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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKEALGEBRAS
PAOLO MANTERO, JASON MCCULLOUGH, AND LANCE EDWARD MILLER
ABSTRACT. Rees-like algebras have played a major role in
settling the Eisenbud-Goto conjecture. This article concerns the
structure of the canonical module ofthe Rees-like algebra and its
class groups. Via an explicit computation based onlinkage, we
provide an explicit and surprisingly well-structured resolution of
thecanonical module in terms of a type of double-Koszul complex.
Additionally,we give descriptions of both the divisor class group
and the Picard group of aRees-like algebra.
1. INTRODUCTION
Rees-like algebras were introduced by I. Peeva and the second
author [9]. Givena homogeneous ideal I in a polynomial ring S =
K[x1, . . . , xn] over a fieldK, theRees-like algebra is RL(I) :=
S[It, t2] ⊆ S[t]. Rees-like algebras provide a ma-chine taking as
input an arbitrary homogeneous ideal I in a standard graded
poly-nomial ring S and producing a homogeneous prime ideal in a
non-standard gradedpolynomial ring. A particularly nice advantage
of the construction is that its defin-ing equations are explicit,
unlike for Rees algebras. Among their applications arethe
construction of graded prime ideals with larger than expected
regularity, whichmay then be homogenized to produce a negative
answer to the Eisenbud-Goto con-jecture [4]. As useful as these
algebras are, there remain many questions as to thegeometry of the
varieties they define. Towards this end, the authors completed
astudy of the singularities of the Rees-like algebras, where again
explicit methodswere used to describe the Jacobian and establish
various normality properties [7].
A fundamental tool to study the properties of finitely generated
algebras over afield is the canonical module. In this paper, we
give a complete description of thecanonical module of the Rees-like
algebra of an ideal of height at least 2 when thecharacteristic of
the base field is not 2. In particular, we give an explicit
presenta-tion of ωRL(I) via linkage theory by fully describing the
minimal free resolutionof ωRL(I), including explicit differential
maps. We show that the resolution hasa surprising self-dual
structure. Moreover, we show that, even though the Rees-like
algebra is not Cohen-Macaulay when I is not principal, its
canonical module,defined as an appropriate Ext module, is
Cohen-Macaulay; see Section 3.
Theorem A. (Theorem 3.12) Suppose k is a field with char(k) 6= 2
and S isthe polynomial ring k[x1, . . . , xn]. Let I = (f1, . . . ,
fm) be an ideal of S with
2010 Mathematics Subject Classification. 13D02,14B05.1
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2 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
ht(I) ≥ 2. The canonical module ωRL(I) of the Rees-like algebra
RL(I) isCohen-Macaulay.
In particular, setting M to be the matrix
M =
[f1t f2t · · · fmt f1 f2 · · · fmf1t
2 f2t2 · · · fmt2 f1t f2t · · · fmt
],
the canonical module of the Rees-like algebraRL(I) isωRL(I) ∼=
coker(M),
and thus type(RL(I)) = 2.The fact that ωRL(I) is Cohen-Macaulay
is not overly surprising given that the
integral closure S[t] of RL(I) is Cohen-Macaulay; RL(I) is
canonically Cohen-Macaulay in the language of Schenzel [12].
Nonetheless, we find the ‘double-Koszul complex’ structure of its
resolution over the presenting polynomial ringrather
interesting.
We next turn our attention to divisor class groups. This is a
somewhat delicatetopic as the literature on class groups primarily
limits itself to normal rings, whileRees-like algebras are never
normal. Nevertheless, Rees-like algebras are Noether-ian domains,
so the codimension-1 Chow group or divisor class group (see e.g.
[3,Section 11.5]) is well-defined. First we prove the following
general result aboutclass groups for which we could find no
reference in the literature.
Theorem B. (Theorem 4.1) Let A be a Noetherian, universally
catenary, integraldomain satisfying Serre’s condition (R1). Let A
denote the integral closure of A.Then
Cl(A) ∼= Cl(A).Because Rees-like algebras of ideals of height at
least two satisfy the (R1) condi-tion [7, Theorem 6], it follows
that the class groups of these Rees-like algebras aretrivial; see
Corollary 4.6.
Finally, we consider the Picard group ofRL(I). The fundamental
approach is toconsider the conductor square, which realizes the
Rees-like algebra as a pullback.This is also called a Milnor
square, and exploiting a fundamental exact sequencerelating Picard
groups and groups of units defined using this square, we show
thePicard group of a Rees-like algebra vanishes precisely when I is
radical.
Theorem C. (Theorem 4.7) For k a field, S = k[x1, . . . , xn] an
S-ideal I isradical if and only if Pic(RL(I)) = 0.
Considering [7, Sec. 5, Thm. 8], where it is shown that I is
radical if and onlyif RL(I) is seminormal, Theorem C supports a
theme suggesting that Rees-likealgebras are best behaved for
radical ideals.
The rest of the paper is structured as follows. In Section 2, we
recall somepreliminary results and definition on Rees-like
algebras. In Section 3, we computea presentation and free
resolution of the canonical module of a Rees-like algebra.Finally,
in Section 4 we study the divisor class group and Picard group of a
Rees-like algebra.
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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 3
2. PRELIMINARIES
We reserve the following notation. Throughout, unless otherwise
stated, k isa field and S = k[x1, . . . , xn] is a standard graded
polynomial ring. We alsoreserve bold letters F•,D•, . . . for chain
complexes of modules with differentialsdF• , d
D• , . . .
For a homogeneous S-ideal I with generators I = (f1, . . . ,
fm), recall the Rees-like algebra of I is S[It, t2] ⊆ S[t], where t
is a new variable. We denote this byRL(I) := S[It, t2], and we
denote by RLP(I) the prime ideal arising as thekernel of the map T
→ RL(I), where T = S[y1, . . . , ym, z] is a non-standardgraded
polynomial ring over S, and the map is determined by sending yi 7→
fitand z 7→ t2. In particular, RL(I) ∼= T/RLP(I), where T has
grading defined bydeg(yi) = deg(fi) + 1 and deg(z) = 2. (Later we
distinguish between differentpresentations of RL(I), depending on
the choice of generators of I .) We quicklyrecall the relevant
structure theorem for Rees-like algebras.
