Unspecified Journal
Volume 00, Number 0, Pages 000-000
S ????-????(XX)0000-0
THE UNSTABLE PARTS FUNCTOR AND INJECTIVE OBJECTS
MARA D. NEUSEL
Abstract.The unstable part functor Unassigns to an arbitrary module over
the Steenrod algebra the largest unstable submodule. We start by showing
some general properties of this functor. Then we study the functor UnO S*
*-1
obtained from Unby precomposition with a localization. We show that UnO
S-1 is an exact functor from the category of unstable noetherian modules
over some unstable noetherian algebra to itself. Along the lines we desc*
*ribe
the injective objects in this category.
1. Introduction
Let F be a finite field of characteristic p. Let H be a graded connected com*
*muta-
tive unstable Noetherian F-algebra over the Steenrod algebra of reduced powers *
*P*.
We denote by MH the category of H P*-modules and by UH the full subcategory
of unstable H P*-modules.
In [1] Dwyer and Wilkerson introducted the functor
Un: MH _ UH, A 7! Un(A)
that maps a H P*-module A to the largest unstable submodule, cf. [5] where the
largest unstable quotient has been studied.
Consider the forgetful functor
F : UH _ MH, M 7! M
that forgets the property of being unstable.
Proposition 1.1. The functor F is left adjoint of Un.
Proof.Let M be an unstable H P*-module, and A an arbitrary H P* module.
We obtain a canonical map
: Hom MH(F(M), A) -! Hom UH(M, Un(A)), OE 7! OE,
which is well-defined, because OE commutes with the P*-action. By construction
is bijective. Thus for any pair of maps f : M -! M0 and g : A -! A0we obtain a
commutative diagram
Hom MH (F(M0), A) F(f)-!HomMH(F(M), A) -g*! Hom MH (F(M), A0)
# # #
* g*
Hom UH(M0, Un(A)) f-! Hom UH(M, Un(A)) - ! Hom UH(M, Un(A0))
____________
Received by the editors June 2, 2007.
2000 Mathematics Subject Classification. Primary 55S10 Steenrod Algebra.
Key words and phrases. Steenrod algebra, unstable algebra over the Steenrod a*
*lgebra, modules
over the Steenrod algebra, unstable submodule, unstable parts.
I thank Bill Dwyer and Craig Huneke for good discussions.
cO0000 (copyright holder)
1
2 MARA D. NEUSEL
with vertical isomorphisms.
From this adjointness property we obtain some immediate corollaries on Un:
Corollary 1.2. Consider the functor Un : MH _ UH. Then
(1)Un is left exact,
(2)Un preserves injectives, and
(3)Un preserves all limits.
Proof.Since F as well as Un are additive and F is exact, the first statement fo*
*llows
from Theorem 2.6.1 in [14]. The second statement follows from adjointness, see,
e.g., Proposition 2.3.10 loc.cit. The last statement follows from Theorem 2.6.*
*10
loc.cit.
We note that Un preserves finite coproducts since it preserves limits. Howev*
*er,
Un preserves all coproducts as we show next.
Lemma 1.3. The functor Un preserves all coproducts.
Proof.Let A = iAi. Then by construction we find
M M
Un(Ai) Un(A) A = Ai.
i i
Let a = iai2 Un(A) of degree d. Then
0 = Pj(a) = iPj(ai)
for all 2j > d (resp. j > d for p = 2) and thus Pj(ai) = 0 for all 2j > d = deg*
*(ai)
(resp. j > d for p = 2).
The next example shows that the functor Un is in general not exact.
Example 1.4. Let F have odd characteristic. Let (H P*)a be a cyclic free module
over H P* with deg(a) = d. Consider the exact sequence
X ` X '
0 -! (H P*)(Pj(a)) ,! (H P*)a -! (H P*)a= (H P*)Pj(a) -! 0.
2j>d 2j>d
We note that the quotient module on the right is by construction unstable. Thus
upon taking unstable parts we obtain
` X '
0 -! 0 ,! 0 -! (H P*)a= (H P*)Pj(a) .
2j>d
Since the map on the right is no longer surjective, we have the desired example.
2.The Functor Un O S-1
Let S H be a multiplicatively closed set. We extend the action of the Stee*
*nrod
algebra to S-1H by setting
(?) P(,)(h_s)P(,)(s) = P(,)(h)
for all h_s2 S-1H, where P(,) denotes the giant Steenrod operation, cf. [13],
Proposition 2.1 in [15], and Section 3.1 in [8]. The action of the Steenrod alg*
*ebra
on the localization S-1H is generally no longer unstable.
