aa r X i v : . [ m a t h . R T ] M a y VIC -modules over noncommutative rings
Andrew Putman ∗ Steven V Sam † Abstract
For a finite ring R , not necessarily commutative, we prove that the category of VIC ( R )-modules over a left Noetherian ring k is locally Noetherian, generalizing a theorem ofthe authors that dealt with commutative R . As an application, we prove a very generaltwisted homology stability for GL n ( R ) with R a finite noncommutative ring. The program of representation stability was introduced by Church and Farb [3, 6]. The ideais that many of the representations that occur in nature depend on a parameter n , and it isuseful to study algebraic structures that encode all of these representations simultaneously.For instance, the cohomology groups of the space Conf n ( R ) of configurations of n labeledpoints in R are representations of the symmetric group S n , which acts by permuting the n points. Individually, these are very hard to understand; however, taken together they havea lot of global structure, especially as n
7→ ∞ . Representations of categories.
This can be encoded in many ways. One of the mostfruitful is Church–Ellenberg–Farb’s [1] theory of FI -modules. Here FI is the category whoseobjects are the finite sets [ n ] = { , . . . , n } and whose morphisms are injections. For acategory C like FI and a ring k , a C -module over k is a functor M from C to k -Mod. Thus M consists of a k -module M c for every object c ∈ C and a k -module map f : M c → M d for every C -morphism f : c → d . For an FI -module M , we will write M n for M [ n ] . The FI -endomorphisms of [ n ] are S n , and these act on M n , making each M n a representation of S n . Example . For a fixed p , we can define an FI -module M over Z with M n = H p (Conf n ( R ); Z ).The induced S n = End FI ([ n ])-action on M n is precisely the S n -action on H p (Conf n ( R ); Z )from the previous paragraph. We therefore get a single object encoding all these representa-tions together with the various ways that they are related as n
7→ ∞ . Homological algebra.
For a category C , the collection of C -modules over k forms anabelian category whose morphisms are natural transformations between functors C → k -Mod.An important insight of Church–Ellenberg–Farb [1] is that one can do commutative andhomological algebra in this category in a way that is very similar to the category k -Mod.For instance, one can construct projective resolutions, take derived functors, etc. ∗ Supported in part by NSF grant DMS-1811322 † Supported in part by NSF grant DMS-1849173 ocal Noetherianity. Perhaps the most important technical result for this is a versionof the Hilbert basis theorem. A C -module M over a ring k is finitely generated if thereexist objects c , . . . , c k ∈ C and elements x i ∈ M c i such that the smallest C -submodule of M containing all the x i is M . In other words, for each c ∈ C the set k M i =1 M f : c i → c f ( x i ) ⊂ M c spans M c . We say that the category of C -modules over k is locally Noetherian if for all finitelygenerated C -modules M over k , all C -submodules of M are finitely generated. Generalizingprevious work that dealt for instance with fields k of characteristic 0, Church–Ellenberg–Farb–Nagpal [2] proved that the category of FI -modules over a left Noetherian ring k islocally Noetherian. VIC-modules.
The category of FI -modules encodes representations of the symmetricgroups, and there has been a huge amount of work developing analogues for other fami-lies of groups (see, e.g., [8, 14, 15, 17, 20]). One particularly important family of groupsare the general linear groups GL n ( R ) over a ring R . Here it is natural to look at categorieswhose objects are the finite-rank free right R -modules R n with n ≥
0. As for the morphisms,there are several potential choices. To help keep the notation for our morphisms straight,we will write [ R n ] when we mean to regard R n as an object of one of our categories and R n when we mean to regard it as an R -module. • The category V ( R ), whose morphisms [ R n ] → [ R m ] are R -linear maps R n → R m .Versions of this go back to work of Lannes and Schwartz and are the focus of the Artinian conjecture (see [11, Conjecture 3.12]), which was resolved independently bythe authors [14] and by Sam–Snowden [17]. • The category VI ( R ), whose morphisms [ R n ] → [ R m ] are injective R -linear maps f : R n → R m that are splittable in the sense that there exists some g : R m → R n with g ◦ f = id. Equivalently, the image of f is a summand of R m . This was introduced byScorichenko in his thesis ([18]; see [7] for a published account). • The category
VIC ( R ), whose morphisms [ R n ] → [ R m ] are pairs ( f , f ), where f : R n → R m is an injective R -linear map and f : R m → R n is a splitting of f , so f ◦ f = id.This was introduced by the authors in [14]. Remark . One motivation for studying
VIC ( R ) is that it is the only one of these categorieswhere there is a functor VIC ( R ) → Groups taking R n ∈ VIC ( R ) to GL n ( R ). For a morphism( f , f ) : [ R n ] → [ R m ], the induced group homomorphism GL n ( R ) → GL m ( R ) is as follows.Set C = ker( f ), so R m = im( f ) ⊕ C . Our homomorphism then takes φ ∈ GL n ( R ) to themap R m → R m obtained from f ◦ φ ◦ f − : im( f ) → im( f ) by extending over C by theidentity. Remark . Our definition of
VIC ( R ) is slightly different from the one in [14], which requiresthat a VIC ( R )-morphism ( f , f ) also have ker( f ) free. For finite (and, more generally,Artinian) rings, this added condition is superfluous: ker( f ) is in any case stably free, andfor Artinian rings finitely generated stably free modules are free (see [13, Example I.4.7.3];rings with this property are called Hermite rings ).2 ain theorem.
Fix a left Noetherian ring k . In [14], it is proven that for a finite commu-tative ring R , the categories of V ( R )- and VI ( R )- and VIC ( R )-modules over k are all locallyNoetherian (see [17] for alternate proofs for V ( R ) and VI ( R ), but not for VIC ( R )). However,in many situations (e.g. in algebraic K-theory), it is important to study GL n ( R ) where R isa noncommutative ring. For instance, R might be a group ring F p [ G ] for a finite group G .Our main theorem addresses this more general situation: Theorem A.
Let R be a finite ring, not necessarily commutative, and let k be a left Noethe-rian ring. Then the categories of V ( R ) -modules and VI ( R ) -modules and VIC ( R ) -modules over k are locally Noetherian.Remark . For the
VIC ( R )-modules VIC ( R ) → k -Mod considered in Theorem A, we allownot just the finite rings R but also the base rings k to be noncommutative, and similarly for V ( R ) and VI ( R ). In fact, for R commutative the proof of Theorem A in [14] works in thatlevel of generality. Remark . For infinite commutative R , the authors proved in [14] that the categories of V ( R )- and VI ( R )- and VIC ( R )-modules over a ring k are not locally Noetherian. The sameargument works for infinite noncommutative R . See [9] for one way to get around this for R = Z . Application: twisted homological stability.
