . . . , q
σ if j = q + 1
In particular, pdE3q+1(S3q+11 ) = q + 1 for q ∈ N0 ⇒ gl. dim(E3q+1) ≥ q + 1 for q ∈ N0.
Proof. The proof of part (a) is similar to the proof given in Lemmas 3.5.4 and 3.5.6, and
proofs of parts (b) and (c) is similar to the proof given in Theorem 3.5.8.
3.5.3 The Module M ′
Let
R1 ( R2 ( · · · ( Rl = k[[t]]
be an ascending chain of rings constructed via the lazy construction. We define
M ′ =l⊕
i=2
Ri and E ′ = EndR2(M′)
Since E is an l× l matrix, the matrix E ′ is (l−1)× (l−1) matrix. Furthermore, R1 ( R2
implies that EndR2(M′) = EndR1(M
′). The matrix E in block form has the following
form;
E =
(R1 (M ′)∗
HomR1(R1,M′) E ′
)
where
HomR1(R1,M′) = HomR1
(R1,
l⊕i=2
Ri
)=
l⊕i=2
HomR1(R1, Ri) =l⊕
i=2
Ri =
R2
R3
...
Rl
Chapter 3. “Lazy” Construction 90
and
(M ′)∗ = HomR1(M′, R1)
= HomR1
(l⊕
i=2
Ri, R1
)
=l⊕
i=2
HomR1(Ri, R1)
=(
HomR1(R2, R1) HomR1(R3, R1) . . . HomR1(Rl, R1))
Let S ′2, S′3, . . . , S
′l be the simple E ′-modules and P ′2, P
′3, . . . , P
′l be the indecomposable
projective E ′-modules. A consequence of this construction is the following result which
we state as a Lemma for future reference.
Lemma 3.5.12. The simple modules S ′i and projective modules P ′i satisfy the following;
S ′id1e =
(k, k, 0, . . . , 0) if i = 2
Si if 3 ≤ i ≤ l
P ′id1e = Pi for 2 ≤ i ≤ l
Since l′i = li − 1 the map π′i : P ′i → S ′i for 2 ≤ i ≤ li has li − 1 coordinates, and it is
given by
(π′i)j =
ξi if i = j + 1
0 if i 6= j + 1
where ξi : Ri → k is the natural map. This leads to the following corollary.
Corollary 3.5.13. pdE(Si) = pdE′(S′i) for 3 ≤ i ≤ l.
Proof. This follows from Lemmas 3.5.12, 2.6.5, and 2.6.6.
In fact, given a minimal projective resolution of S ′i for 3 ≤ i ≤ l, when we apply d1eto its minimal projective resolution we get a minimal projective resolution for Si. We
can also do this in the reverse direction (for this we would need to remove the first row
and column of E). Hence, knowing the minimal resolution of one gives us the minimal
resolution of the other (for 3 ≤ i ≤ l).
Chapter 3. “Lazy” Construction 91
3.5.4 Global Dimension
In this section we focus on computing the global dimension of Ei for a specific set of
starting rings. All the construction are via the lazy construction. To begin, we introduce
some notation.
Notation 3.5.14. Given positive inters a, b with a ≤ b, let
Vi(a, b) = max{pdEi(Sij) : j = a, a+ 1, ..., b}
Lemma 3.5.15. Suppose {Ri1 : i ∈ N} is a family of starting rings constructed in section
2.5. Then,
pdEi(Sij) = pdEi+1
(Si+1j+n
2−1
)for i ≥ 2, 2 ≤ j ≤ li. In particular,
Vi+1
(n2
+ 1, li+1
)= Vi(2, li)
for i ≥ 2.
Proof. Since Rij = Ri+1
j+n2−1 for i ≥ 2, 1 ≤ j ≤ li, we have
Sij
⌈n2− 1⌉
= Si+1j+n
2−1 for i ≥ 2, 2 ≤ j ≤ li (not true if j = 1)
P ij
⌈n2− 1⌉
= P i+1j+n
2−1 for i ≥ 2, 1 ≤ j ≤ li,
and li+1 = li +n
2− 1. In particular, given i ≥ 2 and 2 ≤ j ≤ li, if
0 Sij P ij W i
1 · · · W ia 0
πij f1 f2 fa
is a minimal projective resolution of Sij, then by Lemmas 2.6.5, 2.6.6 and Proposition
Chapter 3. “Lazy” Construction 92
3.5.2,
0 Sij⌈n2− 1⌉
P ij
⌈n2− 1⌉
W i1
⌈n2− 1⌉
0 W ia
⌈n2− 1⌉
· · ·
πij⌈n2− 1⌉
f1
f2
fa
is a minimal projective resolution of Sijdn2−1e = Si+1j+n
2−1. Hence,pdEi(Sij) = pdEi+1
(Si+1j+n
2−1
)for i ≥ 2, 2 ≤ j ≤ li. The second part is a consequence of what we just proved.
