In this paper, we study the asymptotic behavior of lengths of $\tor$ modules
of homologies of complexes under the iterations of the Frobenius functor in
positive characteristic. We first give upper bounds to this type of length
functions in lower dimensional cases and then construct a counterexample to the
general situation. The motivation of studying such length functions arose
initially from an asymptotic length criterion given in [D4] which is a
sufficient condition to a special case of nonnegativity of $\chi_\infty$. We
also provide an example to show that this sufficient condition does not hold in
general.Comment: 11 page

Li, Jinjia

English

arXiv

AbstractIn this paper, we study the asymptotic behavior of lengths of Tor modules of homologies of complexes under the iterations of the Frobenius functor in positive characteristic. We first give upper bounds to this type of length functions in lower dimensional cases and then construct a counterexample to the general situation. The motivation of studying such length functions arose initially from an asymptotic length criterion given in [S.P. Dutta, Intersection multiplicity of modules in the positive characteristics, J. Algebra 280 (2004) 394–411] which is a sufficient condition to a special case of nonnegativity of χ∞. We also provide an example to show that this sufficient condition does not hold in general

Elsevier - Publisher Connector 

Journal of Algebra 285 (2005) 856–867
www.elsevier.com/locate/jalgebra
The upper bound of Frobenius related length
functions
Jinjia Li
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 20 September 2004
Available online 22 January 2005
Communicated by Paul Roberts
Abstract
In this paper, we study the asymptotic behavior of lengths of Tor modules of homologies of com-
plexes under the iterations of the Frobenius functor in positive characteristic. We first give upper
bounds to this type of length functions in lower dimensional cases and then construct a counterexam-
ple to the general situation. The motivation of studying such length functions arose initially from an
asymptotic length criterion given in [S.P. Dutta, Intersection multiplicity of modules in the positive
characteristics, J. Algebra 280 (2004) 394–411] which is a sufficient condition to a special case of
nonnegativity of χ∞. We also provide an example to show that this sufficient condition does not hold
in general.
 2004 Elsevier Inc. All rights reserved.
Keywords: Complex; Homology; Frobenius; Intersection multiplicity
Introduction and notations
In this paper, (A, m, k) will be a complete local ring of characteristic p > 0, m its
maximal ideal, k = A/m and k is perfect. By a free complex we mean a complex F• =
(Fi, di)i0 (· · · → F2 d2−→ F1 d1−→ F0 → 0) of finitely generated free A-modules. We define
codimension of M to be dimA− dimM (denoted by codimM) for any A-module M . The
E-mail address: jinjiali@math.uiuc.edu.0021-8693/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jalgebra.2004.11.015
J. Li / Journal of Algebra 285 (2005) 856–867 857Frobenius endomorphism fA : A → A is defined by fA(r) = rp for r ∈ A. Each iteration
f nA defines a new A-module structure on A, denoted by f
n
A for which a · b = apnb. Write
FnA(M) for M ⊗A f
n
A and FnA(F•) for F• ⊗A f
n
A. We drop the subscript A when there is
no ambiguity.
In [D1], Dutta introduced the following definition of χ∞.
Definition. Let R be a local ring in characteristic p > 0. Let M and N be two finitely
generated modules such that (M ⊗R N) < ∞ and proj dimM < ∞. Define
χ∞(M,N) = lim
n→∞χ
(
Fn(M),N
)
/pn codimM.
χ∞ plays an important role in the study of intersection multiplicity χ defined by
Serre [S], especially in the nonsmooth situation. For example, over complete intersections,
χ∞(M,N) = χ(M,N) when both M and N are of finite projective dimension [D4, Corol-
lary to Theorem 1.2]. Thus the positivity (or nonnegativity) of χ∞ settles the positivity
(respectively nonnegativity) conjecture of χ over complete intersections.
Our main object is to examine the following sufficient condition for the nonnegativity
of χ∞ [D4, Corollary 1 to Theorem 2.2].
Theorem (Dutta). Let R be a local Gorenstein ring in characteristic p > 0. Let M and
N be finitely generated modules of finite projective dimensions such that (M ⊗ N) < ∞.
Suppose dimM + dimN = dimR, dimN = depthN + 1 = s and dimM = depthM +
1 = 2. Then χ∞(M,N) 0, if
lim
n→∞
(
Ext3(N,R) ⊗ H 0m
(
Fn
(
Exts+1(M,R)
))∨)
/pns = 0. (1)
Note here proj dimM = s + 1 and proj dimN = 3 by Auslander–Buchsbaum formula, and
these Exts are the natural duals under the generalized “Matlis” duality.
This study leds us to investigate the asymptotic behavior of (TorAj (Hi(F n(F•)),N)),
where F• is a free complex with homologies of finite length. (F• is not necessarily a
bounded complex here!)
In [D3], Dutta established that