A basic theorem of van der Kallen [19]says that for rings R satisfying mild hypotheses (for instance, all finite rings), the groupsGL n ( R ) satisfy homological stability , i.e. for all p , we haveH p (GL n ( R ); Z ) ∼ = H p (GL n +1 ( R ); Z ) for n ≫ p. In fact, building on ideas of Dwyer [5], van der Kallen is even able to prove this for cer-tain twisted coefficient systems (those that are “polynomial” in an appropriate sense). Forexample, he is able to show for all m ≥ p (GL n ( R ); ( R n ) ⊗ m ) ∼ = H p (GL n +1 ( R ); ( R n +1 ) ⊗ m ) for n ≫ p. In [14, §4], the authors showed how to deduce a much more general version of this for finitecommutative rings from the local Noetherianity of
VIC ( R ). Given our new Theorem A, theexact same argument gives the following result for finite noncommutative rings. For a VIC ( R )-module M , write M n for the value of M on [ R n ] ∈ VIC ( R ). The VIC ( R )-endomorphisms of[ R n ] are GL n ( R ), so M n is a representation of GL n ( R ). Theorem B.
Let R be a finite ring, not necessarily commutative, and let M be a finitelygenerated VIC ( R ) -module over a left Noetherian ring k . Then for all p ≥ , we have H p (GL n ( R ); M n ) ∼ = H p (GL n +1 ( R ); M n +1 ) for n ≫ p .Remark . The proof of Theorem B for commutative rings in [14, §4] uses the more stringentdefinition of
VIC ( R ) discussed in Remark 1.3, which as we discussed there is equivalent toours for finite rings. 3 emark . In [19], van der Kallen also gives an explicit estimate of when this stabilityoccurs. Since we apply our non-effective Noetherianity theorem, we are not able to give suchan estimate.
Ideas from proof.
We will derive Theorem A for V ( R ) and VI ( R ) from the case of VIC ( R ),so we will focus on that category. In [14], this is dealt with for finite commutative R by asort of Gröbner basis argument that was introduced to the theory of representation stabilityin [17] (though the general theorems of [17] do not apply to VIC ( R ); also, we remark thata similar kind of argument appeared much earlier in work of Richter [16]). We do the samething, but the details are far harder. The main issue is that finite noncommutative ringsare much more complicated than finite commutative rings. Indeed, the starting point of theproof in [14] is the fact that finite commutative rings are Artinian, and thus are the productof finitely many local rings. Local rings are not that different from fields, so in the end wecan mostly focus on the case of finite fields. Unfortunately, noncommutative Artinian ringsare not nearly as well-behaved, which greatly complicates the proof. Convention: left vs right modules.
Throughout this paper, we emphasize that columnvectors R n are considered as right R -modules. With this convention, the group GL n ( R )acts on R n on the left by right R -module homomorphisms. If we wanted to deal with left R -modules, then we would have to use row vectors and have GL n ( R ) act on the right. Outline.
We start in §2 by reducing to proving local Noetherianity for an “ordered” versionof
VIC ( R ) called OVIC ( R ). The rest of the paper is devoted to this: in §3, we discuss the struc-ture of finite noncommutative rings, in §4 we define OVIC ( R ) and give its basic properties,and finally in §5 we prove that the category of OVIC ( R )-modules is locally Noetherian. Remark . Some parts of our argument are the same as in [14], but we tried to make thispaper mostly self-contained at least for
VIC ( R ). The fact that we will focus on this singlecategory will allow us to write in a much less abstract way, so one side benefit is that wethink some of the details of the proof here will be a little easier to parse. Acknowledgments.
We would like to thank Benson Farb and Andrew Snowden for helpfulcomments, and Peter Patzt for pointing out a small mistake in an earlier version of thispaper.
VIC
Instead of working with
VIC ( R ) directly, our proof will focus on a subcategory OVIC ( R ). The“O” stands for “ordered”. Its main properties are as follows: Theorem 2.1.
Let R be a finite ring. There exists a subcategory OVIC ( R ) of VIC ( R ) withthe following properties:(a) The objects of OVIC ( R ) are the same as VIC ( R ) : the finite-rank free R -modules R n for n ≥ . b) Every VIC ( R ) -morphism f : [ R d ] → [ R n ] can be factored as [ R d ] f −→ [ R d ] f −→ [ R n ] , where f : [ R d ] → [ R d ] is a VIC ( R ) -morphism and f : [ R d ] → [ R n ] is an OVIC ( R ) -morphism.(c) The category of OVIC ( R ) -modules over a left Noetherian ring k is locally Noetherian. The proof of Theorem 2.1 is spread throughout the rest of the paper: in §3, we discuss somering-theoretic preliminaries, in §4 we construct
OVIC ( R ) and prove part (b) of Theorem 2.1(see Proposition 4.5), and finally in §5 we prove part (c) of Theorem 2.1 (see Proposition5.4). Here we will show how to use Theorem 2.1 to prove Theorem A. Proof of Theorem A, assuming Theorem 2.1.
Let R be a finite ring, not necessarily commu-tative, and let k be a left Noetherian ring. In [14, §2.4], the local Noetherianity of thecategories of V ( R )- and VI ( R )-modules over k for finite commutative rings R are derivedfrom the local Noetherianity of the category of VIC ( R )-modules over k . This derivationdoes not make use of the commutativity of R , so we must just prove that the category of VIC ( R )-modules over k is locally Noetherian.Let M be a finitely generated VIC ( R )-module over k . Our goal is to prove that every VIC ( R )-submodule of M is finitely generated. Theorem 2.1 says that for the subcategory OVIC ( R ) of VIC ( R ), the category of OVIC ( R )-modules over k is locally Noetherian. Viarestriction, we can regard M as an OVIC ( R )-module, so it is enough to prove that M isfinitely generated as an OVIC ( R )-module.We will do this by studying representable VIC ( R )-modules, which function similarly tofree modules. For d ≥
0, let P ( d ) be the VIC ( R )-module defined via the formula P ( d ) n = k [Hom VIC ( R ) ( R d , R n )] ( n ≥ . By Theorem 2.1, every
VIC ( R )-module morphism f : [ R d ] → [ R n ] can be factored as[ R d ] f −→ [ R d ] f −→ [ R n ] , where f : [ R d ] → [ R d ] is a VIC ( R )-morphism and f : [ R d ] → [ R n ] is an OVIC ( R )-morphism.This implies that as an OVIC ( R )-module, P ( d ) is generated by the set Hom VIC ( R ) ( R d , R d ) ⊂ P ( d ) d , which is finite since R is a finite ring.For all x ∈ M d there exists a VIC ( R )-morphism P ( d ) → M taking the element id : [ R d ] → [ R d ] of P ( d ) d = k [Hom VIC ( R ) ( R d , R d )] to x . The image of this VIC ( R )-morphism is the VIC ( R )-submodule spanned by x . Since M is finitely generated, for some d , . . . , d k ≥ x i ∈ M d i such that { x , . . . , x k } generates M . Associated to these x i is a surjective VIC ( R )-morphism k M i =1 P ( d i ) −→ M. Since each P ( d i ) is finitely generated as an OVIC ( R )-module, so is M .5 The structure of Artinian rings
To discuss
OVIC ( R ), we will need some basic facts about finite rings. In fact, the results weneed hold more generally for Artinian rings, so we will state them in this level of generality.A suitable textbook reference is [12]. Throughout this section, R is an Artinian ring. Peirce decomposition, I.