The matrix Ei+1 can be written as follows:
Ei+1 =
P i+11
P i+12
P i+13
···
P i+1n2−1
P i+1n2
P i+1n2+1
···
P i+1li+1
=
P i+11
P i+12
P i+13
···
P i+1n2−1
P i1
⌈n2− 1⌉
P i2
⌈n2− 1⌉
···
P ili
⌈n2− 1⌉
Lemma 3.5.16. Suppose {Ri
1 : i ∈ N} is a family of starting rings constructed in section
2.5, then
gl. dim(Ei+1) = max{Vi+1
(1,n
2
), Vi(2, li)
}for i ≥ 2
Proof. By Theorem 1.1.3 and Lemma 3.5.15,
gl. dim(Ei+1) = maxVi+1(1, li+1)
= max{Vi+1
(1,n
2
), Vi+1
(n2
+ 1, li+1
)}= max
{Vi+1
(1,n
2
), Vi(2, li)
}
Chapter 3. “Lazy” Construction 93
Lemma 3.5.17. Suppose n ≥ 6 is an even number and
R11 = lead
{0, n,
3n
2
},
then gl. dim(E1) = 2, V1(1, l1) = {1, 2} and V1(2, l1) = {2} where l1 =3n
2− 1.
Proof. This follows from Lemma 3.2.4 and its proof.
For the rest of this section
R11 = lead {0, 6, 9} := B1
1
Ri1 = lead {0, 3b, 11 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2
Bi1 = lead {0, 3b, 12 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2
Notice that these two sets of starting rings are of the type which was introduced in section
2.5 with n = 6, C(R21) = 11 and C(B2
1) = 12. We define
M i =
li⊕j=1
Rij and Ei = EndR1(M
i).
Lemma 3.5.18. gl. dim(E2) = 3 and V2(1, l2) = V2(2, l2) = {2, 3} where l2 = 9.
Proof. Notice that l2 = l1 + 1 = 9 and
S1j−1d1e = S2
j for 3 ≤ j ≤ l2
P 1j−1d1e = P 2
j for 2 ≤ j ≤ l2
Lemmas 2.6.5, 2.6.6, and 3.5.17 imply that pdE2(S2j ) = pdE1(S1
j−1) = 2 for 3 ≤ j ≤ 9.
Chapter 3. “Lazy” Construction 94
Moreover, using the resolutions of S1j−1, the minimal resolutions of S2
j are as follows:
0 S2j P 2
j
P 2j−1
⊕P 25+j
P 24+j 0 for 3 ≤ j ≤ 4
π2j (1, t6)
(t6
−1
)
0 S2j P 2
j
P 2j−1
⊕P 29
P 29 0 for 5 ≤ j ≤ 9
π2j (1, t10−j)
(t11−j
−t
)
By Theorem 3.5.8, the minimal projective resolution of S21 is
0 S21 P 2
1
P 25
⊕P 28
P 29 0
π21 (t6, t9)
(t5
−t2
)
A simple computation (like the ones done in section 3.2) shows that that the minimal
projective resolution of S22 is
0 S22 P 2
2 P 21 ⊕ P 2
7 P 25 ⊕ P 2
8
0 P 29
π22 (1, t6)
(−t6 −t9
1 t3
)(−t5
t2
)
completing the proof.
Chapter 3. “Lazy” Construction 95
Notation 3.5.19. Let
1 = inclusion map
In is the n× n identity matrix
ζ = (t6, t9)
ϑ = (1, t3)
θ =
(t9
−t6
)
φ =
(t8
−t5
)
φ1 =
(t5
−t2
)
η =
(t6
−t3
)
τ =
(t9 t12
−t6 −t9
)
Lemma 3.5.20. gl. dim(E3) = 3 and V3(1, l3) = V3(2, l3) = {2, 3} where l3 = 11.