Theorem 2.1 (McCullough and Peeva [9, Theorem 1.6, Proposition
2.9]). TheidealRLP(I) is the sumRLP(I)syz +RLP(I)gen with
generators
RLP(I)syz =
{rj :=
m∑i=1
cijyi |m∑i=1
cijfi = 0
}and
RLP(I)gen = {yiyj − zfifj | 1 ≤ i, j ≤ m}.Moreover,
• eEuler(T/RLP(I)) = 2∏mi=1
(deg(fi) + 1
),
• pdT (T/RLP(I)) = pd(S/I) +m− 1,• ht(RLP(I)) = m,
and in particular, T/RLP(I) is Cohen-Macaulay if and only if m =
1.
In the previous theorem, eEuler(M) denotes the Euler
multiplicity of the posi-tively graded T -moduleM defined as
follows. LetEM (u) =
∑i
∑j(−1)iβTi,j(M)uj ∈
Z[u] denote the Euler polynomial of M . After factoring out a
maximal possiblepower of (1 − u) we write EM = (1 − u)chM (u).
Finally we define the Eulermultiplicity of M to be eEuler(M) = hM
(1). When T is a standard graded poly-nomial ring, this is the
usual degree or multiplicity of M . See [1, Theorem 2.5] forfurther
details.
3. THE CANONICAL MODULE
We start this section with a brief summary of the proof of the
main theoremconcerning the structure of the canonical module.
Recall, the Rees-like algebraS[It, t2] is a quotient of a
polynomial ring T . Set Q := RLP(I). As T/Q is notCohen-Macaulay if
ht(I) ≥ 2, we take as our definition of the canonical moduleωT/Q :=
ExtcT (T/Q, T ), where c = codimQ. To calculate the canonical
module,
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4 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
our approach is based on linkage. Two ideals I and J in S are
said to be linked pro-vided there is a complete intersection C ⊂ I
∩ J so that J = C : I and I = C : J .Many nice properties of ideals
persist on linkage, namely if I and J are linked,then I defines a
Cohen-Macaulay quotient if and only if J does. The applicationfor
us is to compute the canonical module via the following well-known
result; fora proof we refer the reader to [14, Thm. 6.25].
Theorem 3.1. For a polynomial ring T , a prime idealQ of
heightm, and aC ⊂ Qa complete intersection of height m,
ωT/Q ∼= (C : Q)/Q.
The key observation is that among the generators of Q we find a
natural com-plete intersection C to work with. We determine the
primary decomposition ofC in an explicit manner and provide a
Rees-like algebra interpretation for it, seeLemma 3.2(4). We then
compute the minimal generators for C : Q, which alsoform a Gröbner
basis. These generators allow one to relate this calculation ofthe
canonical module to an interesting chain complex, obtained by
combining twoKoszul complexes, which then serves as the claimed
explicit minimal free resolu-tion.
We assume k is a field with char(k) 6= 2. Let S = k[x1, . . . ,
xn] and f1, . . . , fmminimal generators of a homogeneous ideal I .
We also assume that ht(I) ≥ 2.Denote by RLP(f1, . . . , fm) the
Rees-like prime defined in Section 2. There is adistinguished
complete intersection inRLP(f1, . . . , fm), namely,
C =(y21 − zf21 , y22 − zf22 , . . . , y2m − zf2m
).
Note that a different choice of minimal generating set g1, . . .
, gm of I givesa different but isomorphic Rees–like prime in the
same polynomial ring T =S[y1, . . . , ym, z]. For instance,
RLP(f1,−f2, f3, . . . , fm) 6= RLP(f1, . . . , fm),whileRLP(f1,−f2,
f3, . . . , fm) ∼= RLP(f1, . . . , fm).
Lemma 3.2. With the the notation above, we have the
following:(1) For any choice of +− signs, C ⊂ RLP(+−f1,+−f2, . . .
,+−fm).(2) RLP(f1, f2, . . . , fm) = RLP(−f1,−f2, . . . ,−fm).(3)
If m ≥ 2, then for any choice of +− sign as indicated
RLP(f1, f2, . . . , fm) 6= RLP(f1,−f2,+−f3,+−f4, . . .
,+−fm).(4) The complete intersection ideal C defined above is
radical and has the
following primary decomposition
C =⋂RLP(f1,+−f2,+−f3, . . . ,+−fm),
where the intersection is taken over all possible choices of +−
sign.
Proof. (1) One simply observes that when we replace yi by +−fit
and z by t2, we
see that y2i − zf2i becomes (+−fit)2 − t2f2i = 0.
(2) Let φ : T → S[t] be the map sending yi 7→ fit and z 7→ t2.
Then clearly
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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 5
RLP(f1, f2, . . . , fm) = Ker(φ) = Ker(−φ) = RLP(−f1,−f2, . . .
,−fm).(3) The element y1y2 − zf1f2 is in the left-hand ideal but
not the right-hand one.(4) By (3), there are 2m−1 distinct primes
in the intersection above, let us writethem Q1, . . . , Q2m−1 . By
(1), C is a subset of the ideal H =
⋂2m−1j=1 Qj .
Both C and Q are unmixed homogeneous ideals with the grading
deg(xj) = 1,deg(yi) = di + 1 and deg(z) = 2. Since y2i − zf2i is
homogeneous of degree2(deg(fi)+1), we have eEuler(T/C) = 2mD,
whereD =
∏mi=1(di+1). By The-
orem 2.1, eEuler(T/Qi) = 2D for every i = 1, . . . , 2m−1. Then
eEuler(T/C) =eEuler(T/H) = 2
mD. Since C ⊆ H are unmixed ideals of the same Euler
multi-plicity and height, we have C = H . �
Next, we want to obtain an explicit description of the link L =
C : RLP(I),where RLP(I) = RLP(f1, f2, . . . , fm). To do this, we
identify interest-ing candidate generators which posses remarkable
symmetries. For any subsetA ⊆ [m] := {1, 2, . . . ,m} we define the
elements gevenA and goddA as follows. Fora subset S ⊆ A, let yS
denote
∏i∈S yi and set S = A \S. We define two elements
of T ,
gevenA :=
b#A/2c∑i=0
∑S⊆A#S=2i
ySfSzi,
goddA :=
b(#A−1)/2c∑i=0
∑S⊆A
#S=2i+1
ySfSzi,
where #A denotes the cardinality of A. For example, when m = 4
we get
geven[4] = y1y2y3y4 + y1y2f3f4z + y1f2y3f4z + · · ·+ f1f2y3y4z +
f1f2f3f4z2,
godd[4] = y1y2y3f4 + y1y2f3y4 + · · ·+ f1y2y3y4 + y1f2f3f4z + ·
· ·+ f1f2f3y4z.
