A Remark on FI-Module Homology

A Remark on FI-Module Homology

We e L i a n g G a n&L i p i n g L i

Abstract. We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo–Mumford regularity of a finitely generated torsion FI- module is equal to its degree.

1. Introduction

Let R be a commutative ring. Let FI be the category of finite sets and injec- tive maps. An FI-module is a (covariant) functor from FI to the category of R-modules. In [3, (11)], Church, Ellenberg, Farb, and Nagpal defined, for any FI-moduleV, a complexS_−∗V of FI-modules. The same complex was also con- sidered independently by Putman [7,§4], who called it the central stability chain complex. The purpose of our present paper is to give a proof that the homology of the complexS_−∗V is the FI-homologyH_∗^FI(V )ofV, whose definition we now recall.

LetN be the set of nonnegative integers. For eachn∈N, let [n]be the set {1, . . . , n}withnelements; in particular,[0] = ∅. It is convenient to introduce the nonunitalR-algebra

A:=

0≤m≤n

A_m,n,

where A_m,n is the free R-module on the set HomFI([m],[n]). For any α∈ HomFI([m],[n])andβ∈HomFI([r],[s]), their productαβinAis defined by

αβ:=

α◦β ifs=m,

0 else.

Define a two-sided idealJ ofAby

J:=

0≤m<n

A_m,n.

For eachn∈N, lete_n∈A_n,n be the identity endomorphism of[n]. A leftA- moduleMis said to begradedif M=

n∈Ne_nM. IfV is an FI-module, then

n∈NV ([n])has the natural structure of a graded left A-module. The functor

V →

n∈NV ([n])is an equivalence from the category of FI-modules to the cat- egory of graded leftA-modules. Thus, we shall not distinguish between the notion

Received July 14, 2015. Revision received May 7, 2016.

The second author is supported by the National Natural Science Foundation of China 11541002, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037. We thank the referee for many helpful suggestions that improved the exposition of the paper.

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of FI-modules and the notion of graded leftA-modules. By abuse of notation, we shall also denote

n∈NV ([n])byV.

Leta∈N. Following [1] and [2,Definition 2.3.7], the FI-homologyfunctor H_a^FIis, by definition, theath left-derived functor of the right exact functor

H₀^FI:V →V /J V (1)

from the category of FI-modules to itself. In other words,H_a^FI(V ):=Tor^A_a(A/

J, V ).

The main result of this paper is the following.

Theorem1. LetV be anFI-module.For eacha∈N,we have H_a^FI(V )∼=H_a(S_−∗V ).

Let us briefly describe our strategy for proving Theorem1. For eachd∈Nand a finite setX, let M(d)(X):=RHomFI([d], X), the freeR-module on the set of injections from[d]toX. This defines the FI-module M(d). We shall prove that for eachd∈N, we haveH_a(S_−∗M(d))=0 whenevera≥1. This key result implies that fora≥1, the functorV →H_a(S_−∗V )is coeffaceable because every FI-module is a quotient of

i∈IM(d_i)for somed_i∈N. It is easy to see that the collection of functorsV →Ha(S_−∗V ) for a≥0 form a homologicalδ-functor withH0(S_−∗V )∼=V /J V, and hence Theorem1follows by a standard result in homological algebra.

We give an application of Theorem1.

For any FI-moduleV, itsdegreeis defined to be deg(V ):=sup{n∈N|V ([n])=0},

where by convention the supremum of an empty set is−∞. TheCastelnuovo–

Mumford regularityreg(V ) of V is defined to be the infimum of the set of all c∈Zsuch that

deg(H_a^FI(V ))≤c+a for eacha≥1, where by convention the infimum of an empty set is∞.

Theorem2. LetV be a finitely generatedFI-module withdeg(V ) <∞.Then reg(V )=deg(V ).

After recalling the construction of the complexS_−∗V in the next section, we prove the key result we need in Section 3. The proof of Theorem1 will be given in Section4, and the proof of Theorem2will be given in Section5.