Proof. By Lemma 3.5.15, V3(4, 11) = V2(2, 9) = {2, 3}. By Theorem 3.5.10, the minimal
projective resolution of S31 is
0 S31 P 3
1
P 35
⊕P 37
P 311 0
π31 ζ φ
(3.19)
We now show pdE3(S32) = pdE3(S3
3) = 3, the proofs of which are given in great detail for
future reference. Using the definition introduced in section 3.5.3 we let
(M3)′ =
l3⊕j=2
R3j , (E3)′ = EndR3
2((M3)′) = EndR3
1((M3)′)
Since R32 = B2
1 , by Theorem 3.5.8 the minimal projective resolution of (S32)′ after renum-
Chapter 3. “Lazy” Construction 96
bering the subscripts is
0 (S32)′ (P 3
2 )′(P 3
6 )′
⊕(P 3
9 )′(P 3
11)′ 0
(π32)′ ζ η
(3.20)
where the above modules are (E3)′-modules. Applying d1e to the above exact sequence
makes the modules above into E3-modules, and Lemmas 2.6.5 and 2.6.6 yield the follow-
ing minimal projective resolution:
0 (S32)′d1e (P 3
2 )′d1e(P 3
6 )′d1e⊕
(P 39 )′d1e
(P 311)′d1e 0
(π32)′d1e ζ η
where
((π32)′d1e)j = 0 for 3 ≤ j ≤ l3,
and it is the natural map when j = 1, 2. More specifically,
0 (S32)′d1e P 3
2
P 36
⊕P 39
P 311 0
(π32)′d1e ζ η
(3.21)
is a minimal projective resolution. We can think of the sequences in (3.19) and (3.21)
as complexes by extending by zero’s on both sides. We have the following commutative
Chapter 3. “Lazy” Construction 97
diagram
0 S31 P 3
1
P 35
⊕P 37
P 311 0
0 (S32)′d1e P 3
2
P 36
⊕P 39
P 311 0
π31 ζ φ
(π32)′d1e ζ η
1 1 I2 t2
(3.22)
Taking the mapping cone gives us the following exact sequence (Lemma 1.5.1);
0
0
⊕(S3
2)′d1e
S31
⊕P 32
P 31
⊕P 36
⊕P 39
P 35
⊕P 37
⊕P 311
0
P 311
⊕0
(0 0
1 (π32)′d1e
) (−π3
1 0
1 ζ
) (−ζ 0
I2 η
)
(−φ 0
t2 0
)
Chapter 3. “Lazy” Construction 98
which in turn yields the exact sequence
0 (S32)′d1e
S31
⊕P 32
P 31
⊕P 36
⊕P 39
P 35
⊕P 37
⊕P 311
0 P 311
(1, (π32)′d1e)
(−π3
1 0
1 ζ
) (−ζ 0
I2 η
)
(−φt2
)
Let
γ0 = (1, (π32)′d1e)
γ1 =
(−π3
1 0
1 ζ
)
γ2 =
(−ζ 0
I2 η
)
γ3 =
(−φt2
)
δj =
0 if j = 1
identity if 2 ≤ j ≤ l3
Chapter 3. “Lazy” Construction 99
We have the following commutative diagram with exact columns;
0 0 0 0 0
0 S31 S3
1 0 0 0 0
0 (S32)′d1e
S31
⊕P 32
P 31
⊕P 36
⊕P 39
P 35
⊕P 37
⊕P 311
P 311 0
0 S32 P 3
2
P 31
⊕P 36
⊕P 39
P 35
⊕P 37
⊕P 311
P 311 0
0 0 0 0 0
id
γ0 γ1 γ2 γ3
π32 (1, ζ) γ2 γ3
1
(1
0
)
(0, 1)δ I3 I3 id
(3.23)
Since the top two rows are exact the bottom row is exact. Moreover, since the sequences
in (3.19) and (3.21) are minimal and the third, fourth and fifth columns in (3.22) map
into the Jacobian radical of their codomains (or target space), the bottom row in (3.23)
is a minimal projective resolution of S32 .