The elements geven[j] and godd[j] are invariant under an
Sj-action which permutes
the variables yi, and they satisfy the following useful
identities.
Lemma 3.3. For 1 ≤ j ≤ m and 1 ≤ h ≤ j, we have
godd[j] = yhgodd[j]r{h} + fhg
even[j]r{h},
geven[j] = yhgeven[j]r{h} + zfhg
odd[j]r{h},
yhgeven[j] = zfhg
odd[j] +
(y2h − zf2h
)geven[j]r{h},
fhgeven[j] = yhg
odd[j] −
(y2h − zf2h
)godd[j]r{h}.
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6 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
Proof. The proof of the first two identities are similar to each
other as are the proofsof the last two. We provide the reasoning
for the first and third identities and leavethe other two for the
interested reader.
To prove the first identity, we fix h and isolate the terms
involving yh to obtain
godd[j] =
b(j−1)/2c∑i=0
∑S⊆{1,...,j}#S=2i+1
ySfSzi.
= yh
b(j−1)/2c∑i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i+1
ySfSzi + fh
b(j−1)/2c∑i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i
ySfSzi
= yh
b(j−1)/2c∑i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i+1
ySfSzi + fhgeven[j]r{h}
= yhgodd[j]r{h} + fhg
even[j]r{h}.
To see the last equality holds note the following observations.•
If j is even, then b(j − 1)/2c = b(j − 2)/2c, so∑b(j−1)/2c
i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i+1
ySyhfSzi = godd[j]r{h}.
• If j is odd, for i = b(j − 1)/2c there is only one subset S ⊆
[j] with#S = 2i + 1, namely S = [j]. For this value of i and the
only possibleassociated S, the variable yh does not divide ySfSzi =
f1f2 · · · fjzi. Thus
b(j−1)/2c∑i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i+1
ySfSzi =
b(j−2)/2c∑i=0
∑S⊆{1,...,ĥ,...,j}
#S=2i+1
ySfSzi = godd[j]r{h}.
Thus the first identity holds.
As for the third identity, we have
yhgeven[j] = y
2hg
even[j]r{h} + yhzfhg
odd[j]r{h}
= y2hgeven[j]r{h} + zfh
(godd[j] − fhg
even[j]r{h}
)=(y2h − zf2h
)geven[j]r{h} + zfhg
odd[j] ,
where the first equality follows from the second identity, and
the middle equalityfrom the first identity and the last simply
rearranges the terms.
�
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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 7
To simplify the notation, in what follows we simply write goddj
for godd[j] and
gevenj for geven[j] .
Lemma 3.4. If Q = RLP(f1,−f2,+−f3,+−f4, . . . ,+−fm), then
gevenm , g
oddm ∈ Q.
Proof. We show that gevenj , goddj ∈ Q by induction on 2 ≤ j ≤
m. First note
that geven2 = y1y2 + zf1f2 = y1y2 − zf1(−f2) ∈ Q and, similarly,
godd2 =y1f2 + y2f1 = y2f1 − y1(−f2) ∈ Q,
Now let j > 2 and suppose gevenj−1 , goddj−1 ∈ Q. Then, by
Lemma 3.3, gevenj =
yjgevenj−1 + zfjg
oddj−1 ∈ Q and, similarly, goddj = yjgoddj−1 + fjgevenj−1 ∈ Q.
�
Corollary 3.5. If Q = RLP(+−f1,+−f2, . . . ,+−fm), then gevenm ,
g
oddm ∈ Q for any
choice of +− signs except for Q = RLP(f1, . . . , fm) = RLP(−f1,
. . . ,−fm).
Proof. By the symmetry of gevenm , goddm , we can assume that
the signs on f1 and f2
are different. Then the statement follows from Lemma 3.4 and
Lemma 3.2(2). �
Our next goal is to prove thatC : RLP(f1, . . . , fm) =
C+(gevenm , g
oddm
). From
now on we adopt the following notation
Notation 3.6. Let I = (f1, . . . , fm) ⊆ S, and let Q = RLP(f1,
. . . , fm) ⊆ T beits Rees-like prime. We set L := C : Q ⊆ T , and
J := C +
(gevenm , g
oddm
)⊆ T .
Proving L = J will require a sequence of lemmas. First we
construct two usefulshort exact sequences.
Lemma 3.7. With Notation 3.6, we have short exact sequences
0→ T/Q ·goddm−−−→ T/C → T/(C + (goddm ))→ 0,
and
0→ T/(IT + (y1, . . . , ym))·gevenm−−−→ T/(C + (goddm ))→ T/J →
0.
In particular, Q = C : (goddm ) and IT + (y1, . . . , ym) = (C +
(goddm )) : (g
evenm ).
Proof. The first short exact sequence is explained by the
equality C : (goddm ) = Q,which follows by Lemma 3.2(4) and
Corollary 3.5.Analogously, for the second sequence we need to show
(C + (goddm )) : (g
evenm ) =
IT + (y1, . . . , ym). First note that by the third and fourth
equalities in Lemma 3.3we see that fh and yh lie in (C + (goddm ))
: (g
evenm ) for every 1 ≤ h ≤ m, so
IT + (y1, . . . , ym) ⊆ (C + (goddm )) : (gevenm ).Since IT ⊂ (C
+ (goddm )) : (gevenm ), it suffices to consider the reverse
inclusion
modulo IT . Let a ∈ T be such that a · gevenm ∈ (C + (goddm ))
modulo IT . Sincegevenm ≡ y1y2 · · · ym modulo IT and (C + (goddm
)) ≡ (y21, . . . , y2m) modulo IT ,we have ay1 · · · ym ∈ (y21, . .