We begin with some generalities (see [12, §21]). Assume that { e , . . . , e µ } are idempotent elements of R that are orthogonal (i.e. e i e j = 0 for distinct1 ≤ i, j ≤ µ ) and satisfy 1 = e + · · · + e µ . Each e i Re j is an additive subgroup of R , and we have the Peirce decomposition R = µ M i,j =1 e i Re j . (3.1)To make this a ring isomorphism, view elements of the right hand side as µ × µ matriceswhose ( i, j )-entries lie in e i Re j . Using the fact that( e i Re k )( e k Re j ) ⊂ e i Re j , we can multiply these matrices as usual, turning the right hand side of (3.1) into a ringand (3.1) into a ring isomorphism. Since e i Re j ⊂ R , we can view (3.1) as an embeddingΦ : R ֒ → Mat µ ( R ) that we will call the Peirce embedding . Peirce decomposition, II.
Continue with the notation of the previous paragraph. A moreconceptual way to think about the Peirce embedding is as follows. Each e i R is a right R -module, and letting R R denote R considered as a right R -module we have R R = µ M i =1 e i R. The ring R acts on the left on R R by right R -module endomorphisms, and in fact R ∼ =End( R R ). We thus have R = End( R R ) = µ M i,j =1 Hom( e j R, e i R ) . (3.2)For all 1 ≤ i, j ≤ µ , we have Hom( e j R, e i R ) = e i Re j , where φ ∈ Hom( e j R, e i R ) correspondsto the element φ ( e j ) ∈ e i Re j . Making these identifications turns (3.2) into (3.1). This makesit clear that the Peirce embedding reflects the left action of R on R R ; indeed, using R R = µ M i =1 e i R, we can embed R R into the set of length- µ column vectors R µ , which is itself a right R -module.The matrices Mat µ ( R ) act on R µ , and we have a commutative diagram R ∼ = −−−→ End( R R ) Φ y y Mat µ ( R ) ∼ = −−−→ End( R µ ) . acobson radical. Let J ( R ) be the Jacobson radical of R . By definition, J ( R ) consists ofall y ∈ R such that for all x, z ∈ R , the element 1 − xyz is a unit. Since R is Artinian, J ( R )can also be characterized as the largest ideal of R that is nilpotent, i.e. such that J ( R ) k = 0for k ≫ R = R/J ( R ). For x ∈ R , let x ∈ R be its image.Also, for a matrix M ∈ Mat n,m ( R ), let M ∈ Mat n,m ( R ) be its image. The following simplefact will be very important for us. Lemma 3.1.
Let R be a ring and let M ∈ Mat n ( R ) for some n ≥ . Then M is invertibleif and only if M is invertible.Proof. We have J (Mat n ( R )) = Mat n ( J ( R )) (see [12, p. 61]), so Mat n ( R ) = Mat n ( R ). Theresult now follows from the fact that for any ring R , an element x ∈ R is invertible if andonly if x ∈ R is invertible. Artin–Wedderburn.
The fact that R is Artinian implies that R is semisimple (see [12,Theorem 4.14]), which by the Artin–Wedderburn Theorem [12, Theorem 3.5] means that R ∼ = Mat µ ( D ) × · · · × Mat µ q ( D q ) (3.3)for division rings D , . . . , D q . We remark that when R is finite as it is in most of thispaper, Wedderburn’s Little Theorem [12, Theorem 13.1] implies that the D k are actually(commutative) fields. The decomposition (3.3) arises from orthogonal idempotents e ki ∈ R for 1 ≤ k ≤ q and 1 ≤ i ≤ µ k satisfying1 = ( e + · · · + e µ ) + · · · + ( e q + · · · + e qµ q ) and e ki R ∼ = D µ k k . (3.4)Here D µ k k denotes the right R -module consisting of length- µ k column vectors with entriesin D k . Setting µ = µ + · · · + µ q , the Peirce embedding associated to (3.4) is precisely theembedding Φ : R ֒ → Mat µ ( R ) taking an element of R to the matrices in (3.3), arranged asdiagonal blocks in Mat µ ( R ). Lifting idempotents.
Since J ( R ) is nilpotent, idempotents in R can be lifted to R (see[12, Theorem 21.28]; we remark that a ring R such that R is semisimple and all idempotentsin R can be lifted to R is called semiperfect ). Combined with [12, Proposition 21.25] andthe proof of [12, Theorem 23.6], this implies we can find orthogonal idempotents e ki ∈ R for1 ≤ k ≤ q and 1 ≤ i ≤ µ k lifting the e ki such that1 = ( e + · · · + e µ ) + · · · + ( e q + · · · + e qµ q ) . (3.5)What is more, by [12, Proposition 21.21], we have e ki R ∼ = e k ′ i ′ R ⇔ e ki R ∼ = e k ′ i ′ R ⇔ k = k ′ . (3.6)For 1 ≤ h, k ≤ q , let L hk = e h Re k ∼ = Hom( e k R, e h R ) . We thus have L kk = D k and L hk = 0 for h = k. L kk are thus local rings, and the L hk are additive subgroups of J ( R ). Summary.
Recall that µ = µ + · · · + µ q . Using the isomorphisms (3.6), the Peirceembedding Φ : R ֒ → Mat µ ( R ) associated to (3.5) can be identified with a ring homomorphismthat takes x ∈ R to a q × q block matrix of the formΦ( x ) = (Φ hk ( x )) qh,k =1 with Φ hk ( x ) ∈ Mat µ h ,µ k ( L hk ) . Moreover,Φ( x ) = Φ( x ) = (cid:16) Φ hk ( x ) (cid:17) qh,k =1 with Φ hh ( x ) ∈ Mat µ h ( D h ) and Φ hk ( x ) = 0 for h = k. We will call this the
Artin–Wedderburn embedding of R . VIC : definition and basic properties
This section defines the subcategory
OVIC ( R ) of VIC ( R ) and proves some basic facts aboutit. We do this in two steps: in §4.1, we deal with semisimple rings, and in §4.2 we deal withArtinian rings (and thus general finite rings). VIC for semisimple rings
We start by introducing the notation we will use in this section. Let R be a semisimple ring,so R ∼ = Mat µ ( D ) × · · · × Mat µ q ( D q ) (4.1)for µ , . . . , µ q ≥ D , . . . , D q . Set µ = µ + · · · + µ q and let Φ : R ֒ → Mat µ ( R ) be the Artin–Wedderburn embedding of R . For x ∈ R , the matrix Φ( x ) thusconsists of the matrices in (4.1), arranged as diagonal blocks in Mat µ ( R ). Decomposing maps.