(
TorAj
(
Hi
(
Fn(F•)
)
,N
))
 CijpndimN
when codimN = 1 [D3, Proposition 1.3]. Naturally, one can ask whether this inequality is
still valid when N has higher codimension. Investigation of the length condition (1) raises
the same question. The expectation was that the same inequality should hold in general for
any N , namely, (TorAj (Hi(F
n(F•)),N))CijpndimN . A positive answer to this question
in codimension 3 would yield an affirmative answer for (1). However, our investigation
revealed that one can only extend this for codimN  2.
The following result in Section 1 shows that one can extend this inequality for
codimN  2.
858 J. Li / Journal of Algebra 285 (2005) 856–867Theorem (Corollary 1.3 in Section 1). Let F• be a free complex with homologies of finite
length over a Cohen–Macaulay local ring A. Let N be a finitely generated A-module such
that codimN  2. Then there exist constants Cij ’s, such that

(
TorAj
(
Hi
(
Fn(F•)
)
,N
))
 CijpndimN
for all i, j  0.
When codimN = 3, we provide a counterexample in Section 2. This counterexample in
turn leads to us our main theorem in Section 3.
Main Theorem (Theorem 3.2 in Section 3). Let R = K[[X,Y,U,V ]]/(XY −UV ) where
K is a field of characteristic p > 0 and X, Y , U , V are indeterminates. There exist finitely
generated modules M , N over R as in the above theorem with s = 1, such that the sufficient
condition (1) for nonnegativity of χ∞ fails to hold.
Nevertheless, this counterexample does not give a negative χ∞.
1.
We first state a proposition due to Seibert [Se, Proposition 1, Section 3] which plays a
crucial role in our proof.
Proposition 1.1 (Seibert). Let F• be a free complex over A with homologies of finite length
and N be any finitely generated A-module. Then there exist constants Ci ’s such that

(
Hi
(
Fn(F•) ⊗A N
))
 CipndimN.
The following is our first result which generalizes a result due to Dutta [D3, Proposi-
tion 1.3].
Proposition 1.2. Let F• be a free complex with homologies of finite length over A. Let N
be A/xA or A/(x, y) where {x} or {x,y}, respectively, forms a regular sequence. Then
there exist constants Cij ’s, such that

(
TorAj
(
Hi
(
Fn(F•)
)
,N
))
 CijpndimN (2)
for all i, j  0.
The following special lemma has been used repeatedly in the proof of Proposition 1.2.
We leave the proof as an exercise for the reader.
J. Li / Journal of Algebra 285 (2005) 856–867 859Special Lemma. Let A be a local ring and M be a module over A such that (M) < ∞.
Suppose x is A-regular. Then

(
TorA1 (M,A/xA)
)= (M ⊗A (A/xA)).
Proof of Proposition 1.2. We write A¯ = A/xA and F¯• = F• ⊗A A¯.
Case 1. N = A/xA. This case has already been demonstrated in [D3] in a more general
set up. (See the proof of Proposition 1.3 in [D3], although the official statement there is in
the form of limit.) We give a simple proof of this case anyway for completeness.
Since proj dimN = 1,
TorAj
(
Hi
(
Fn(F•)
)
,N
)= 0
for j  2 and by the special lemma