. , y2m) in T/IT . Because y1, . . . , ym is a regularsequence on
T/IT , we get a ∈ (y1, . . . , ym)+IT . Therefore IT+(y1, . . . ,
ym) =(C + (goddm )) : (g
evenm ).
�
Next, we compute the initial ideal of J .
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8 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
Lemma 3.8. Fix y1 > y2 > · · · > ym > z > x1 >
· · · > xn and let < be the lexorder < on T . Then y21 −
zf21 , . . . , y2m − zf2m, gevenm , goddm form a Gröbner basis ofJ
with respect to
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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 9
As a step toward proving J is unmixed, we next show that (y1,
y2, . . . , ym, z) isnot an associated prime of T/J .
Lemma 3.10. Let p = (y1, y2, . . . , ym, z). Then p /∈
Ass(T/J).
Proof. First we show that Qp is a complete intersection. Recall
that we have adecomposition Q = RLP(I)syz + RLP(I)gen as in Theorem
2.1. The idealRLP(I)syz is generated by elements of the form
∑i siyi such that
∑i sifi = 0 in
S. In particular, the following elements corresponding to Koszul
syzygies of I arein (RLP(I)syz)p: y1 − f1fm ym, y2 −
f2fmym, . . . , ym−1 − fm−1fm ym. For brevity, set
y′i = yi −fifmym. Since y2m − zf2m ∈ RLP(I)gen, it follows that
Qp is generated
by the regular sequence y′1, y′2, . . . , y
′m−1, y
2m − zf2m. (These elements, along with
ym, form a regular system of parameters of the regular local
ring Sp.)Now we compute the link Lp = Cp : Qp. Set yi = yi + fifm
ym, so that
y2i − zf2i = yiy′i +f2if2m
(y2m − zf2m).
Therefore[y21 − zf21 , . . . , y2m − zf2m
]= D
[y′1, . . . , y
′m−1, y
2m − zf2m
]T, where
D =
y1 0 · · · 0 f21 /f2m0 y2 · · · 0 f22 /f2m0 0
. . . 0...
0 0 0 ym−1 f2m−1/f
2m
0 0 0 0 1
.By [13, Theorem A.140], Lp = (C + (detD))p. Note that
det(D) =m−1∏i=1
yi
=m−1∏i=1
(yi +fifm
ym)
=∑
S⊆{1,...,m−1}
ySfS
f|S|m
y|S|m
=
b(m−1)/2c∑i=0
∑S⊆{1,...,m−1}
#S=2i
ySfS
f2imy2im +
∑S⊆{1,...,m−1}
#S=2i+1
ySfS
f2i+1my2i+1m
≡b(m−1)/2c∑
i=0
∑S⊆{1,...,m−1}
#S=2i
ySfSzi +∑
S⊆{1,...,m−1}#S=2i+1
ySfSziymfm
(mod Cp)= gevenm−1 +
ymfm
goddm−1,
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10 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
where the third line follows from expanding the product, the
fourth line separatesthe even and odd terms, and the fifth line
follows since z − y
2mf2m∈ Cp. Finally note
thatfm det(D) ≡ fmgevenm−1 + ymgoddm−1 ≡ goddm (mod Cp).
It follows that Lp = (C + goddm )p. Since
fmgevenm = ymg
oddm − (y2m − zf2m)goddm−1 ∈ C + (goddm ),
we haveLp = (C + (g
oddm ))p = Jp.
Since Qp is a complete intersection, in particular Tp/Qp is
Cohen-Macaulay.Since Jp = Cp : Qp is a link of Qp, then by linkage,
e.g., [11, Prop. 2.6]. AlsoTp/Jp is Cohen-Macaulay. In particular,
Jp is unmixed of height m; thereforepTp /∈ Ass(Tp/Jp) and so p /∈
Ass(T/J). �
We can now prove the following:
Proposition 3.11. In Notation 3.6, one has L = J , i.e. C : Q =
C+(goddm , gevenm ).
Proof. The containment L ⊇ J follows from Lemma 3.2 and
Corollary 3.5. Nextwe show Jun = L. Since C = Q ∩ L ⊆ J ⊆ L, since
all these ideals have heightm, and since Q,L are unmixed, we have C
⊆ Jun ⊆ L. Since C ⊆ Jun areunmixed of the same height, Ass(T/Jun)
⊆ Ass(T/C), so, by Lemma 3.2(4), allassociated primes of T/Jun have
the formRLP(f1,±f2, . . . ,±fm). By Theorem2.1 (or the proof of
Lemma 3.10) they are all contained in p = (y1, . . . , ym, z).Since
Jp = Lp, by Lemma 3.10, then JQi = LQi for eachQi ∈ Ass(T/Jun).
Thisproves Jun = L.
It then suffices to prove that J is unmixed. We observe that for
any associatedprime q of T/J we have ht(q) ≤ m+ 1, because
ht(q) ≤ pd(T/J) ≤ pd(T/in 1, the only
possibility is that z ∈ p, and therefore p = (y1, . . . , ym,
z). But this possibility isruled out by Lemma 3.10. �
Claim 2. We may assume ym is regular on T/J .
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CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 11
Proof of Claim 2. By Claim 1 there is a linear form 0 6= ` ∈
k[y1, . . . , ym] thatis regular on T/J . By possibly multiplying
by a unit and permuting the variables,we may assume that ` = ym
+
∑m−1i=1 αiyi, where αi ∈ k. We consider the
automorphism ψ of T that fixes all variables except it sends ym
7→ `. It is easycheck that ψ−1(J) has the same generators as J
except that every instance of fmis replaced by fm +
∑m−1i=1 αifi. This then corresponds to choosing a different
minimal set of generators of I before constructing the Rees-like
prime. Since `is not in any associated prime of J , ym is not in
any associated prime of ψ−1(J). �
We now conclude the proof of Proposition 3.11. Since ym is
regular onT/J and y2m − zf2m ∈ J , then also fm is regular on T/J .