Remark3. Theorem1 was first stated without proof in [3, Remark 2.21]; af- ter our present paper was written, we learnt that a different proof was given by Church and Ellenberg in [1,Proposition 4.9]. Theorem2was first conjectured in [6,Corollary 5.22].

2. Reminder on Koszul Complex of an FI-Module

The complexS_−∗V defined in [3,(11)] is a Koszul complex [5,§12.4]. To state its definition, we need a few notations.

For any finite setI, letRI be the freeR-module with basisI, and det(I )the freeR-module_|I|RI of rank one. In particular, ifI = ∅, then det(I )=R. If I= {i₁, . . . , i_a}, theni₁∧ · · · ∧i_ais a basis for det(I ).

LetV be an FI-module, andXa finite set. SupposeT ⊂Xandi∈X\T. For anyv∈V (T ), we shall writei(v)for the elementι(v)∈V (T ∪ {i}), whereιis the inclusion map fromT toT ∪ {i}. For anya∈N, let

(S₋aV )(X):=

I⊂X

|I|=a

V (X\I )⊗Rdet(I ). (2)

The differentiald:(S₋aV )(X)→(S₋a+1V )(X)is defined on each direct sum- mand by the formula

d(v⊗i₁∧ · · · ∧i_a):=

a

p=1

(−1)^pi_p(v)⊗i₁∧ · · ·i_p· · · ∧i_a,

where v∈V (X\I ), I = {i₁, . . . , i_a}, and i_p means thati_p is omitted in the wedge product. If X is a finite set and α:X→X is an injective map, then α:(S₋aV )(X)→(S₋aV )(X)is defined on each direct summand by the formula

α(v⊗i₁∧ · · · ∧i_a):=α(v)⊗α(i₁)∧ · · · ∧α(i_a),

wherev∈V (X\I ),α(v)∈V (X\α(I )), andI= {i₁, . . . , i_a}. This defines, for eacha∈N, an FI-moduleS₋aV. Thus, we obtain a complexS_−∗Vof FI-modules:

· · · −→S₋2V −→S₋1V −→S0V −→0. (3) For eacha∈N, the homologyH_a(S_−∗V )of the complex (3) is an FI-module.

Observe that for eachn∈N, the image ofd :(S₋1V )([n])→(S0V )([n])is theR-submodule ofV ([n])spanned by all elements of the formα(v)for some injective mapα: [n−1] → [n]andv∈V ([n−1]), so

H0(S_−∗V )∼=V /J V . (4) Remark4. In [3], the complexS_−∗V is constructed via another complexB_∗V defined in [3,(10)]. We have given a direct construction ofS_−∗V.

3. A Key Fact

Recall that for each d ∈N, the FI-module M(d) is defined by M(d)(X):=

RHomFI([d], X)for each finite setX.

Lemma5. Letd∈N.Ifn=d,thenH₀((S_−∗M(d))([n]))=0.

Proof. The complex(S_−∗M(d))([n])is

· · · −→

T⊂[n]

|T|=n−1

RHomFI([d], T )⊗Rdet([n] \T )−→RHomFI([d],[n])−→0.

If n < d, then HomFI(d,[n])= ∅. If n > d, then the image of any injective map from[d]to[n]lies in some subsetT ⊂ [n]with|T| =n−1. The lemma

follows.

The following result plays a crucial role in this paper.

Proposition6. Letd∈N.Ifa≥1,thenHa(S_−∗M(d))=0.

Proof. We have to show that for eachn∈N, the homologyH_a((S_−∗M(d))([n])) is 0 for alla≥1. We do this by induction onn. Ifa > n, then for any FI-module V, we have (S₋aV )([n])=0. Therefore, if a > n, then (S₋aM(d))([n]) is 0 for all d∈N. Thus, then=0 case is trivial. Now suppose thatn >0 and that H_a((S_−∗M(d))([n−1]))=0 for alla≥1 andd∈N.