(
TorA1
(
Hi
(
Fn(F•)
)
,N
))= (Hi(Fn(F•))⊗A N).
Thus it suffices to prove the result for j = 0.
If i = 0, since H0(F n(F•)) ⊗ N = H0(F n(F•) ⊗ N), we get the desired inequality by
Proposition 1.1.
If i  1, since FnA(F•) ⊗A A¯ = FnA¯(F¯•), there is a short exact sequence of complexes
0 → Fn(F•) x−→ Fn(F•) → FnA¯(F¯•) → 0.
Taking the associated long exact sequence of homologies, we get
· · · → Hi
(
Fn(F•)
) x−→ Hi(Fn(F•))→ Hi(FnA¯(F¯•))→ Hi−1(Fn(F•))→ ·· · .
It yields the following short exact sequence:
0 → Hi
(
Fn(F•)
)⊗ A/xA → Hi(FnA¯(F¯•))→ (0 : x)Hi−1(F n(F•)) → 0 (3)
for i  1. So,

(
Hi
(
Fn(F•)
)⊗ A/xA) (Hi(FnA¯(F¯•)))
and again, the desired inequality follows from Proposition 1.1 with N = A¯.
Case 2. N = A/(x, y). In this case, since proj dimN = 2,
TorAj
(
Hi
(
Fn(F•)
)
,N
)= 0
860 J. Li / Journal of Algebra 285 (2005) 856–867for j  3. By a result due to Serre [S, Theorem 1, Chapter IV],
2∑
j=0
(−1)j (TorAj (Hi(Fn(F•)),N))= χ(Hi(Fn(F•)),N)= 0.
Hence, it is enough to prove the result for j = 0 and 1.
Tensor (3) with A/(x, y) ( A¯/yA¯) over A¯. We obtain
· · · → TorA¯1
(
(0 : x)Hi−1(F n(F•)), A¯/yA¯
)→ Hi(Fn(F•))⊗A A/(x, y)
→ Hi
(
Fn
A¯
(F¯•)
)⊗A¯ A¯/yA¯ → (0 : x)Hi−1(F n(F•)) ⊗A¯ A¯/yA¯ → 0 (4)
for i  1. It follows that

(
Hi
(
Fn(F•)
)⊗A A/(x, y))
 
(
TorA¯1
(
(0 : x)Hi−1(F n(F•)), A¯/yA¯
))+ (Hi(FnA¯(F¯•))⊗A¯ A¯/yA¯).
Notice that by the special lemma,

(
TorA¯1
(
(0 : x)Hi−1(F n(F•)), A¯/yA¯
))= ((0 : x)Hi−1(F n(F•)) ⊗A¯ A¯/yA¯)
and from the above long exact sequence (4),

(
(0 : x)Hi−1(F n(F•)) ⊗A¯ A¯/yA¯
)
 
(
Hi
(
Fn
A¯
(F¯•)
)⊗A¯ A¯/yA¯).
Hence

(
Hi
(
Fn(F•)
)⊗A A/(x, y)) 2(Hi(FnA¯(F¯•))⊗A¯ A¯/yA¯).
Therefore by Case 1, we are done for j = 0.
Finally, for j = 1, we use the following spectral sequence obtained by base change:
TorA¯p
(
TorAq
(
Hi
(
Fn(F•)
)
, A¯
)
, A¯/yA¯
) ⇒
p
TorAp+q
(
Hi
(
Fn(F•)
)
,A/(x, y)
)
.
It follows that

(
TorA1
(
Hi
(
Fn(F•)
)
,A/(x, y)
))
 
(
TorA¯1
(
TorA0
(
Hi
(
Fn(F•)
)
, A¯
)
, A¯/yA¯
))+ (TorA¯0 (TorA1 (Hi(Fn(F•)), A¯), A¯/yA¯))
= (Hi(Fn(F•))⊗ A/(x, y))+ (TorA1 (Hi(Fn(F•)), A¯)⊗ A/(x, y)).
The last equality here is by the special lemma again.
J. Li / Journal of Algebra 285 (2005) 856–867 861Since x is A-regular, TorA1 (Hi(F
n(F•)), A¯)  (0 : x)Hi(Fn(F•)). Therefore by (3), we
have a surjection
Hi+1
(
Fn
A¯
(F¯•)
)⊗ A/(x, y) TorA1 (Hi(Fn(F•)), A¯)⊗ A/(x, y).
Thus

(
TorA1
(
Hi
(
Fn(F•)
)
,A/(x, y)
))
 