To prove J is unmixedit then suffices to show Jfm is unmixed in the
localization Tfm . Since fm is a unitin Tfm and fmg
evenm = ymg
oddm − (y2m − zf2m)goddm−1 ∈ (C + (goddm ))fm , the ideal
Jfm = (C + (goddm ))fm is an almost complete intersection of
height m.
Now, in the ring Tfm we have(2)
J + (ym) = (y21 − zf21 , . . . , y2m−1 − zf2m−1, y2m − zf2m, ym,
goddm )
= (y21 − zf21 , . . . , y2m−1 − zf2m−1, zf2m, ym, goddm
)(because fm is a unit) = (y21 − zf21 , . . . , y2m−1 − zf2m−1, z,
ym, goddm )
= (y21, y22, . . . , y
2m−1, ym, z, g
oddm )
(by definition of goddm ) = (y21, y
22, . . . , y
2m−1, ym, z, y1 · · · ym−1).
Since M = (y21, . . . , y2m−1, y1y2 · · · ym−1) is (y1, . . . ,
ym−1)-primary and
extended from k[y1, . . . , ym−1], then M is Cohen-Macaulay of
heightm − 1. Since ym, z is a regular sequence on (T/M)fm , the
ideal(y21, y
22, . . . , y
2m−1, y1 · · · ym−1, ym, z)fm = (J+(ym))fm is Cohen-Macaulay
too.
Since ym is regular on T/J and fm is regular on T/J , ym is also
regular on(T/J)fm , and thus (T/J)fm is Cohen-Macaulay. In
particular, Jfm is unmixedand then so is J . �
We are now able to construct a finite T -free resolution of the
canonical moduleof any Rees-like algebraRL(I) = S[It, t2] =
T/RLP(I), assuming char(k) 6= 2and I has height at least 2. It is
built from an amalgamation of the Koszul com-plexes on the
generators f1, . . . , fm of I and the variables y1, . . . ,
ym.
Theorem 3.12. Suppose k is a field with char(k) 6= 2. Let S =
k[x1, . . . , xn]and let I = (f1, . . . , fm) be an ideal of S with
ht(I) ≥ 2. Then the canonicalmodule ωRL(I) of the Rees-like algebra
RL(I) is a maximal Cohen-MacaulayRL(I)-module. In particular, if M
is the matrix
M =
[y1 y2 · · · ym f1 f2 · · · fmzf1 zf2 · · · zfm y1 y2 · · ·
ym
],
then the canonical module of the Rees-like algebraRL(I) isωRL(I)
∼= coker(M),
as T -modules, and thus type(RL(I)) = 2.
-
12 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
Proof. As usual let T = S[y1, . . . , ym, z]. Let K•(y) denote
the Koszul complexon y1, . . . , ym over T with differential maps
d
y
i : Ki(y) → Ki−1(y), and letK•(f) denote the Koszul complex on
f1, . . . , fm over T with differential maps
df
i : Ki(f) → Ki−1(f). Define a new complex of free T -modules D•
withDi = T
2(mi ) for 0 ≤ i ≤ m with differential given as a matrix by
dDi =
dyi dfiz ·dfi d
y
i
.It is easy to check that dDi−1 ◦ dDi = 0 and thus D• is a
complex. We also have thefollowing short exact sequences of
complexes
0→ D•z−→ D• → D•/zD• → 0.
and0→ K•(y)⊗T T/zT → D•/zD• → K•(y)⊗T T/zT → 0.
Because K•(y)⊗T T/zT is acyclic, it follows from the long exact
sequence of ho-mology associated to the second short exact sequence
that D•/zD• is also acyclic.Now from the long exact sequence
associated to the first short exact sequence wesee that
multiplication by z induces an isomorphism on Hi(D•) for i > 0;
then byNakayama’s Lemma we get Hi(D•) = 0 for i > 0. Note that
dD1 = M.
Now define dD0 : D0 →C:QC as follows. By Proposition 3.11,
C:QC is minimally
generated by gevenm and −goddm . Since D0 = T 2, we map the
first basis element togevenm and the second basis element to −goddm
. By Lemma 3.3, we have
ymgevenm + zfm(−goddm ) = gevenm−1
(y2m − zf2m
)∈ C
andfmg
evenm + ym(−goddm ) = −goddm−1
(y2m − zf2m
)∈ C.
Therefore Im(dD1 ) = Im(M) ⊆ Ker(dD0 ). To show the reverse
inclusion, supposethat a, b ∈ T such that dD0 [a, b]T = 0 ∈
C:QC ; that is,
a · gevenm + b(−goddm ) ∈ C.
Then by Lemma 3.7, a ∈ (C + (goddm )) : (gevenm ) = IT + (y1, .
. . , ym). Since theentries in the first row of M generate IT +
(y1, . . . , ym), we can use the columnsof M to rewrite a and b and
we may assume that a = 0. But then b ∈ C : (goddm ) =Q. By Theorem
2.1, every element of Q is a linear combination of the elementsyiyj
− zfifj , where 1 ≤ i ≤ j ≤ m and
∑j cjyj , where
∑j cjfj = 0. Note that[
0yiyj − zfifj
]= yj
[fiyi
]− fi
[yjzfj
]∈ Im(dD1 ),
and [0∑j cjyj
]=∑j
cj
[fjyj
]∈ Im(dD1 ),
-
CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 13
where∑
j cjfj = 0. Therefore [0, b]T ∈ Im(dD1 ), for any b ∈ Q. It
follows that
Im(dD1 ) = Ker(dD0 ), and that D• is a minimal T -free
resolution of
C:QC . Finally,
we have ωRL(I) ∼= C:QC , e.g. by [6, Lemma 3.1].�
In retrospect, the fact that the canonical module is
Cohen-Macaulay should notbe surprising since the integral closure
of S[It, t2] is a polynomial ring, and thusa finite Cohen-Macaulay
module over the non-Cohen-Macaulay Rees-like algebraRL(I). Yet, we
find the self-dual nature of the T -free resolution of the
canon-ical module in the previous theorem interesting, especially
given that one of theconstituent Koszul complexes, K•(f) need not
be exact.