UNSTABLE PARTS 3
Remark 2.1. It is easy to see by long division that with definition (?) the alg*
*ebra
S-1H remains closed under the action of the Steenrod algebra, no matter what S
is.
The localization S-1M ~=S-1H HM of an unstable H P*-module thus inherits
a Steenrod algebra action and S-1M becomes an object in MH.
We want to study the composite functor
Un O S-1 : Ufg,H_ UH,
where Ufg,Hdenotes the category of unstable H P*-modules that are finitely
generated as H-modules. This functor appears in several works, e.g., [2], [7], *
*[11],
and [16]. In particular we want to mention the following motivating results.
Denote by Unalgthe functor from the category of algebras over the Steenrod
algebra to the category of unstable algebras over the Steenrod algebra by assig*
*ning
to an object the largest unstable subalgebra. Let us consider the functor Unalg*
*OS-1
as a functor from the category K of unstable algebras to itself. Then for any r*
*educed
Noetherian object H we find that the unstable part of the localization
______
UnalgO S-1(H) = HS-1H
coincides with the integral closure of H in the localization S-1H, see Proposit*
*ion
1.2 in [16] for the case of integral domains H and [7] for the general case.
Remark 2.2. We note that in this case Un O S-1(H) = UnalgO S-1(H) where we
consider in the first expression H as a free module over itself generated by 1.*
* This
can be seen as follows: We have a inclusion of sets UnalgO S-1(H) UnO S-1(H).
Take an element
1_h 2 UnO S-1(H).
s
Then unstability as a module tells us that P(,)(s)|P(,)(h). Comparing highest
coefficients shows that the highest term of P(,)(h_s) is (h_s)q,deg(h)-deg(s), *
*where
q = |F|.
We want to mention also the following result: Consider the special case of
Un O S-1 : Ufg,F[V_]UF[V ]
where S = F[V ] \ p is the complement of a P*-invariant prime ideal. In [2] Dwy*
*er
and Wilkerson showed that this functor coincides with a certain component of
Lannes's T-functor and thus inherits its properties, in particular exactness. I*
*n the
next section we show that the functor
UnO S-1 : Ufg,H_ UH
is exact independent of the choice of S or H. In Section 4 we prove more proper*
*ties
of Un O S-1: in particular we prove that Un O S-1(M) remains noetherian if M is
noetherian. Along the lines we describe the injective objects in Ufg,H. It turn*
*s out
that many classical properties of injective objects in the category of all noet*
*herian
H-modules carry over nicely to the category of unstable noetherian H-modules.
3.Exactness
By Corollary 1.2 the functor Un is left-exact. Since localization is exact,*
* the
composite Un O S-1 is left-exact. Thus in order to show that Un O S-1 exact it *
*is
enough to show that Un O S-1 is exact on injective modules.
We start with an explicit calculation for the case H = F[V ].
4 MARA D. NEUSEL
Example 3.1. Let V = Fn be the n-dimensional vector space over F. Denote by
F[V ] the symmetric algebra over the dual space V *. We want to show that the
functor
Un O S-1 : Ufg,F[V_]UF[V ]
is exact. We go back to the classification of injective objects in the category*
* Ufg,F[V,]
see [4], and find that the indecomposable injectives look like
(o) E[V, W, k] = F[V ] F[V=W]JF[V=W](k)
for some W V and k 2 N0. By Lemma 8.5.3 in [12] the modules E[V, W, k] are
annihilated by some power of the prime ideal pW defined by
pW = ker(F[V ] -! F[W ]).
Thus we find
(
(*) S-1E[V, W, k] = 0 if S \ p 6= ?
S-1F[V ] F[V=W]JF[V=W](k)otherwise.
Since UnO S-1 commutes with coproducts by Lemma 1.3, and injective modules in
Ufg,F[Va]re finite direct sums of indecomposable injectives (o), we are done.
The preceding example suggests that we look for the injective objects in the
category Ufg,H. Those have been classified in [6] and thus we could proceed by *
*an
explicit calculation as above. However, their description is a bit cumbersome, *
*so we
prefer to start with a quick characterization of them. In doing so we do not re*
*fer to
the original classification except that we assume that injective hulls in Ufg,H*
*exist.
Let M be a module in Ufg,H. Denote by E(M) its injective hull in Ufg,H.
Lemma 3.2. Let p H be a P*-invariant prime ideal. Then the injective hull
E(H=p) is indecomposable.