(
Hi
(
Fn(F•)
)⊗ A/(x, y))+ (Hi+1(FnA¯(F¯•))⊗ A/(x, y)).
Both of the terms on the right-hand side of the above inequality are bounded by a constant
times pndimN by the j = 0 case, and so we are done for j = 1 which finishes our proof. 
Corollary 1.3. Let A be a Cohen–Macaulay local ring and let F• be as in Proposition 1.2.
Let N be a finitely generated A-module such that codimN  2. Then there exist constants
Cij ’s, such that

(
TorAj
(
Hi
(
Fn(F•)
)
,N
))
 CijpndimN
for all i, j  0.
Proof. Suppose codimN = h, h = 1 or 2. Then AnnA N contains an A-regular sequence
{x1, . . . , xh}. We have the following short exact sequence:
0 → Q → (A/(x1, . . . , xh))t → N → 0.
Tensoring the above short exact sequence with Hi(Fn(F•)), we get a long exact sequence
· · · → TorA1
((
A/(x1, . . . , xh)
)t
,Hi
(
Fn(F•)
))
→ TorA1
(
N,Hi
(
Fn(F•)
))→ Q ⊗ Hi(Fn(F•))
→ (A/(x1, . . . , xh))t ⊗ Hi(Fn(F•))→ N ⊗ Hi(Fn(F•))→ 0.
By Proposition 1.2 and induction on j , we obtain the desired inequality. 
Remark 1.4. If A is a regular local ring, since the functor Fn(−) is exact [K, Theorem 3.3],
TorAj (Hi(F
n(F•)),N)  TorAj (Fn(Hi(F•)),N). Thus by Proposition 1.1, the inequality in
Proposition 1.2 holds for any finitely generated A-module N .
862 J. Li / Journal of Algebra 285 (2005) 856–8672.
Now, we demonstrate an example to show that the inequality (2) in Proposition 1.2, as
well as the one in Corollary 1.3, can fail when codimN = 3.
We first state two standard facts in commutative algebra which will be used in the proof
of Proposition 2.4.
Fact 2.1. Let R be a finitely generated algebra over a field K and M be a finitely generated
R-module. Let m be a maximal ideal of R. Suppose SuppM = {m} and K  R/m via the
natural map. Then R(M) = dimK M . Here dimK M denote the dimension of M as a K-
vector space.
Fact 2.2. Let R be a commutative ring and M be a finitely generated R-module. Let R̂m be
the m-adic completion of Rm where m is a maximal ideal of R. If SuppR M = {m}, then
R(M) = Rm(Mm) = R̂m(M̂m).
Lemma 2.3. Let R = K[X,Y,U,V ]/(XY − UV ) where K is a field of characteristic
p > 0 and X, Y , U , V are indeterminates. Consider K as a module over R in the obvious
way. Then HomR(K,FnR(K)) is a K-vector space and
dimK HomR
(
K,FnR(K)
)
 pn.
Proof. To simplify our notations, we use x, y,u, v to denote the images of X, Y , U , V
respectively in any quotient ring of K[X,Y,U,V ] if there is no confusion about that am-
bient quotient ring. HomR(K,FnR(K)) is a K-vector space consisting of all the elements
of FnR(K) which are killed by the maximal ideal (x, y,u, v). Let A= {xp
n−1yiupn−1−i |
0 i  pn − 1}, which is a subset of
FnR(K) =
K[X,Y,U,V ]
(X
pn
, Y
pn
,U
pn
,V
pn
,XY − UV ) .
It is easy to verify that A ⊂ HomR(K,FnR(K)). We will show that elements in A are
linearly independent over K which gives us the desired inequality.
Let {λi}0ipn−1 be elements in K such that
pn−1∑
i=0
λix
pn−1yiupn−1−i = 0 ∈ FnR(K). (5)
Let
S = K[X,Y,U,V ]
pn pn pn pn
.
(X ,Y ,U ,V )
J. Li / Journal of Algebra 285 (2005) 856–867 863Then R = S/(xy − uv). Lift the relation (5) to a relation in S. Since S is a K-vector space
with basis {xiyjukvl | 0 i, j, k, l  pn − 1}, we obtain
pn−1∑
i=0
λix
pn−1yiupn−1−i =
( ∑
0i,j,k,lpn−1
µ
i,j,k,l
xiyjukvl
)
(xy − uv) ∈ S, (6)
where the µ
i,j,k,l
are elements of K . Define
λ
i,j,k,l
=
{
λj , if i = pn − 1, j + k = pn − 1 and l = 0,
0, otherwise.
We also define µ
i,j,k,l
= 0 if one of i, j , k, l is negative.
By comparing the coefficients on both sides of (6), we obtain that
λ
i,j,k,l
= µ
i−1,j−1,k,l − µi,j,k−1,l−1 , ∀i, j, k, l  pn − 1.
Using the above formula repeatedly, noticing that λ
i,j,k,l
= 0 if i < pn − 1, we get
λi = λpn−1,i,pn−1−i,0
= µpn−2,i−1,pn−1−i,0 + 0
= µpn−3,i−2,pn−i,1
= µpn−4,i−3,pn−i+1,2
...
= µpn−i−1,0,pn−2,i−1
= 0
for all i = 0,1, . . . , pn − 1. 
The following is an example where the inequality (2) in Proposition 1.2 fails when
codimN = 3. The complex F• is taken to be a free resolution of K and i = 0.
Proposition 2.4. Let R = K[[X,Y,U,V ]]/(XY −UV ) where K is a field of characteristic
p > 0 and X, Y , U , V are indeterminates. Then