By Lemma 3.5.18 or Theorem 3.5.8 the minimal projective resolution of S21 is (where
all modules are E2-modules);
0 S21 P 2
1
P 25
⊕P 28
P 29 0
π21 ζ φ1
Chapter 3. “Lazy” Construction 100
By Lemmas 2.6.5 and 2.6.6 the sequence
0 S21d1e P 2
1 d1eP 25 d1e⊕
P 28 d1e
P 29 d1e 0
π21d1e ζ φ1
(3.24)
is a minimal projective resolution of S21d1e. Since l3 = l2 + 2 the modules in the above
sequence are (E3)′-modules. Then (3.20) and (3.24) give the following commutative
diagram with exact rows;
0 (S32)′ (P 3
2 )′(P 3
6 )′
⊕(P 3
9 )′(P 3
11)′ 0
0 S21d1e P 2
1 d1eP 25 d1e⊕
P 28 d1e
P 29 d1e 0
(π32)′ ζ η
π21d1e ζ φ1
1 1 I2 t
Taking the mapping cone and using a similar argument given for the bottom row of (3.23)
shows that the minimal projective resolution for (S33)′ is as follows;
0 (S33)′ (P 2
1 )d1e
(P 32 )′
⊕P 25 d1e⊕
P 28 d1e
(P 36 )′
⊕(P 3
9 )′
⊕P 29 d1e
0 (P 311)′
(π33)′ (1, ζ)
(−ζ 0
I2 φ1
)
(−ηt
)
(3.25)
Applying d1e to (3.25), Lemmas 2.6.5, 2.6.6 and 3.5.12, Proposition 3.5.2, and the fact
Chapter 3. “Lazy” Construction 101
that (π33)′d1e = π3
3 shows that
0 S33 P 3
3
P 32
⊕P 37
⊕P 310
P 36
⊕(P 3
9
⊕P 311
0 P 311
π33 (1, ζ)
(−ζ 0
I2 φ1
)
(−ηt
)
is the minimal projective resolution of S33 . Hence, gl. dim(E3) = 3 and V3(1, l3) =
V3(2, l3) = {2, 3}.
Lemma 3.5.21. gl. dim(E4) = 3 and V4(1, l4) = V4(2, l4) = {2, 3} where l4 = 13.
Proof. By Lemma 3.5.15, V4(4, 13) = V3(2, 11) = {2, 3}. A similar proof to the one given
Chapter 3. “Lazy” Construction 102
in Lemma 3.5.20 shows that the minimal resolution of S41 , S
42 , and S4
3 is as follows;
0 S41 P 4
1
P 45
⊕P 47
P 412 0
π41 ζ θ
0 S42 P 4
2
P 41
⊕P 46
⊕P 48
P 45
⊕P 47
⊕P 413
0 P 412
π42 (1, ζ)
(−ζ 0
I2 θ
)
(−θ1
)
0 S43 P 4
3
P 42
⊕P 47
⊕P 49
P 46
⊕P 48
⊕P 413
0 P 413
π43 (1, ζ)
(−ζ 0
I2 φ
)
(−θt
)
and the result follows.
Lemma 3.5.22. gl. dim(E5) = 4 and V5(1, l5) = V5(2, l5) = {2, 3, 4} where l5 = 15.
Proof. By Lemma 3.5.15, V5(4, 15) = V4(2, 13) = {2, 3}. A similar proof to the one given
Chapter 3. “Lazy” Construction 103
in Lemma 3.5.20 shows that the minimal resolution of S51 , S
52 , and S5
3 is as follows;
0 S51 P 5
1
P 55
⊕P 57
P 511
⊕P 514
P 515 0
π51 ζ τ φ1
0 S52 P 5
2
P 51
⊕P 56
⊕P 58
P 55
⊕P 57
⊕P 513
0 P 515
P 511
⊕P 514
π52 (1, ζ)
(−ζ 0
I2 θ
)
(−τϑ
)
−φ1
0 S53 P 5
3
P 52
⊕P 57
⊕P 59
P 56
⊕P 58
⊕P 514
0 P 513
π43 (1, ζ)
(−ζ 0
I2 θ
)
(−θ1
)
and the result follows.
Notation 3.5.23. Let
ε = C(R21)−
3n
2, ε1 = C(R2
1)− n, ε2 = C(R21)−
n
2
We are now in position to prove the second main result of this thesis.
Chapter 3. “Lazy” Construction 104
Theorem 3.5.24. Let
R11 = lead {0, 6, 9}
Ri1 = lead {0, 3b, 11 + 3(i− 2) : b = 2, 3, ..., i+ 1} for i ≥ 2
where for each i the chain
Ri1 ( Ri
2 ( . . . ( Rili
= k[[t]],
the module M i and the ring Ei are constructed via the lazy construction. Then,