Modules for elementary abelian groups and hypersurface singularities

24 Pages·2015·0.25 MB·English

by  David J. Benson

CommutativeAlgebraandNoncommutativeAlgebraicGeometry,II
MSRIPublications
Volume68,2015
Modules for elementary abelian groups and
hypersurface singularities
DAVID J. BENSON
ThispaperisaversionofthelectureIgaveattheconferenceon“Representa-
tionTheory,HomologicalAlgebraandFreeResolutions”atMSRIinFebruary
2013,expandedtoincludeproofs. Mygoalsinthislectureweretoexplain
toanaudienceofcommutativealgebraistswhyafinitegrouprepresentation
theoristmightbeinterestedinzerodimensionalcompleteintersections,andto
giveaversionoftheOrlovcorrespondenceinthiscontextthatiswellsuited
tocomputation. Inthecontextofmodularrepresentationtheory,thisgivesan
equivalencebetweenthederivedcategoryofanelementaryabelian p-group
ofrankr,andthecategoryof(graded)reducedmatrixfactorisationsofthe
polynomialy Xp+···+y Xp. Finally,Iexplaintherelevancetosomerecent
1 1 r r
jointworkwithJuliaPevtsovaonrealisationofvectorbundlesonprojective
spacefrommodularrepresentationsofconstantJordantype.
1. Introduction 20
2. TheOrlovcorrespondence 24
3. Thefunctors 26
4. Anexample 28
5. ThebidirectionalKoszulcomplex 29
6. Abimoduleresolution 32
7. Theadjunction 34
8. Theequivalence 34
9. Thetrivialmodule 35
10. Computeralgebra 37
11. Cohomology 38
12. ModulesofconstantJordantype 40
Acknowledgments 41
References 41
ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationundergrantno.
0932078000,whiletheauthorwasinresidenceattheMathematicalScienceResearchInstitute
(MSRI)inBerkeley,California,duringtheSpringsemesterof2013.
19
20 DAVIDJ.BENSON
1. Introduction
Mygoalhereistoexplainwhyafinitegrouprepresentationtheoristmightbe
interestedincommutativealgebra,andinparticulartheOrlovcorrespondence
[Orlov 2006]. I will then give an exposition of the Orlov correspondence for
anarbitraryzero-dimensionalcompleteintersection. Ratherthangodownthe
samerouteasOrlov,mydescriptionwillbebettersuitedtocomputationandwill
havetheaddedadvantageofgivingalift ofthiscorrespondencefromthestable
categorytothederivedcategory. FinallyIshallexplaintherelevancetosome
recentjointworkwithJuliaPevtsova[BensonandPevtsova2012]onrealisation
of vector bundles on projective space from modular representations of constant
Jordantype.
I should point out that Theorem 2.4, the main theorem of this paper, is a
specialcaseofTheorem7.5ofBurkeandStevenson[2015];eventhefunctors
realisingtheequivalencesinthetheoremarethesame. Theproofpresentedhere
usesaminimumofheavymachinery,takingadvantageofthespecialsituationin
hand to reduce to an explicit computation involving the “bidirectional Koszul
complex”,introducedinSection5.
Let G be a finite group and k a field of characteristic p. Recall that many
featuresoftherepresentationtheoryandcohomologyarecontrolledbyelemen-
taryabeliansubgroupsof G,thatis,subgroupsthatareisomorphictoadirect
product E ∼=((cid:90)/p)r ofcyclicgroupsoforder p. Thenumberr ofcopiesof(cid:90)/p
iscalledtherank of E.
Forexample,Chouinard’stheorem[1976]statesthatakG-moduleisprojective
ifandonlyifitsrestrictiontoeveryelementaryabelian p-subgroup E of G is
projective.
A theorem of Quillen [1971a; 1971b] states that mod p cohomology of G
isdetectedupto F-isomorphismbytheelementaryabelian p-subgroupsof G.