(
TorR3
(
Fn(K),R/(x, y,u + v))) pn.
Proof. Since {x, y, u + v} forms an R-sequence, it follows that
TorR3
(
Fn(K),R/(x, y,u + v)) HomR(R/(x, y,u + v),F n(K)).
864 J. Li / Journal of Algebra 285 (2005) 856–867Since there is a surjection R/(x, y,u+v)K , by applying HomR(−,F n(K)), we obtain
an injection
Hom R
(
K,Fn(K)
)
↪→ HomR
(
R/(x, y,u + v),F n(K)).
From Facts 2.1, 2.2 and Lemma 2.3, we have

(
HomR
(
K,Fn(K)
))
 pn.
Therefore,

(
TorR3
(
Fn(K),R/(x, y,u + v))) pn. 
Remark 2.5. Using the same method, one can show that over the hypersurface ring R =
K[[X1, . . . ,Xt , Y1, . . . , Yt ]]/(∑ti=1 XiYi), (HomR(K,Fn(K))) is unbounded.
3.
In [D4], Dutta gave an asymptotic length condition over Gorenstein local rings of pos-
itive characteristic for the nonnegativity of χ∞(M,N) when dimM = 2. In this section,
we will construct examples to show that over the local hypersurface R discussed in Corol-
lary 2.4, this length condition fails to hold.
Let R be a local ring in characteristic p > 0. Let M and N be two finitely generated
modules such that (M ⊗R N) < ∞, dimM + dimN  dimR and proj dimM < ∞. In
[D1], Dutta defined
χ∞(M,N) = lim
n→∞χ
(
Fn(M),N
)
/pn codimM.
For properties of χ∞, see [D1,D2,R,Se]. Dutta [D4] established the following criterion for
nonnegativity of χ∞ over a local Gorenstein rings of positive characteristic.
Theorem 3.1 (Dutta). Let R be a local Gorenstein ring in characteristic p > 0. Let M and
N be finitely generated modules of finite projective dimension such that (M ⊗ N) < ∞.
Suppose dimM + dimN = dimR, dimN = depthN + 1 = s and dimM = depthM +
1 = 2. Then χ∞(M,N) 0, if
lim
n→∞
(
Ext3(N,R) ⊗ H 0m
(
Fn
(
Exts+1(M,R)
))∨)
/pns = 0.
Here (−)∨ denotes the Matlis duality HomR(−,E) where E is the injective hull of the
residue field of R.
The following is an example where the length criterion in Theorem 3.1 fails.
J. Li / Journal of Algebra 285 (2005) 856–867 865Theorem 3.2. Let R = K[[X,Y,U,V ]]/(XY − UV ) where K is a field of characteristic
p > 0 and X, Y , U , V are indeterminates. There exist finitely generated modules M , N
over R satisfying the conditions in Theorem 3.1 such that
lim
n→∞
(
Ext3(N,R) ⊗ H 0m
(
Fn
(
Ext2(M,R)
))∨)
/pn > 0.
Proof. We are going to construct modules M and N satisfying the conditions in Theo-
rem 3.1 with s = 1, such that Ext2R(M,R)  K and Ext3R(N,R)  K .
Let x, y, u, v denote the images of X, Y , U , V in R. Take a minimal free resolution of
K over R
· · · → Rt ψ−→ R4 φ−→ R → K → 0
where φ can be written as a matrix [x, y,u, v] with respect to the standard bases for R4
and R. Let (−)∗ denote HomR(−,R). Apply (−)∗ to the above exact sequence. Since
depthR = 3, K∗ = 0 and we obtain the following exact sequence
0 → R φ∗−→ R4 → M ′ → 0