Consider an R -linear map h : R m → R n . Via (4.1), we can identify h with a collection of D k -linear maps h k : D µ k mk → D µ k nk for 1 ≤ k ≤ q . The matrix of h k is asubmatrix of the matrix corresponding to the R -linear map Φ( h ) : R µm → R µn obtained byapplying Φ to each entry of the matrix representing h . Distinguished bases.
We will need notation for the collections of basis elements of R µm and R µn corresponding to these submatrices. The distinguished basis of R µm is defined asfollows. For each 1 ≤ k ≤ q , let { ~v ( k ) , . . . , ~v ( k ) µ k m } be the portion of the standard basisof R µm corresponding to the columns of Φ( h ) whose nonzero entries are required to be in D k , arranged in their natural increasing order. In its natural ordering, the standard basisfor R µm is thus ~v (1) , . . . , ~v (1) µ , ~v (2) , . . . , ~v (2) µ , . . . , ~v ( q ) , . . . , ~v ( q ) µ q followed by ~v (1) µ +1 , . . . , ~v (1) µ + µ , ~v (2) µ +1 , . . . , ~v (2) µ + µ , . . . , ~v ( q ) µ q +1 , . . . , ~v ( q ) µ q + µ q , ~v (1) ( m − µ +1 , . . . , ~v (1) ( m − µ + µ , . . . , ~v ( q ) ( m − µ q +1 , . . . , ~v ( q ) ( m − µ q + µ q . Similarly, the distinguished basis of R µn is defined by letting { ~w ( k ) , . . . , ~w ( k ) µ k n } for 1 ≤ k ≤ q be the portion of the standard basis of R µn corresponding to the rows of Φ( h ) whosenonzero entries are required to be in D k , arranged in their natural increasing order. For all1 ≤ k ≤ q and 0 ≤ j ≤ µ k m , we thus haveΦ( h )( ~v ( k ) j ) ⊂ µ k n M i =1 ~w ( k ) i · D k . Surjective maps.
Now assume that h : R m → R n is a surjective R -linear map. The maps h k : D µ k mk → D µ k nk discussed above are thus also surjective. Recall that linear algebra overdivision rings is very similar to linear algebra over fields. In particular, notions of basis,dimension, etc. make sense in this noncommutative context. Considerations of dimensionshow that there exists some subset S ⊂ { , . . . , µ k m } such that { Φ( h )( ~v ( k ) i ) | i ∈ S } is abasis for the D k -submodule of R µ k n spanned by { ~w ( k ) , . . . , ~w ( k ) µ k n } . Order µ k n -elementsubsets of { , . . . , µ k m } with the lexicographic order, and define S ( h, k ) to be the smallestsuch S . The following lemma gives an alternate characterization of S ( h, k ): Lemma 4.1.
Let h : R m → R n be a surjective R -linear map. For ≤ k ≤ q , write S ( h, k ) = { j < j < · · · < j µ k n } . Then the j i are the unique elements of { , . . . , µ k m } satisfying thefollowing two conditions: • { Φ( h )( ~v ( k ) j ) , . . . , Φ( h )( ~v ( k ) j µkn ) } is a basis for the D k -module µ k n M i =1 ~w ( k ) i · D k . • Consider ≤ j ≤ µ k m , and let ≤ i ≤ µ k n be the largest index such that j i ≤ j .Then Φ( h )( ~v ( k ) j ) ∈ i M i =1 Φ( h )( ~v ( k ) j i ) · D k . Proof.
Immediate.
Column-adapted maps.
This allows us to make the following definition. A surjective R -linear map h : R m → R n is column-adapted if it satisfies the following condition for each1 ≤ k ≤ q . Write S ( h, k ) = { j < j < · · · < j µ k n } . We then require that Φ( h )( ~v ( k ) j i ) = ~w ( k ) i for all 1 ≤ i ≤ µ k n . One should regard these matrices as being generalizations ofupper triangular matrices. This class of maps is closed under composition: Lemma 4.2.
Let h : R m → R n and h : R n → R ℓ be column-adapted maps. Then h ◦ h : R m → R ℓ is column-adapted. roof. Let ~v ( k ) i and ~w ( k ) i and ~u ( k ) i be the distinguished bases for R µm and R µn and R µℓ ,respectively. Fix some 1 ≤ k ≤ q , and write S ( h , k ) = { j < j < · · · < j µ k n } , S ( h , k ) = { j ′ < j ′ < · · · < j ′ µ k ℓ } . For 1 ≤ i ≤ µ k ℓ , define j ′′ i = j j ′ i . We thus have { j ′′ < j ′′ < · · · < j ′′ µ k ℓ } (4.2)and h ◦ h ( ~v ( k ) j ′′ i ) = h ◦ h ( ~v ( k ) j j ′ i ) = h ( ~w ( k ) j ′ i ) = ~u ( k ) i . From this, it is easy to see that (4.2) satisfies the criterion of Lemma 4.1, so S ( h ◦ h , k )equals (4.2) and h ◦ h is column-adapted. Ordered VIC, semisimple case.
From the above, it makes sense to define
OVIC ( R ) to bethe subcategory of VIC ( R ) whose objects are all the R n with n ≥ f : [ R n ] → [ R m ] are all the VIC ( R )-morphisms f = ( f ′ , f ′′ ) such that f ′′ is column-adapted.Since the only column-adapted maps R n → R n are the identity, it follows that the identity isthe only OVIC ( R )-endomorphism of [ R n ]. In the next section, we will show how to generalizeall of this to the case of Artinian R , and thus in particular to all finite R . VIC for general Artinian rings
Let R be an Artinian ring. The structure of R was discussed in §3. The quotient ring R = R/J ( R ) is semisimple, so R ∼ = Mat µ ( D ) × · · · × Mat µ q ( D q )for µ , . . . , µ q ≥ D , . . . , D q . Set µ = µ + · · · + µ q . Let Φ : R ֒ → Mat µ ( R )and Φ : R ֒ → Mat µ ( R ) be the Artin–Wedderburn embeddings of R and R , so Φ( x ) = Φ( x )for all x ∈ R . Also, for 1 ≤ h, k ≤ q let L hk ⊂ R be as defined in §3, so the L kk are localrings and L kk = D k and L hk = 0 for h = k. The Artin–Wedderburn embedding Φ :
R ֒ → Mat µ ( R ) can then be decomposed into a q × q block matrix of the formΦ( x ) = (Φ hk ( x )) qh,k =1 with Φ hk ( x ) ∈ Mat µ h ,µ k ( L hk ) , andΦ( x ) = Φ( x ) = (cid:16) Φ hk ( x ) (cid:17) qh,k =1 with Φ hh ( x ) ∈ Mat µ h ( D h ) and Φ hk ( x ) = 0 for h = k. Distinguished bases.