For eacha∈N, letC_−a be theR-submodule of(S₋aM(d))([n])spanned by the elements α⊗i1∧ · · · ∧ia where α is an injective map from [d] to [n] \ {i1, . . . , ia}andn /∈ {i1, . . . , ia}; thus, C₋a is spanned by the direct summands of (2) forV =M(d)andX= [n]withn /∈I. (Note that the definition of C_−a depends ond andn, but we suppress them from our notation. Moreover,C_−ais not anS_n-submodule of(S₋aM(d))([n]).) It is clear thatC_−∗is a subcomplex of (S_−∗M(d))([n]).

For each basis elementα⊗i₁∧· · ·∧i_a∈C_−a, we have eithern /∈α([d])orn= α(r)for exactly oner∈ [d]; ifn /∈α([d]), then we can considerαas an injective map from[d]to[n−1] \ {i1, . . . , ia}, and soα⊗i1∧ · · · ∧iacan be considered as an element of(S₋aM(d))([n−1]), whereas ifr∈ [d]andn=α(r), thenα|_[d]\{r}

can be considered as an injective map from[d−1]to[n−1] \ {i₁, . . . , i_a}(by fixing an identification of[d]\{r}with[d−1]), and soα|_[d]\{r}⊗i₁∧· · ·∧i_acan be considered as an element of(S₋aM(d−1))([n−1]). Hence, we can identify the complexC_−∗with

(S_−∗M(d))([n−1])⊕ d

r=1

(S_−∗M(d−1))([n−1])

.

The quotient complex of (S_−∗M(d))([n]) by C_−∗ can be identified with (S_−∗M(d))([n−1]), where is the shift functor on complexes such that (K)_−a=K_−a+1for any complexK; specifically, each basis elementα⊗i₁∧

· · · ∧i_a−1 of (S₋a+1M(d))([n−1])(where αis an injective map from [d]to [n−1]\{i1, . . . , ia−1}) can be identified with the elementα⊗i1∧· · ·∧ia−1∧n∈ (S₋aM(d))([n]).

Therefore, we have a short exact sequence of complexes

0−→C_−∗−→(S_−∗M(d))([n])−→(S_−∗M(d))([n−1])−→0.

This gives a long exact sequence in homology

· · · −→Ha(C_−∗)−→Ha((S_−∗M(d))([n]))−→Ha−1((S_−∗M(d))([n−1]))

−→H_a−1(C_−∗)−→ · · ·.

According to the induction hypothesis, ifa≥2, then we haveH_a(C_−∗)=0 and H_a−1((S_−∗M(d))([n−1]))=0. ConsequentlyH_a((S_−∗M(d))([n]))=0. By the induction hypothesis we have H₁(C_−∗)=0 as well. Moreover, by Lemma 5 we have H0((S_−∗M(d))([n−1]))=0 whenever d =n−1. Hence, we have H1((S_−∗M(d))([n]))=0 wheneverd=n−1, and ifd=n−1, then the complex (S_−∗M(d))([n])is

0−→

T⊂[n]

|T|=n−1

RHomFI([n−1], T )⊗Rdet([n] \T )

−→RHomFI([n−1],[n])−→0.

Since the last sequence is exact, we haveH₁((S_−∗M(n−1))([n]))=0.

4. Proof of Theorem1

We will now prove Theorem1.

Proof of Theorem1. For any FI-moduleV anda∈N, we shall write H_a(V ):=H_a(S_−∗V );

our aim is to prove thatH_a^FI(V )∼=H_a(V ).

Any short exact sequence 0→V→V →V→0 of FI-modules gives a short exact sequence 0→S_−∗V→S_−∗V →S_−∗V→0 of complexes. It fol- lows that the systemH=(H_∗)is a homologicalδ-functor. For any FI-moduleV, there is a surjective homomorphism

i∈IM(d_i)→V for somed_i∈N. Hence, by Proposition6the functorH_ais coeffaceable for eacha≥1. By [4,Proposi- tion 2.2.1] it follows thatHis a universal δ-functor. Since, by (1) and (4), we have

H₀(V )∼= V

J V ∼=H₀^FI(V ),

it follows thatH_a(V )∼=H_a^FI(V )for alla.