As a corollary, we get the following surprising self-duality
statement:
Corollary 3.13. Using the notation above,
ωRL(I) ∼= ExtmT (T/Q, T ) ∼= ExtmT(ωRL(I), T ).
Proof. Because K•(y) and K•(f) are self-dual, it follows from
the definition thatD• is self-dual as well, i.e. D• ∼= HomT (D•, T
).
�
Example 3.14. Let S = k[x1, x2] and set I = (x1, x2)2. We
construct the res-olution of the canonical module of the Rees-like
algebra RL(I). As such, setT = S[y1, y2, y3, z] and let Q =
RLP(x21, x1x2, x22). By the previous theorem,ωRL(I) ∼= C:QC , where
C = (y
21 − zx41, y22 − zx21x22, y23 − zx42) and
C : Q = C + (godd3 , geven3 ),
where
geven3 = y1y2y3 + x1x32y1z + x
21x
22y2z + x
31x2y3z,
godd3 = x22y1y2 + x1x2y1y3 + x
21y2y3 + x
31x
32z.
Moreover, as a T -module, ωRL(I) has T -free resolution:
T 2 T 6d1oo T 6
d2oo T 2d3oo 0,oo
-
14 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
where
d1 =
[y1 y2 y3 x
21 x1x2 x
22
zx21 zx1x2 zx22 y1 y2 y3
]
d2 =
−y2 −y3 0 −x1x2 −x22 0y1 0 −y3 x21 0 −x220 y1 y2 0 x
21 x1x2
−zx1x2 −zx22 0 −y2 −y3 0zx21 0 −zx22 y1 0 −y30 zx21 zx1x2 0 y1
y2
d3 =
−y3 −x22y2 x1x2−y1 −x21−zx22 −y3zx1x2 y2−zx21 −y1
.
4. CLASS GROUPS
We now turn our attention to the investigation of class groups
of Rees-like alge-bras. A main complication comes from the fact
that Rees-like algebras are nevernormal. On the other hand, the
integral closure of S[It, t2] is the UFD S[t], and,when the height
of I is at least 2, S[It, t2] satisfies Serre’s (R1) condition by
[7,Thm. 5.1]. We leverage these two facts in our computations.
4.1. Divisor class group. We first review class groups in the
generality we con-sider; for details we refer the reader to [3,
Section 11.5]. Denote for a ringR the setof height 1 primes by
Spec1R. Let R be a Noetherian domain. A Weil divisor isa formal
finite Z-linear combination
∑p∈Spec1(R) np[p] of height 1 primes. These
naturally form an abelian group Div(R).If R is normal, then Rq
would be a DVR for all height 1 primes q, leading
to the usual notion of linear equivalence. Note however that for
any (possiblynon-normal) domain R, the ring Rq is still a one
dimensional domain. Thusfor any nonzero x ∈ R, the Rq-module Rq/xRq
has finite length which we de-note ordq(x) := λ (Rq/xRq). When Rq
is a DVR, ordq(x) agrees with theq-adic valuation of x and so this
recovers the more familiar definition of classgroup. This extends
in the natural fashion to Frac(R) and yields a well-definedmap divR
: Frac(R) → Div(R) sending x/y ∈ Frac(R) with x, y ∈ R to∑
q∈Spec1R (ordq(x)− ordq(y)) [q]. Elements in the image Prin(R)
of this mapare called principal divisors and the divisor class
group or codimension-1 Chowgroup is the quotient
Cl(R) := Div(R)/Prin(R).
There are few computations in the literature of class groups of
non-normal do-mains.
To compute the class group of a Rees-like algebra we prove a
much more generaltheorem providing sufficient conditions under
which the class group of an algebra
-
CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 15
is isomorphic to the one of its integral closure (Theorem 4.1).
Since the integralclosure of a Rees-like algebra is a polynomial
ring, it follows that the class groupof a Rees-like algebra is
trivial, under mild hypotheses.
Theorem 4.1 is likely unsurprising for experts, but we could not
locate its state-ment or proof in the literature, so we provide a
proof along with examples to il-lustrate the necessity of the
hypothesis. The proof of [16, Chapter V, Section 5,Remark, p. 269]
makes essentially similar claims and one could deduce a
quickerargument accepting those, but we opted to provide a more
detailed argument.
We work first in the following general setup. Let A be a
Noetherian integraldomain, and let A denote its integral
closure.
Theorem 4.1. Let A be a Noetherian, universally catenary,
integral domain satis-fying Serre’s condition (R1). Let A denote
the integral closure of A. Then
Cl(A) ∼= Cl(A).
Proof. The proof follows by showing that contraction of primes
along the inclu-sion A → A induces a bijection between the sets of
height one primes Spec1(A)and Spec1(A). Let ϕ : Div(A) → Div(A) be
the function obtained by lin-early extending ϕ(P ) := P ∩ A. This
map is clearly a group homomorphism.In the following, we will
demonstrate an equality of rings AP = Aϕ(P ). AsFrac(A) = Frac(A),
any principal divisor divA(f) =
∑ai[Pi] in Div(A) has
image∑ai[Pi ∩ A] = divA(f), which then will guarantee that Cl(A)
∼= Cl(A).
We establish these in the following claims.
Claim 1. If P ∈ Spec1(A), then p := P ∩A ∈ Spec1(A).
Since dim(A) = dim(A), then trdegA(A) = 0. Also, this
forcestrdegκ(p)(κ(P )) = 0. Finally, the dimension equality [8,
Theorem 15.6] holds,so one has
ht(P ) = ht(p) + trdegA(A)− trdegκ(p)(κ(P )).It follows that
ht(P ) = ht(p) = 1.
Claim 2. For every p ∈ Spec1(A), there is P ∈ Spec1(A) with p =
P ∩A.