Proof.Assume that E(H=p) = E1 E2 is decomposable as an H P*-module,
where E1, E2 are (necessarily injective) nontrivial modules. Since
H=p ,! E(H=p) = E1 E2
is essential we have that Ei\ H=p 6= 0 for i = 1, 2. Let hi2 Ei\ H=p, hi6= 0. T*
*hen
h1h2 2 (E1\ H=p) \ (E2\ H=p) H=p.
However H=p is an integral domain. This is a contradiction.
P
Remark 3.3. More generally, let1 Ht ~= H=p be a cyclic module in Ufg,H, then
X X
E(Ht) ~=E( H=p) ~= E(H=p)
is indecomposable.
P
Proposition 3.4. If E is an indecomposable injective in Ufg,Hthen E ~= E(H=p)
for some P*-invariant prime ideal p H.
____________
1A P without any indexing denotes a (possibly higher) suspension.
UNSTABLE PARTS 5
Proof.Since E is an object in Ufg,Hthe set of associated prime ideals consists
of P*-invariant prime ideals and is in particular finite, see [9]. Let p H be*
* an
associated prime of E, then we find a map of H P*-modules
Ht ,! E
P
where Ht ~= H=p, see [10]. Since E is indecomposable it follows that
X X
E ~=E(Ht) ~=E( H=p) ~= E(H=p).
P
Proposition 3.5. The set of associated prime ideals of E(H=p) consists solely
of p.
Proof.Since the set of associated prime ideals of a noetherian module M and its
suspensions coincide, it is enough to prove the result for E(H=p).
We denote by I(M) the injective hull of the module M in the category of H-
modules.
Consider the diagram
H=p ,! E(H=p)
\ .
I(H=p)
Since E(H=p) is an object in Ufg,Hthe set of its associated prime ideals consis*
*ts of
(finitely many) P*-invariant prime ideals, see [9]. Let q H be in this set. S*
*ince
the extension H=p ,! E(H=p) is essential we find that
0 6= H=p \ H=q H=p I(H=p).
Since I(H=p) is indecomposable, see, e.g., Theorem 3.3.7 in [3], we have by sym*
*me-
try
I(H=p) = I(H=p \ H=q) = I(H=q).
Thus p = q by Theorem 3.3.8 in [3].
Proposition 3.6. E(H=p) is annihilated by some power of p.
Proof.Let m 2 E(H=p) be a nonzero element. Thus Hm ~=H=Ann (m) E(H=p).
By the preceding result p is the only associated prime of E(H=p). Thus Ann(m) *
* p
and hence p is associated to Hm. Indeed, p is the unique minimal element in the
support of Hm, because the modules involved are noetherian as H-modules. Thus
p is the radical of Ann(m). Since p is finitely generated, pt annihilates m for*
* some
large t 2 N. Since E(H=p) is finitely generated, ps annihilates E(H=p) for some
s 2 N.
Remark 3.7. We note that an element h 2 H \ p induces a monomorphism of
H-modules
~h : E(H=p) -! E(H=p), m 7! hm
since p is the only associated prime ideal. In other words, if S H is a multi*
*plica-
tively closed set with S \ p = ?, then the canonical map
E(H=p) ,! S-1E(H=p)
is a monomorphism.
6 MARA D. NEUSEL
Proposition 3.8. Let M be an object in Ufg,H. Then its injective hull is given *
*by
M X
E(M) = E( H=pi) ni
i
where the sum runs over a finite number of P*-invariant prime ideals p H, and
ni2 N.
Proof.We proceed by induction on the length of M. By [10] an unstable module
M admits a finite prime filtration of unstable H P*-modules
0 ,! M1 ,! M2 ,! M3 ,! . .,.! Mn-1 ,! Mn = M
P
such that Mi=Mi-1~= (H=pi) (as H P*-modules) for some P*-invariant prime
ideal pi H. Thus inductively we obtain
E(M) E(Mn-1) E(M=Mn-1)
X
= E(Mn-1) E( (H=pn))
= . . .
X X
E( (H=p1)) . . .E( (H=pn))
a direct sum of indecomposablePinjectives. Thus E(M) is the direct sum of inde-
composable injectives E( (H=pi)) for certain i 2 {1, . .,.n}.
Remark 3.9. Indeed, the set of prime ideals pi appearing in the injective hull *
*of a
noetherian module M coincides with the set of prime ideals in H associated to M.
This characterization enables us to prove the general theorem.
Theorem 3.10. Let H be an unstable Noetherian F-algebra over the Steenrod al-
gebra. Then the composite functor
UnO S-1 : Ufg,H_ UH
is exact.