Moreprecisely,themap
H∗(G,k)→limH∗(E,k)
←−
E
isan F-isomorphism,wheretheinverselimitistakenoverthecategorywhose
objects arethe elementary abelian p-subgroups E of G and themaps are given
byconjugationsin G followedbyinclusions. Tosaythatamapof(cid:70) -algebras
p
isan F-isomorphismmeansthatthekernelisnilpotent, andgivenanyelement
of the target, some p-power power of it is in the image. This is equivalent to
thestatementthatthecorrespondingmapofprimeidealspectraintheopposite
direction is a homeomorphism in the Zariski topology. The role of the coho-
mologyringintherepresentationtheoryof G hasbeeninvestigatedextensively.
MODULESFORELEMENTARYABELIANGROUPS 21
We refer the reader in particular to [Alperin 1987; Benson et al. 1997; 2011;
Linckelmann1999].
Soourgoalwillbetounderstandthestablemodulecategorystmod(kG). This
categoryhasasitsobjectsthefinitelygeneratedkG-modules,anditsarrowsare
givenby
Hom (M,N)=Hom (M,N)/PHom (M,N),
kG kG kG
wherePHom (M,N)denotesthelinearsubspaceconsistingofthosehomomor-
kG
phismthatfactorthroughsomeprojectivekG-module. NotethatthealgebrakG
isself-injective,meaningthattheprojectiveandinjectivekG-modulescoincide.
Thecategorystmod(kG)isanotanabeliancategory,butratheratriangulated
category. Thisistrueforanyfinitedimensionalself-injectivealgebra,ormore
generally for the stable category of any Frobenius category. The details can
be found in [Happel 1988]. This triangulated category is closely related to
the boundedderivedcategory Db(kG). Let perf(kG) be the thick subcategory
of Db(kG) consisting of the perfect complexes, namely those complexes that
are isomorphic in Db(kG) to finite complexes of finitely generated projective
kG-modules.
Theorem1.1. Thereisacanonicalequivalencebetweenthequotient
Db(kG)/perf(kG)
andthestablemodulecategorystmod(kG).
This theorem appeared in the late 1980s in the work of several people and
in several contexts; see, for example, [Buchweitz 1986; Keller and Vossieck
1987; Rickard 1989, Theorem 2.1]. It motivated thefollowing definition for any
ring R.
Definition1.2. Let R bearing,andletDb(R)betheboundedderivedcategory
offinitelygenerated R-modules. Thenthesingularitycategoryof RistheVerdier
quotient
D (R)=Db(R)/perf(R).
sg
Likewise,if Risagradedring,wedenotebyDb(R)theboundedderivedcategory
offinitelygeneratedgraded R-modulesandD (R)thequotientbytheperfect
sg
complexesofgradedmodules.
Warning 1.3. In commutative algebra, this definition is much better behaved
forGorensteinringsthanformoregeneralcommutativeNoetherianrings. For
a Gorenstein ring, the singularity category is equivalent to the stable category
of maximal Cohen–Macaulay modules [Buchweitz 1986], but the following
exampleistypicalofthebehaviourfornon-Gorensteinrings.
22 DAVIDJ.BENSON
Example1.4. Let R bethering
k[X,Y]/(X2,XY,Y2).
Thenthe radicalof R isisomorphic tok⊕k,so thereisa shortexactsequence
of R-modules
0→k⊕k → R→k →0.
This means that in D (R) the connecting homomorphism of this short exact
sg
sequencegivesanisomorphismk ∼=k[1]⊕k[1]. Wehave
k ∼=k[1]⊕2∼=k[2]⊕4∼=k[3]⊕8∼=··· ,
andsok isaninfinitelydivisiblemodule. ItsendomorphismringEndDsg(R)(k)is
thecolimitof
k →Mat (k)→Mat (k)→Mat (k)→··· ,
2 4 8
where each matrix ring is embedded diagonally into a product of two copies,
sitting in the next matrix ring. In fact, this endomorphism ring is an example
of a von Neumann regular ring. For a generalisation of this example to finite
dimensionalalgebraswithradicalsquarezero,see[Chen2011].