The existence of a prime P ∈ Spec(A) contracting to p is
guaranteed by thelying-over property of integral extensions. By the
dimension formula
ht(P ) ≤ ht(P ) + trdegκ(p)(κ(P )) = ht(p) + trdegA(A) = ht(p) =
1.
As P is a nonzero ideal of the domain A, ht(P ) = 1.
Claim 3. We have an equality of rings Ap = AP inside their
common fractionfield.
First, observe that Ap is a DVR, so we can write pAp = fAp for
some f ∈ Ap,and every element a ∈ Ap has the form a = wf t for some
unitw ∈ Ap and t ∈ N0.If the equality does not hold, then there
exists x ∈ AP with x /∈ Ap. Since A ⊆ A
-
16 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
is birational, then AP ⊆ Frac(AP ) = Frac(Ap), so we can write x
= a1/a2 witha1, a2 ∈ Ap. By the above, there exist r1, r2 ∈ N0 and
units u1, u2 ∈ Ap suchthat ai = uf ri for i = 1, 2, so x = uf r for
some unit u = u1/u2 ∈ Ap andr = r1 − r2 ∈ Z. Since x /∈ Ap, then r
< 0, and since u−1 and f lie in Ap ⊆ AP ,then also f−1 ∈ AP . We
use it to prove that AP is a field: any non-zero elementy ∈ AP can
be written, as above, in the form y = vfs, where v is a unit in Ap
ands ∈ Z. By the above, both v and f s are units in AP , thus y is
a unit, and thereforeAP is a field. This is a contradiction, so Ap
= AP as claimed.
By the above, for every prime p ∈ Spec1(A), there is a prime P ∈
Spec1(A)lying over p. We now show that P is unique. Let P ′ ∈
Spec(A) be another heightone prime with P ′ ∩ A = p, then by the
arguments above AP ′ = Ap = AP . Lety ∈ P ′, and write y = ab ,
with a, b ∈ AP and b /∈ P . Since y ∈ P
′AP ′ = PAP ,we have by = a ∈ P and thus y ∈ P . It follows that
P ′ ⊆ P and then, bysymmetry, P ′ = P , which concludes the proof.
�
In particular, we obtain that any birational, integral extension
of a k-algebra Asatisfying (R1) has the same class group as A.
Corollary 4.2. Let k be a field and letA ⊆ B be a birational,
integral extension offinitely generated k-algebra domains such that
A satisfies Serre’s condition (R1).Then
Cl(A) ∼= Cl(B).
Proof. Being k-algebras, both A and B are universally catenary
and, by assump-tion, they have the same integral closureA. We then
prove thatB satisfies the (R1)condition. Since the proof of Claim 2
in the previous theorem does not requireA tobe (R1), one has the
every prime ideal p′ ofB is contracted from a height one primeideal
P ofA. One then has natural inclusionsAP∩A ⊆ Bp ⊆ AP . SinceA is
(R1),the proof of Claim 3 of the previous theorem implies that AP∩A
= Bp = AP , soB is (R1).
The conclusion now follows from the previous theorem, because
Cl(A) andCl(B) are both isomorphic to Cl(A). �
The next examples show the necessity of each assumption in the
previous result.
Example 4.3 (Necessity of birationality). Let A = k[x3, x2y,
xy2, y3] be the thirdVeronese of B = k[x, y]. The ring B is
regular, A is (R1), and A → B is anintegral but not birational
extension. One has Cl(B) = 0, but Cl(A) ∼= Z/3Z isnon-zero.
Example 4.4 (Necessity of integrality). LetA = k[x, y, xt, yt]
be the Rees algebraof (x, y) in k[x, y] and B = k[x, y, t]. The
ring B is regular and hence a UFD,A satisfies Serre’s (R1)
property, and A → B is a birational extension but not anintegral
extension. Thus all assumptions of Corollary 4.2 apply except
integrality.
Clearly Cl(B) = 0 but Cl(A) 6= 0 as A is an integrally-closed
non-UFD. Infact, Cl(A) ∼= Z. Thus we cannot remove the integral
extension hypothesis inCorollary 4.2.
-
CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 17
Example 4.5 (Necessity of (R1)). Let A = k[x, xt, t2] be the
Rees-like algebra of(x) in K[x], and let B = k[x, t] be its
integral closure. Then the ring B is regular,A→ B is an integral,
birational extension, but A is not (R1). A is the coordinatering of
a Whitney Umbrella variety which is a semi-normal hypersurface that
is notnormal and thus not (R1). Here we verify that Cl(A) 6= 0.
Since A is not (R1), some care must be taken with computing the
class group.Consider the height one prime ideal P = (x, xy) = xB ∩
A of A. If Cl(A) = 0,then ordP (f) = 1 for some f ∈ Frac(A) =
Frac(B). Writing f = aa′ fora, a′ ∈ A, we have 1 = ordP (f) = λ(AP
/aAP )− λ(AP /a′AP ).
We now find a contradiction by proving that λ(AP /cAP ) is an
even integer forall c ∈ A.
Observe that A ∼= k[u, v, w]/(v2 − u2w), where we identify x ↔
u, xy ↔ v,and y2 ↔ w. It is easy to see that the multiplicity of
the one-dimensional ringAP is 2 and AP has Hilbert-Samuel function
λ(AP /P i+1AP ) = 1 + 2i. Let0 6= c ∈ AP , and write c = αud +
βud−1v + higher order terms in u and vfor some integer d ≥ 0 and
some α, β units in AP not both 0. Thenλ(AP /cAP ) = e(AP /cAP ) =
e(grPAP (AP /cAP )). Since grPAP (AP /cAP )
∼=K(W )[U, V ]/(V 2−U2W,αUd+ βUd−1V ) is defined by a complete
intersectionof degrees 2 and d, it follows that λ(AP /cAP ) = 2d.
This gives a contradiction,so Cl(A) 6= 0.
We now prove that Rees-like algebras have trivial class
groups.
Corollary 4.6. Let S = k[x1, . . . , xn]. If I ⊆ S is an ideal
of height at least two,then Cl(S[It, t2]) = 0.