The following corollary was first proved in [3,Proposition 2.25]; let us show that it is an immediate consequence of Theorem1and [3,Theorem A].

Corollary7. SupposeRis noetherian andV is a finitely generatedFI-module.

For eacha∈N,we havedeg(Ha(S_−∗V )) <∞.

Proof. By [3, Theorem A] there is a projective resolution · · · →P2→P1→ P0 of V, where each Pa is a finitely generated projective FI-module. Apply- ing the functor (1) to the complex P_∗, we obtain a complex P_∗/JP_∗ where each P_a/JPa is a finitely generated FI-module. But for any finitely gener- ated FI-module W with J W =0, we have W ([n])=0 for all n sufficiently

large; indeed, if W is generated by a finite set of elements w_i ∈W ([n_i]) for i=1, . . . , m, then W ([n])=0 for n >max{n1, . . . , nm} since α(wi)=0 for everyα∈HomFI([n_i],[n])withn > n_i. Therefore, we have deg(Pa/JPa) <∞.

By Theorem 1 we have H_a(S_−∗V )∼=H_a^FI(V )∼=H_a(P_∗/JP_∗). Consequently,

deg(Ha(S_−∗V )) <∞.

5. An Application The goal of this section is to prove Theorem2.

Proof of Theorem2. The V =0 case is trivial, so assume thatV =0. Setp= deg(V ).

Leta∈N. Ifn > p+a, thenV ([n−a])=0, so(S₋aV )([n])=0, and hence by Theorem1we haveH_a^FI(V )([n])=0. We claim thatH_a^FI(V )([p+a])=0 for allasufficiently large; this would imply the theorem.

We have:

(S₋(a+1)V )([p+a])∼=V ([p−1])^⊕(^p+a_a+1), (S₋aV )([p+a])∼=V ([p])^⊕(^p+_a^a), (S₋(a−1)V )([p+a])∼=0,

whereV ([p−1])is 0 ifp=0. By Theorem1, to prove our claim, it suffices to show the following statement:for all a sufficiently large, there is no surjective R-linear map

V ([p−1])^⊕(^p+a_a+1)−→V ([p])^⊕(^p+_a^a).

IfR is a field, then the statement is clear since dim(V ([p−1])^⊕(^p+a_a+1))is a polynomial ina of degree at mostp−1, whereas dim(V ([p])^⊕(^p+_a^a))is a poly- nomial ina of degreep. Suppose now thatR is any commutative ring and set M=V ([p]); we can reduce the statement to the case whereRis a field using the following observations:

• SinceMis a nonzeroR-module, there exists a maximal idealqofRsuch that the localizationMqis nonzero; thus, we reduce to the case whereRis a ring.

• IfR is a local ring with unique maximal ideal q, then since M is a nonzero finitely generatedR-module, it follows by Nakayama’s lemma thatM/qMis nonzero; thus, we reduce to the case whereRis a field.

This concludes the proof of the theorem.

We believe that the conclusion of this theorem holds for torsion modules that might not be finitely generated. However, the technique used here (comparing dimensions) does not work in this general case. It would be interesting to find a proof valid for all torsion modules.

References

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[3] T. Church, J. Ellenberg, B. Farb, and R. Nagpal,FI-modules over Noetherian rings, Geom. Top. 18 (2014), no. 5, 2951–2984,arXiv:1210.1854v2.

[4] A. Grothendieck,Sur quelques points d’algèbre homologique,Tôhoku Math. J. (2) 9 (1957), 119–221.

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W. L. Gan

Department of Mathematics University of California Riverside, CA 92521 USA

wlgan@math.ucr.edu

L. Li

Key Laboratory of High Performance Computing and Stochastic

Information Processing (Ministry of Education)

College of Mathematics and Computer Science Hunan Normal University Changsha

Hunan 410081 China

lipingli@hunnu.edu.cn