Proof.Since our functor is left-exact it is enough to show that it is exact on *
*injec-
tives. By our above characterization of the injective indecomposable modules we
find that (
X 0 if S \ p 6= ?
S-1E( (H=p)) = P
S-1E( (H=p)) otherwise,
cf. Equation (*) above and Theorem 3.3.8 in [3]. Thus an exact sequence
M X M X M X
0 -! E( (H=pi)) bi,! E( (H=pi)) ai- ! E( (H=pi)) ci-! 0
i i i
with necessarily ai= bi+ ciyields an exact sequence
M X M X
0 -! S-1E( (H=pi)) bi,! S-1E( (H=pi)) ai
i,S\pi=? i,S\pi=?
M X
-! S-1E( (H=pi)) ci-! 0.
i,S\pi=?
UNSTABLE PARTS 7
Since Un commutes with coproducts we obtain that
M X M X
0 -! UnO S-1E( (H=pi)) bi,! UnO S-1E( (H=pi)) ai
i,S\pi=? i,S\pi=?
M X
-! UnO S-1E( (H=pi)) ci-! 0.
i,S\pi=?
remains exact.
4.Further Results and Corollaries
By Dwyer and Wilkerson's result UnO S-1(M) is noetherian for any module M
in Ufg,F[Vw]hen S = F[V ]\p is the complement of a P*-invariant prime ideal. Th*
*is
remains true for any H and any S as we see next.
Theorem 4.1. Let H be an unstable graded connected commutative noetherian
algebra. Let M be an object in Ufg,H. Let S H be a multiplicatively closed se*
*t.
Then Un O S-1(M) remains noetherian (as an H-module).
Proof.We proceed by induction on the length of a prime filtration
0 ,! M1 ,! M2 ,! . .,.! Mn = M.
P
If n = 1, then M = Ht ~= H=p. Since the Steenrod algebra acts trivially on t,
see the proofs of Theorems 3.2 and 4.1 in [10], we have that
(______
-1H t if S \ p = ?
Un O S-1(M) = UnO S-1(Ht) = UnO S-1(H)t = HS
0 otherwise,
______
where_HS-1H_denotes the integral closure of H in S-1H. Since H is noetherian, so
is HS-1H, see [7], proving the induction start.
To complete the proof, note that for n > 1 we have a short excat sequence
0 -! Mn-1 ,! Mn i Mn=Mn-1 -! 0
P
with Mn=Mn-1 ~= H=p. By Theorem 3.10 this yields an exact sequence
0 -! UnO S-1Mn-1 ,! UnO S-1Mn i UnO S-1Mn=Mn-1 -! 0
where by induction the two outer modules are noetherian. Thus Un O S-1Mn is
noetherian.
We want to close with a few results on Un O S-1 and injective objects.
Proposition 4.2. Let M be an object in Ufg,H. Let S be a multiplicatively closet
subset of H such that it contains no zero divisors on M. Then the extension
M ,! UnO S-1M
is essential in the category Ufg,H.
Proof.Observe that the additional assumption on S guarantees that the canonical
map M -! S-1M is an inclusion. Furthermore, by the preceding result, UnOS-1M
is noetherian and thus we have an extension
M ,! UnO S-1M
8 MARA D. NEUSEL
in the category Ufg,H. Let N UnO S-1M be an unstable submodule. Let n 2 N.
By construction we can write n as
X hs
n = __ms
s s
with s 2 S, hs 2 H, and ms 2 M. Then
Y
( s)n 2 N \ M.
s
In particular N \ M 6= 0.
Corollary 4.3. Let M be an object in Ufg,H. Let S be a multiplicatively closet
subset of H such that it contains no zero divisors on M. Then
E(M) = E(Un O S-1M).
In particular, if M = E is injecive we have
E = UnO S-1E.
Proof.This is immediate from the preceding Proposition 4.2.
Corollary 4.4. Let E be an injecive module in Ufg,H. Then the module UnOS-1E
is always injective, independent of the choice of S.
Proof.Since E is a finite coproduct of indecomposable injectives, and Un O S-1
commutes with coproducts, it is enough to show this result for E = E(H=p). By
the preceding corollary UnO S-1E(H=p) is injective for S \ p = ?. If S \ p 6= ?*
*, it
is zero.
Remark 4.5. We note that (
S-1I(H=p) = I(H=p) if S \ p = ?
0 otherwise,
see, e.g., Theorem 3.3.8 in [3]. Thus the preceding two corollaries reflect nic*
*ely this
property for the category Ufg,H.
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Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lu*
*b-
bock, Texas 79409
E-mail address: Mara.D.Neusel@ttu.edu