Consider an R -linear map h : R m → R n . Let h : R m → R n be theinduced map, and let Φ( h ) : R µm → R µn and Φ( h ) : R µm → R µn be the maps obtained by10pplying Φ and Φ to the entries of matrices representing h and h , respectively. For 0 ≤ k ≤ q ,let { ~v ( k ) , . . . , ~v ( k ) µ k m } and { ~w ( k ) , . . . , ~w ( k ) µ k n } (4.3)be the distinguished bases for R µm and R µn discussed in §4.1. These were introduced tomake sense of Φ( h ). We will need the exact same bases for R µm and R µn , so let { ~v ( k ) , . . . , ~v ( k ) µ k m } and { ~w ( k ) , . . . , ~w ( k ) µ k n } be the subsets of the standard bases for R µm and R µn that map to (4.3) under the maps R µm → R µm and R µn → R µn . For all 1 ≤ k ≤ q and 1 ≤ j ≤ µ k m , we thus haveΦ( h )( ~v ( k ) j ) ∈ q M h =1 µ h n M i =1 ~w ( h ) i · L hk ! . (4.4) S-function.
Given a surjective map h : R m → R n , the induced map h : R m → R n is alsosurjective. For 1 ≤ k ≤ q , we define S ( h, k ) = S ( h, k ) ⊂ { , . . . , µ k m } , so | S ( h, k ) | = µ k n . Column-adapted maps.
A surjective map h : R m → R n is said to be column-adapted if itsatisfies the following two conditions:(i) The map h : R m → R n is column-adapted in the sense of §4.1.(ii) For each 1 ≤ k ≤ q , write S ( h, k ) = { j < j < · · · < j µ k n } . We then require thatΦ( h )( ~v ( k ) j i ) = ~w ( k ) i for all 1 ≤ i ≤ µ k n .This class of maps is closed under composition: Lemma 4.3.
Let h : R m → R n and h : R n → R ℓ be column-adapted maps. Then h ◦ h : R m → R ℓ is column-adapted.Proof. By Lemma 4.2, the map h ◦ h = h ◦ h is column-adapted, so condition (i) issatisfied for h ◦ h . The same argument used in the proof of Lemma 4.2 then shows thatcondition (ii) is satisfied for h ◦ h . The lemma follows. Canonical splittings.
One of the key features of column-adapted maps is the followinglemma. We will call the map g constructed in it the canonical splitting of h ; as the lemmasays, it only depends on S ( h, k ) for 1 ≤ k ≤ q . Lemma 4.4.
For each ≤ k ≤ q , let S ( k ) ⊂ { , . . . , µ k m } be an µ k n -element set. Therethen exists an R -linear map g : R n → R m such that if h : R m → R n is a column-adapted mapwith S ( h, k ) = S ( k ) for all ≤ k ≤ q , then h ◦ g = id. roof. Let ~v ( k ) i and ~w ( k ) i be the distinguished bases for R µm and R µn , respectively. For1 ≤ k ≤ q , write S ( k ) = { j ( k ) , . . . , j ( k ) µ k n } . Define G : R µn → R µm via the formula G ( ~w ( k ) i ) = ~v ( k ) j ( k ) i (1 ≤ k ≤ q, ≤ i ≤ µ k n ) . Since for all 1 ≤ k ≤ q and 1 ≤ i ≤ µ k n we trivially have G ( ~w ( k ) i ) ∈ q M h =1 µ h m M j =1 ~v ( h ) j · L hk , it follows that there exists some g : R n → R m with Φ( g ) = G . If h : R m → R n is a column-adapted map with S ( h, k ) = S ( k ) for all 1 ≤ k ≤ q , then for all 1 ≤ k ≤ q and 1 ≤ i ≤ µ k n we have Φ( h ) ◦ Φ( g )( ~w ( k ) i ) = Φ( h )( ~v j ( k ) i ) = ~w ( k ) i , so Φ( h ) ◦ Φ( g ) = id and thus h ◦ g = id. Ordered VIC, Artinian case.
From the above, it makes sense to define
OVIC ( R ) to bethe subcategory of VIC ( R ) whose objects are all the R n with n ≥ f : [ R n ] → [ R m ] are all the VIC ( R )-morphisms f = ( f ′ , f ′′ ) such that f ′′ is column-adapted.Since the only column-adapted maps R n → R n are the identity, it follows that the identityis the only OVIC ( R )-endomorphism of [ R n ]. Factoring VIC-morphisms.
The following proposition verifies part (b) of Theorem 2.1:
Proposition 4.5.
Let R be an Artinian ring. Every VIC ( R ) -morphism f : [ R d ] → [ R n ] canbe factored as [ R d ] f −→ [ R d ] f −→ [ R n ] , where f : [ R d ] → [ R d ] is a VIC ( R ) -morphism and f : [ R d ] → [ R n ] is an OVIC ( R ) -morphism.Proof. Write f = ( f ′ , f ′′ ), where f ′ : R d → R n is an injection and f ′′ : R n → R d is a splittingof f ′ , so f ′′ ◦ f ′ = id.Let ~v ( k ) i and ~w ( k ) i be the distinguished bases of R µn and R µd , respectively. Also, write S ( f ′′ , k ) = { j ( k ) < · · · < j ( k ) µ k d } ⊂ { , . . . , µ k n } . Define G : R µd → R µd via the formula G ( ~w ( k ) i ) = Φ( f ′′ )( ~v ( k ) j ( k ) i ) (1 ≤ k ≤ q, ≤ i ≤ µ k d ) . Using (4.4) for h = f ′′ , for 1 ≤ k ≤ q and 1 ≤ i ≤ µ k d we have G ( ~w ( k ) i ) ∈ q M h =1 µ h d M j =1 ~w ( h ) j · L hk . g : R d → R d such that G = Φ( g ).Since the columns of Φ( g ) are a basis for R µd , it follows that g is an isomorphism, soby Lemma 3.1 it follows that g is an isomorphism. By construction, the map g − ◦ f ′′ iscolumn-adapted, so f = ( f ′ ◦ g, g − ◦ f ′′ ) is an OVIC ( R )-morphism. Setting f = ( g − , g ),the map f is a VIC ( R )-morphism and f = f ◦ f , as desired. Free and dependent rows.
Consider an
OVIC ( R )-morphism f : [ R n ] → [ R m ] with f =( f ′ , f ′′ ). The condition that f ′′ is column-adapted is a condition on the columns of Φ( f ′′ ) ∈ Mat µn,µm ( R ). We now discuss the rows of Φ( f ′ ) ∈ Mat µm,µn ( R ). We will call the rows ofΦ( f ′ ) that lie in S ( f ′′ , k ) ⊂ { , . . . , µ k m } for some 1 ≤ k ≤ q the dependent rows , and allthe other rows will be called the free rows . The reason for this terminology is the followinglemma: Lemma 4.6.