Proof. By [7, Theorem 6], S[It, t2] satisfies Serre’s condition
(R1). It is easy tocheck that the hypothesis char(k) 6= 2 stated in
[7] is not necessary. Its integralclosure S[t] is a UFD and so has
Cl(A) = 0. Thus Cl(S[It, t2]) = 0 by Theo-rem 4.1. �
4.2. Picard group. Finally we consider the Picard group, i.e.,
the group Pic(R)of invertible fractional ideals modulo principal
fractional ideals ofR. In the normalcase, the Picard group is a
subgroup of the divisor class group, and so if we werein a normal
setting it would be reasonable to expect the Picard group of a
Rees-likealgebra S[It, t2] to also be trivial. However, in our
setting the situation is moreinteresting, as we show that Pic(S[It,
t2]) = 0 if and only if I is radical.
Our approach uses Milnor squares; see e.g. [15, Ex. 2.6]. We use
the followingsetup. Suppose A → B is an inclusion of rings and let
c := AnnA(B/A) be theconductor ideal. The ideal c is the largest
ideal of A that is also an ideal of B. Inthis situation, A is the
pullback of the diagram
A �
//
����
B
����
A/c �
// B/c
-
18 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER
Spec(S[It, t2]) An+1
V (I)× A12 : 1 map
V (I)× A1
FIGURE 1. Pushout diagram for Spec(S[It, t2]) - Pictured withI =
(x) ⊂ k[x]
It is easy to verify that the conductor ideal for the Rees-like
algebra S[It, t2] isAnnS[It,t2](S[t]/S[It, t
2]) = I + It and the corresponding Milnor square is:
S[It, t2] �
//
����
S[t]
����
(S/I)[t2] �
// (S/I)[t].
Dual to this diagram is a pushout of affine varieties, which
provides some per-spective on the geometry of Rees-like algebras.
Fix an ideal I in S = k[x1, . . . , xn]and consider the cylinder V
(I) × A1k inside A
n+1k . Identifying (a, b) ∼ (a,−b) ∈
V (I)× A1K creates a 2 : 1 gluing which, when extended to all of
An+1k , yields the
affine variety Spec(S[It, t2]). The pinch point, or Whitney
umbrella, is the varietyassociated to the Rees-like algebra of the
ideal (x) ⊆ k[x]. Thus one can view va-rieties defined by Rees-like
algebras as higher dimensional analogues of the pinchpoint surface.
See Figure 1.
To compute the Picard group of a Rees-like algebra, we apply the
Units-Picexact sequence associated to the Milnor square for
Rees-like algebra.
Theorem 4.7. Let I ⊆ S = k[x1, . . . , xn] be a ideal. The
Picard groupPic(S[It, t2]) = 0 if and only if I is radical.
-
CANONICAL MODULES AND CLASS GROUPS OF REES-LIKE ALGEBRAS 19
Proof. Given the Milnor square above, the Units-Pic exact
sequence [15, Thm.3.10] is as follows
1 // S[It, t2]× // S[t]× × (S/I)[t2]× // (S/I)[t]×∂
// Pic(S[It, t2]) // Pic(S[t])× Pic((S/I)[t2]) //
Pic((S/I)[t]).
As S[It, t2] and S[t] are standard graded domains, both have
units groups iso-morphic to k×. If I is not radical, then S/I has a
nonzero nilpotent element, sayη ∈ S/I . Since 1 + ηt ∈ (S/I)[t]× r
(S/I)[t2]×, it follows that coker(∂) 6= 0,whence Pic(S[It, t2]) 6=
0.
If on the other hand I is radical, then (S/I)[t]× = (S/I)[t2]× =
k× and so∂ = 0. As S is regular, Pic(S[t]) = 0. The inclusion
(S/I)[t2] → (S/I)[t]is a free extension and hence the natural map
Pic((S/I)[t2]) → Pic((S/I)[t]) isinjective. It follows from the
above sequence that Pic(S[It, t2]) = 0. �
Remark 4.8. The Rees-like algebra S[It, t2] is seminormal if and
only if I is radi-cal, by [7, Corollary 4]. The proof cited states
that I is homogeneous, however thiscondition is not needed. In
general, it is not true that the Picard group of everyseminormal
ring is trivial. Clearly, any number ring with class number
greaterthan 1 is a counterexample. Specifically, the Dedekind
domain R = Z[
√−5] R is
of course normal, whence seminormal, but Pic(R) = Cl(R) ∼= Z/2Z
6= 0.
Remark 4.9. When I is not radical, Pic(S[It, t2]) is not just
nonzero but infinite.Here we consider the case I = (x2) ⊂ k[x]
again. By the proof of Theorem 4.7,Pic(S[It, t2]) ∼= Im(∂) =
coker((S/I)[t2]× → (S/I)[t]×). The units group of(S/I)[t2]
decomposes as k× ⊕
⊕i≥1 k with (α0, α1, α2, . . .) ∈ k× ⊕
⊕i≥1 k
corresponding to α0(1 + α1xt+ α2xt2 + · · · ) ∈ (S/I)[t]×. A
similar calculationworks for (S/I)[t2]×, with the copies of k
appearing in even degrees only. Itfollows that Pic(S[It, t2])
∼=
⊕i∈N k.
The same computation works for I = (x2, y) ⊂ k[x, y], when ht(I)
= 2 andCl(S[It, t2]) = 0.
ACKNOWLEDGEMENTS
The second author was supported by a grant from the Simons
Foundation(576107, JGM) and NSF grant DMS-1900792.
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UNIVERSITY OF ARKANSAS, DEPARTMENT OF MATHEMATICAL SCIENCES,
FAYETTEVILLE,AR 72701
E-mail address: [email protected]
IOWA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS, AMES, IA
50011E-mail address: [email protected]
UNIVERSITY OF ARKANSAS, DEPARTMENT OF MATHEMATICAL SCIENCES,
FAYETTEVILLE,AR 72701
E-mail address: [email protected]
1. Introduction2. Preliminaries3. The canonical module 4. Class
groups4.1. Divisor class group4.2. Picard group
AcknowledgementsReferences