The reason why D (R) is called the “singularity category” is that it only
sg
“sees”thesingularlocusof R.
Example 1.5. If R is a regular ring then R has finite global dimension. So
Db(R)=perf(R)andhenceD (R)=0.
sg
Moregenerally,wehavethefollowing.
Definition 1.6. Let R be a [graded] Noetherian commutative ring. Then the
singularlocus of R is the set of [homogeneous] prime ideals p of R such that
the[homogeneous]localisation R isnotregular.
p
Remark 1.7. Provided that R satisfies a mild technical condition known as
“excellence”,thesingularlocusisaZariskiclosedset,sothatitisoftheform
V(I) for some [homogeneous] radical ideal I of R. Thus a ∈ I if and only if
R[a−1]isregular. Quotientsofpolynomialrings,forexample,areexcellent.
Theorem1.8. Let R bea[graded]NoetheriancommutativeringoffiniteKrull
dimensionwhosesingularlocusisaZariskiclosedset. ThenD (R)isgenerated
sg
by [the graded shifts of] the modules R/p where p is a [homogeneous] prime
idealinthesingularlocusof R.
Remark1.9. Intheungradedcase,thetheoremofSchoutens[2003]impliesthe
abovetheorem,butitisstronger,andtheproofismorecomplicated. Schoutens’
theoremalsoholdsinthegradedcase,withminoradjustmentstotheproof.
MODULESFORELEMENTARYABELIANGROUPS 23
I’dliketothankSrikanthIyengarforsuggestingthesimpleproofpresented
here. TheideabehindthisargumentalsoappearsinLemma2.2ofHerzogand
Popescu[1997]. ThistheoremwillbeusedintheproofofProposition8.1.
ProofofTheorem1.8. Letd =dimR andlet V(I)bethesingularlocusof R. It
sufficestoshowthatif M isafinitelygenerated[graded] R-modulethen M is
inthethicksubcategoryofDb(R)generatedby[gradedshiftsof] R andof R/p
withp∈V(I),thatis,withp⊇ I.
Thefirststepistoreplace M byitsd-thsyzygy(cid:127)d(M),thatis,thed-thkernel
inany[graded]resolutionof M byfinitelygeneratedfree[graded] R-modules
(weallowfreegradedmodulestobesumsofshiftsof R). Thuswemayassume
that M isad-thsyzygy.
Weclaimthatifa isa[homogeneous]elementof I thenforsomen>0,an
annihilates Ext1(M,(cid:127)(M)). This is because R[a−1] has global dimension at
R
most d, so the fact that M is a d-th syzygy implies that M[a−1] is also a d-th
syzygyandishenceprojectiveasan R[a−1]-module. So
Ext1(M,(cid:127)(M))[a−1]=Ext1 (M[a−1],(cid:127)(M[a−1]))=0.
R R[a−1]
Apply this to the extension 0→(cid:127)(M)→ F → M →0, with F a finitely
generatedfree[graded] R-module. Multiplyingthisextensionbyan amountsto
formingthepullback X inthefollowingdiagram:
0 (cid:47)(cid:47) (cid:127)(M) (cid:47)(cid:47) X (cid:47)(cid:47) M (cid:47)(cid:47) 0
an
(cid:15)(cid:15) (cid:15)(cid:15)
0 (cid:47)(cid:47) (cid:127)(M) (cid:47)(cid:47) F (cid:47)(cid:47) M (cid:47)(cid:47) 0
The resulting extension splits, so X ∼= M⊕(cid:127)(M). The snake lemma implies
thatthemiddleverticalarrowgivesrisetoanexactsequence
0→Ker(an,M)→ X → F → M/anM →0.