Let R be an Artinian ring. Consider OVIC ( R ) -morphisms f , f : [ R n ] → [ R m ] with f i = ( f ′ i , f ′′ i ) . Assume that f ′′ = f ′′ and that the free rows of Φ( f ′ ) and Φ( f ′ ) are equal.Then f = f .Proof. What this lemma is saying is that the dependent rows of Φ( f ′ i ) are determined bythe free rows together with the fact that f ′′ i ◦ f ′ i = id. This is a simple fact about matrixmultiplication that is easier to grasp from an example rather than a formal proof: if forinstance we haveΦ( f ′′ i ) = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ and Φ( f ′ i ) = ∗ ∗ ∗⋄ ⋄ ⋄⋄ ⋄ ⋄∗ ∗ ∗⋄ ⋄ ⋄∗ ∗ ∗ , then the ⋄ entries are the dependent rows, and are determined by the ∗ entries together withthe fact that Φ( f ′′ i ) ◦ Φ( f ′ i ) = id. VIC : local Noetherianity
The goal of this section is to prove that the category of
OVIC ( R )-modules over k is locallyNoetherian for a finite ring R and a left Noetherian ring k . This is proved in §5.3, which ispreceded by two preliminary sections: §5.1 discusses well partial orders, and §5.2 constructsa specific ordering that is needed for the proof. Let ( P , (cid:22) ) be a poset. We say that P is well partially ordered if for any infinite sequence p , p , p , . . . ( p i ∈ P ) ,
13e can find indices i < i < i < · · · such that p i (cid:22) p i (cid:22) p i (cid:22) · · · . (5.1)In fact, it is enough to just prove thatthere exist indices i < j with p i (cid:22) p j . (5.2)Here’s a quick proof of this. Letting I = { i | there does not exist j > i with p j (cid:23) p i } , if I isinfinite then it provides a sequence of elements of P violating (5.2), so I must be finite andwe can find the sequence (5.1) starting with any index larger than all the indices in I .We will need the following specific well partial ordering. Fix a finite set Σ, and let Σ ∗ be the set of words s · · · s n whose letters s i are in Σ. Define a partial ordering on Σ ∗ bysaying that s · · · s n (cid:22) t · · · t m if there exists a strictly increasing function f : { , . . . , n } →{ , . . . , m } with the following two properties: • s i = t f ( i ) for 1 ≤ i ≤ n , and • for all 1 ≤ j ≤ m , there exists some 1 ≤ i ≤ n such that f ( i ) ≤ j and t f ( i ) = t j .We then have the following theorem, which is a variant on Higman’s Lemma [10]. Lemma 5.1 ([17, Proposition 8.2.1]) . For all finite sets Σ , the ordering (Σ ∗ , (cid:22) ) is a wellpartial ordering.Remark . An alternate proof of Lemma 5.1 can be found in [4, Proof of Prop. 7.5].
The key to our proof that the category of
OVIC ( R )-modules over k is locally Noetherian fora finite ring R and a left Noetherian ring k is the following lemma. Lemma 5.3.
Let R be a finite ring and let d ≥ . Define P ( d ) = ∞ G n =0 Hom
OVIC ( R ) ( R d , R n ) . There then exists a well partial ordering (cid:22) on P ( d ) along with an extension ≤ of (cid:22) to atotal ordering such that the following holds. Consider OVIC ( R ) -morphisms f : [ R d ] → [ R n ] and g : [ R d ] → [ R m ] with f (cid:22) g . There then exists an OVIC ( R ) -morphism φ : [ R n ] → [ R m ] with the following two properties:(i) g = φ ◦ f , and(ii) if h : [ R d ] → [ R n ] is an OVIC ( R ) -morphism such that h < f , then φ ◦ h < φ ◦ f = g. Proof.
The notation will be as in §4.2. Our finite ring R is Artinian, so R = R/J ( R ) issemisimple and R ∼ = Mat µ ( D ) × · · · × Mat µ q ( D q )for µ , . . . , µ q ≥ D , . . . , D q . Set µ = µ + · · · + µ q . Let Φ : R ֒ → Mat µ ( R )and Φ : R ֒ → Mat µ ( R ) be the Artin–Wedderburn embeddings of R and R , so Φ( x ) = Φ( x )for all x ∈ R . 14 tep 1. We construct the total order ≤ on P ( d ) . Fix an arbitrary total order on R µd . Consider OVIC ( R )-morphisms f : [ R d ] → [ R n ] and g : [ R d ] → [ R m ] in P ( d ). Write f = ( f ′ , f ′′ ) and g = ( g ′ , g ′′ ). We then determine if f ≤ g viathe following procedure: • If n < m , then f < g . • Otherwise, assume that n = m . For each 1 ≤ k ≤ q , we have the µ k d -element subsets S ( f ′′ , k ) and S ( g ′′ , k ) of { , . . . , µ k m } . Order µ k d -element subsets of { , . . . , µ k m } using the lexicographic order, and then further order tuples ( I , . . . , I q ) with I k a µ k d -element subset of { , . . . , µ k m } using the lexicographic ordering. If( S ( f ′′ , , . . . , S ( f ′′ , q )) < ( S ( g ′′ , , . . . , S ( g ′′ , q ))using this order, then f < g . • Otherwise, assume that n = m and that S ( f ′′ , k ) = S ( g ′′ , k ) for all 1 ≤ k ≤ q .Compare the columns of Φ( f ′′ ) ∈ Mat µd,µn ( R ) using our fixed total order on R µd andthe lexicographic order. If under this ordering the columns of Φ( f ′′ ) are less than thecolumns of Φ( g ′′ ), then f < g . • Otherwise, assume that n = m and that f ′′ = g ′′ . Compare the free rows of Φ( f ′ ) ∈ Mat µn,µd ( R ) and Φ( g ′ ) ∈ Mat µn,µd ( R ) using our fixed total order on R µd and thelexicographic order. If under this ordering the free rows of Φ( f ′ ) are less than the rowsof Φ( g ′ ), then f < g .It is clear that this determines a total order ≤ on P ( d ). Step 2.