where M ′ = cokerφ∗. Let {e1, e2, e3, e4} be a standard basis for R4, it follows that M ′ =
R4/R(xe1 + ye2 + ue3 + ve4). Note that if r ∈ AnnR M ′, then there exists an a ∈ R such
that
r(e1 + e2 + e3 + e4) = a(xe1 + ye2 + ue3 + ve4).
It follows that ax = ay = r . But R is a domain and x = y in R, thus r = 0. Therefore
AnnR M ′ = (0) whence dimM ′ = dimR = 3. Moreover, since Ext1R(K,R) = 0, M ′ =
Imψ∗, which is a submodule of Rt and therefore torsion-free. Hence, x ∈ R is a nonzero
divisor on M ′.
Let M = M ′/xM ′ = R4/(xR4 +R(xe1 +ye2 +ue3 +ve4)). It follows that dimM = 2.
One can also prove that proj dimM = 2 since proj dimM ′ = 1 and x is both M ′-regular and
R-regular. Therefore by Auslander–Buchsbaum formula, depthM = 1. Moreover,
Ext2R(M,R)  Ext1R(M ′,R)  K.
In order to construct N , let R¯ = R/(y,u + v). Then dim R¯ = 1, depth R¯ = 1. Take a
minimal resolution of K over R¯
R¯2
ζ−→ R¯ → K → 0.
Apply HomR¯(−, R¯). Let N = coker ζ ∗ and we obtain a free resolution of N over R¯
0 → R¯ ζ ∗−→ R¯2 → N → 0.
Use a similar argument as before, AnnR¯ N = (0). Hence dimR¯ N = 1, proj dimR¯ N = 1
and depth ¯ N = 0. Therefore dimR N = 1, depthR N = 0 and proj dimR N = 3. Note thatR
866 J. Li / Journal of Algebra 285 (2005) 856–867(M ⊗R N) < ∞ since the annihilator of M ⊗R N contains (x, y,u+ v) which is primary
to the maximal ideal (x, y,u, v). Moreover, Ext3R(N,R)  Ext1R¯(N, R¯)  K .
Finally, to check
lim
n→∞
(
Ext3(N,R) ⊗ H 0m
(
Fn
(
Ext2(M,R)
))∨)
/pn > 0,
it is enough to notice that

(
Ext3(N,R) ⊗ H 0m
(
Fn
(
Ext2(M,R)
))∨)
= ((Ext3(N,R) ⊗ H 0m(Fn(Ext2(M,R)))∨)∨)
= (Hom(Ext3(N,R),H 0m(Fn(Ext2(M,R)))))
= (Hom(Ext3(N,R),H 0m(Fn(K))))
= (Hom(K,Fn(K)))
 pn. 
Remark 3.3. Although the length criterion does not hold in general, there do exist local
Gorenstein rings such that the length criterion holds for all M and N . It would be nice to
have a general method to identify such rings.
Acknowledgments
I am indebted to my thesis advisor Sankar Dutta for his direction and many inspiring
discussions on the subject of this paper. I also thank the referee for the valuable suggestions
and comments.
References
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[D3] S.P. Dutta, Ext and Frobenius, II, J. Algebra 186 (1995) 724–735.
[D4] S.P. Dutta, Intersection multiplicity of modules in the positive characteristics, J. Algebra 280 (2004) 394–
411.
[K] E. Kunz, Characterization of regular local rings for characteristic p, Amer. J. Math. 91 (1969) 772–784.
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The upper bound of Frobenius related length functions 

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The Upper Bound of Frobenius Related Length Functions

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