We construct the partial order (cid:22) such that ≤ is a refinement of (cid:22) . Consider
OVIC ( R )-morphisms f : [ R d ] → [ R n ] and g : [ R d ] → [ R m ] in P ( d ). We then saythat f ≺ g if n < m and there exists a sequence f = h , h , . . . , h m − n = g, where for i ≥ h i +1 : [ R d ] → [ R n + i +1 ] is an OVIC ( R )-morphism related to h i : [ R d ] → [ R n + i ] as follows: • Write h i = ( h ′ i , h ′′ i ). Regard h ′ i and h ′′ i as ( n + i ) × d and d × ( n + i ) matrices, respectively.Pick some 1 ≤ a ≤ b ≤ n + i satisfying the following condition: – Let I = { ( a − µ + 1 , . . . , aµ } be the columns of Φ( h ′′ i ) ∈ Mat µd,µ ( n + i ) ( R ) cor-responding to the a th column of h ′′ i . Then I is disjoint from S ( h ′′ i , k ) for all1 ≤ k ≤ q .Writing h i +1 = ( h ′ i +1 , h ′′ i +1 ), we then have the following: – h ′′ i +1 ∈ Mat d,n + i +1 ( R ) is obtained from h ′′ i ∈ Mat d,n + i ( R ) by inserting a copy ofthe a th column of h ′′ i after the b th column. – h ′ i +1 ∈ Mat n + i +1 ,d ( R ) is obtained from h ′ i ∈ Mat n + i,d ( R ) by inserting a copy of the a th row of h ′ i after the b th row, and then possibly changing the dependent rows toensure that h ′′ i +1 ◦ h ′ i +1 = id.This clearly defines a partial ordering (cid:22) on P ( d ), and since f ≺ g required n < m it refines ≤ . Step 3.
We prove that (cid:22) is a well partial order.
15e will embed ( P ( d ) , (cid:22) ) into a poset (Σ ∗ , (cid:22) ) of words, where Σ is a finite set of lettersand (cid:22) is as in Lemma 5.1. That lemma says that (Σ ∗ , (cid:22) ) is a well partial ordering, so thiswill imply that ( P ( d ) , (cid:22) ) is as well.First, define b R = R ⊔ {♣} , where ♣ is a formal symbol. Though b R is not a ring, it still makes sense to speak about theset of matrices with entries in b R . DefineΣ = { ( M , M ) | M ∈ Mat µ,µd ( b R ) and M ∈ Mat d, ( R ) } . We then define a map ι : P ( d ) → Σ ∗ in the following way.Consider some element f : [ R d ] → [ R n ] of P ( d ). Write f = ( f ′ , f ′′ ). Let c , . . . , c n ∈ Mat d, ( R ) be the columns of the matrix representing f ′′ . Next, via the following procedurewe build modified versions b r , . . . , b r n ∈ Mat µ,µd ( b R ) of the rows of the matrix representing f ′ so as to ignore the dependent rows. Start withΦ( f ′ ) ∈ Mat µn,µd ( R ) . Define [ Φ( f ′ ) ∈ Mat µn,µd ( b R ) to be the matrix obtained from Φ( f ′ ) by replacing each entryin the rows whose numbers lie in S ( f ′′ , k ) by ♣ for all 1 ≤ k ≤ q . These are preciselythe dependent rows. We then define b r , . . . , b r n by letting b r ∈ Mat µ,µd ( b R ) be the submatrixof [ Φ( f ′ ) consisting of the first µ rows, letting b r ∈ Mat µ,µd ( b R ) be the submatrix of [ Φ( f ′ )consisting of the second µ rows, etc. Having done this, we define ι ( f ) = ( b r , c )( b r , c ) · · · ( b r n , c n ) ∈ Σ ∗ . This is an injection since knowing ι ( f ), we can reconstruct f ′′ and all the free rows ofΦ( f ′ ), and this determines f by Lemma 4.6. That ι is order-preserving is immediate fromthe definitions. Step 4.
We construct the φ satisfying (i). Consider
OVIC ( R )-morphisms f : [ R d ] → [ R n ] and g : [ R d ] → [ R m ] with f (cid:22) g . Our goalis to construct an OVIC ( R )-morphism φ : [ R n ] → [ R m ] such that g = φ ◦ f . Examining thedefinition of the partial ordering (cid:22) in Step 2, we see that it is enough to deal with the casewhere m = n + 1 since the general case can be dealt with by iterating this m − n times.Write f = ( f ′ , f ′′ ) and g = ( g ′ , g ′′ ). By definition, there exists some 1 ≤ a ≤ b ≤ n + 1such that the following three things hold: • Let I = { ( a − µ + 1 , . . . , aµ } be the columns of Φ( f ′′ ) ∈ Mat µd,µn ( R ) correspondingto the a th column of f ′′ . Then I is disjoint from S ( f ′′ , k ) for all 1 ≤ k ≤ q . • g ′′ ∈ Mat d,n +1 ( R ) is obtained from f ′′ ∈ Mat d,n ( R ) by inserting a copy of the a th column of f ′′ after the b th column. • g ′ ∈ Mat n +1 ,d ( R ) is obtained from f ′ ∈ Mat n,d ( R ) by inserting a copy of the a th rowof f ′ after the b th row, and then possibly changing the dependent rows to ensure that g ′′ ◦ g ′ = id. 16et ψ : R d → R n be the canonical splitting of f ′′ (see Lemma 4.4). Let c ∈ R d be the a th column of the matrix representing f ′′ , and set b c = ψ ( c ) ∈ R n . We then define φ = ( φ ′ , φ ′′ ) inthe following way: • φ ′′ : R n +1 → R n is represented by the matrix obtained by inserting b c after the b th column of id : R n → R n . • φ ′ : R n → R n +1 is represented by the matrix obtained by first subtracting b c from the a th column of id : R n → R n , and then inserting the row (0 , . . . , , , , . . . ,
0) with a 1in position a after the b th row.For example, for n = 7 and a = 3 and b = 4 we would have φ ′′ = b c b c b c b c b c b c b c φ ′ = − b c − b c − b c − b c − b c − b c − b c . It is clear that φ ′′ ◦ φ ′ = id and that the matrix representing f ′′ ◦ φ ′′ : R n +1 → R d is obtained byinserting f ′′ ( b c ) = c after the b th column of the matrix representing f ′′ . Moreover, examiningthe construction of the canonical splitting in Lemma 4.4, we see that the entries of Φ( b c ) ∈ R µn lying in the free rows of Φ( f ′ ) are all 0, so the matrix corresponding to φ ′ ◦ f ′ is obtainedby first inserting a copy of the a th row of the matrix representing f ′ after the b th row of thatmatrix, and then possibly modifying the dependent rows. Step 5.
We prove that the φ we constructed satisfy (ii). Just like in the previous step, it is enough to deal with the case where m = n + 1, so f : [ R d ] → [ R n ] and φ : [ R n ] → [ R n +1 ]. Consider some OVIC ( R )-morphism h : [ R d ] → [ R n ]such that h < f . Our goal is to prove that φ ◦ h < φ ◦ f . Write f = ( f ′ , f ′′ ) and h = ( h ′ , h ′′ )and φ = ( φ ′ , φ ′′ ).Examining the construction of the total ordering ≤ in Step 1, we see that there are threecases we have to deal with. The first is where( S ( h ′′ , , . . . , S ( h ′′ , q )) < ( S ( f ′′ , , . . . , S ( f ′′ , q )) , where the µ k d -element subsets of { , . . . , µ k n } are ordered using the lexicographic orderingand these tuples are further ordered using the lexicographic ordering. The key fact now isthat given any column-adapted maps ζ , ζ : R n → R d with S ( ζ , k ) < S ( ζ , k ) in the lexico-graphic order, we have S ( ζ ◦ η, k ) < S ( ζ ◦ η, k ) for all column-adapted maps η : R n +1 → R n (see the proof of Lemma 4.2). It follows that( S ( h ′′ ◦ φ ′′ , , . . . , S ( h ′′ ◦ φ ′′ , q )) < ( S ( f ′′ ◦ φ ′′ , , . . . , S ( f ′′ ◦ φ ′′ , q )) , so φ ◦ h < φ ◦ f .The second case is where S ( h ′′ , k ) = S ( f ′′ , k ) for all k , but the columns of Φ( h ′′ ) are lessthan the columns of Φ( f ′′ ) in the lexicographic ordering (using our fixed total ordering on17 µd ). In this case, it follows from our construction of φ that the matrix representing h ′′ ◦ φ ′′ is obtained from the matrix representing h ′′ by inserting a copy of the a th column of thematrix representing f ′′ after the b th column, and similarly for f ′′ ◦ φ ′′ . This implies that thecolumns of Φ( h ′′ ◦ φ ′′ ) remain less than the columns of Φ( f ′′ ◦ φ ′′ ), so φ ◦ h < φ ◦ f .The final case is where h ′′ = f ′′ , but the free rows of Φ( h ′ ) are less than the free rows ofΦ( f ′ ) in the lexicographic ordering. In this case, Φ( φ ′ ◦ h ′ ) is obtained from Φ( h ′ ) by takinga bunch of free rows and duplicating them lower in the matrix, and similarly for Φ( φ ′ ◦ f ′ )(with the same rows). It follows that the free rows of Φ( φ ′ ◦ h ′ ) remain less than the freerows of Φ( φ ′ ◦ f ′ ), so φ ◦ h < φ ◦ f . We now prove the following, which verifies part (c) of Theorem 2.1:
Proposition 5.4.
Let R be a finite ring and let k be a left Noetherian ring. Then thecategory of OVIC ( R ) -modules over k is locally Noetherian.Proof. Just like in the proof of Theorem A in §2, we will prove this by studying representablemodules. For d ≥
0, let P ( d ) be the OVIC ( R )-module defined via the formula P ( d ) n = k [Hom OVIC ( R ) ( R d , R n )] ( n ≥ . As we discussed in the proof of Theorem A, every finitely generated
OVIC ( R )-module over k is the surjective image of a direct sum of finitely many P ( d ) (for differing choices of d ). Toprove that every submodule of such a finitely generated module is finitely generated, it isthus enough to prove this for P ( d ).We start with some preliminaries. Let (cid:22) and ≤ be the orderings on P ( d ) = ∞ G n =0 Hom
OVIC ( R ) ( R d , R n )provided by Lemma 5.3. For a nonzero x ∈ P ( d ) n , define the initial term of x , denotedinit( x ), as follows. Write x = α f + · · · + α k f k with α , . . . , α k ∈ k \ { } and f , . . . , f k ∈ Hom
OVIC ( R ) ( R d , R n ) . Order these terms such that f < f < · · · < f k . Then init( x ) = α k f k .Next, for an OVIC ( R )-submodule M of P ( d ), define the initial module I ( M ) • of M to bethe ordered sequence of k -modules defined via the formula I ( M ) n = k { init( x ) | x ∈ M n } ( n ≥ . Be warned that this need not be an
OVIC ( R )-submodule of P ( d ). However, we do have thefollowing. Claim. If N and M are OVIC ( R ) -submodules of P ( d ) with N ⊂ M and I ( N ) • = I ( M ) • ,then N = M . roof of claim. Assume otherwise, and let n ≥ N n ( M n . Let f : [ R d ] → [ R n ]be the ≤ -minimal element of the set { f | there exists x ∈ M n \ N n and α ∈ k such that init( x ) = αf } . Let x ∈ M n \ N n satisfy init( x ) = αf with α ∈ k . By assumption, there exists some y ∈ N n such that init( y ) = αf . The αf terms cancel in x − y , so init( x − y ) = βg with β ∈ k and g < f . Since x ∈ M n \ N n and y ∈ N n , we have x − y ∈ M n \ N n , so this contradicts theminimality of f .We now commence with the proof that every OVIC ( R )-submodule of P ( d ) is finitelygenerated. Assume otherwise, so there exists a strictly increasing chain M ( M ( M ( · · · of OVIC ( R )-submodules of P ( d ). By the above claim, the sequences I ( M i ) • must all bedistinct, so for all i ≥ n i ≥ α i f i ∈ I ( M i ) n i \ I ( M i − ) n i with α i ∈ k and f i : [ R d ] → [ R n i ] . Let x i ∈ M in i be an element with init( x i ) = α i f i .Since (cid:22) is a well partial ordering, there exists some increasing sequence i < i < i < · · · of indices such that f i (cid:22) f i (cid:22) f i (cid:22) · · · . Since k is a left Noetherian ring, there exists some m ≥ α m +1 is in the left k -idealgenerated by α i , . . . , α i m , i.e. we can write α m +1 = c α i + · · · + c m α i m with c , . . . , c m ∈ k . For 1 ≤ j ≤ m , the fact that f i j (cid:22) f i m +1 implies by part (i) of Lemma 5.3 that there existssome OVIC ( R )-morphism φ j : [ R n ij ] → [ R n im +1 ] such that f i m +1 = φ j ◦ f i j . Conclusion (ii) ofLemma 5.3 implies that init( φ j ◦ x ) = α j f i m +1 . Setting y = m X j =1 c j ( φ j ◦ x i j ) ∈ M i m n im +1 , we thus see that init( y ) = m X j =1 c j α j f i m +1 = α m +1 f i m +1 = init( x i m +1 ) . This contradicts the fact thatinit( x i m +1 ) ∈ I ( M i m +1 ) n im +1 \ I ( M i m ) n im +1 . The proposition follows. 19 eferences [1] T. Church, J. S. Ellenberg and B. Farb, FI-modules and stability for representations of symmetricgroups, Duke Math. J. 164 (2015), no. 9, 1833–1910, arXiv:1204.4533v4 .[2] T. Church, J. S. Ellenberg, B. Farb and R. Nagpal, FI-modules over Noetherian rings, Geom. Topol.18 (2014), no. 5, 2951–2984, arXiv:1210.1854v2 .[3] T. Church and B. Farb, Representation theory and homological stability, Adv. Math. 245 (2013),250–314, arXiv:1008.1368v3 .[4] J. Draisma and J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Math. J. 163(2014), no. 1, 35–63, arXiv:1103.5336v2 .[5] W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980),no. 2, 239–251.[6] B. Farb, Representation stability, Proceedings of the 2014 International Congress of Mathematicians.Volume II, 1173–1196, arXiv:1404.4065v1 .[7] V. Franjou and A. Touzé (eds.),
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