Now Ker(an,M) and M/anM are annihilated by an. So X, and hence M, is
in the thick subcategory of Db(R) generated by R and modules supported on
R/aR. Nowinducting ona finiteset ofgenerators forthe ideal I, wesee that
M isinthethicksubcategoryofDb(R)generatedby R andmodulessupported
on V(I). Thelatterareinturngeneratedbythe R/pwithp∈V(I). (cid:3)
Example1.10. Considerthegradedring A=R/(f)where R=k[X ,...,X ],
1 n
each X isgivensomenonnegativedegree,and f issomehomogeneouselement
i
ofpositivedegree. ThenBuchweitz[1986]showedthat
D (A)(cid:39)MCM(A),
sg
24 DAVIDJ.BENSON
the stable category of (finitely generated, graded) maximal Cohen–Macaulay
A-modules. Eisenbud [1980] showed that this category is equivalent to the
categoryofreducedmatrixfactorisationsof f over R.
IfweletY =ProjA,thequasiprojectivevarietyofhomogeneousprimeideals
of A, then the category Coh(Y) of coherent sheaves on Y is equivalent to the
quotient of thecategorymod(A)of finitelygenerated graded A-modules bythe
Serre subcategory of modules which are only nonzero in a finite number of
degrees. We write D (Y) for the corresponding singularity category, namely
sg
thequotientDb(Coh(Y))/perf(Y),whereperf(Y)denotestheperfectcomplexes.
Thuswehave
D (Y)(cid:39)MCM(A),
sg
whereMCM(A)isthequotientofMCM(A)bythemaximalCohen–Macaulay
approximationsofmoduleswhichareonlynonzeroinafinitenumberofdegrees.
Gradingconventions. Wegradeeverythinghomologically,sothatthedifferen-
tialdecreasesdegree. Whenwetalkofcomplexesofgradedmodules,thereare
twosubscripts. Thefirstsubscriptgivesthehomologicaldegreeandthesecond
givestheinternaldegree. IfC isacomplexofgradedmoduleswithcomponents
Ci,j thenwewriteC[n]forthehomologicalshift: C[n]i,j =Ci+n,j,andC(n)
fortheinternalshift: C(n)i,j =Ci,j+n.
2. TheOrlovcorrespondence
In this section we give a version of the Orlov correspondence for a complete
intersection of dimension zero. Let C = k[X ,...,X ]/(f ,..., f ) where
1 r 1 r
f ,..., f is a regular sequence contained in the square of the maximal ideal
1 r
(X ,...,X ).
1 r
Example2.1. Let
E =(cid:104)g ,...,g (cid:105)∼=((cid:90)/p)r
1 r
andletk beafieldofcharacteristic p. LetkE bethegroupalgebraof E overk,
andlet
X =g −1∈kE.
i i
ThenkE =k[X ,...,X ]/(Xp,...,Xp)isacompleteintersectionofcodimen-
1 r 1 r
sionr anddimensionzero.
Let R =k[X ,...,X ] and let R=k[y ,...,y ]⊗ R . We regard R as a
0 1 r 1 r k 0
gradedpolynomialringwiththe y indegreeoneandthe X indegreezero. Let
i i
f = y f +···+y f ∈ R,
1 1 r r
MODULESFORELEMENTARYABELIANGROUPS 25
anelementofdegreeone. Let
A= R/(f) and B = R/(f ,..., f )=k[y ,...,y ]⊗ C.
1 r 1 r k
Wehaveadiagram
(cid:111)(cid:111)(cid:111)(cid:111) (cid:111)(cid:111)(cid:111)(cid:111)
B(cid:79)(cid:79) A (cid:102)(cid:102) R(cid:79)(cid:79)
C (cid:111)(cid:111)(cid:111)(cid:111) k[X ,...,X ]
1 r
Taking Proj of these graded rings, we get the diagram in Section 2 of [Orlov
2006]:
Z(cid:31)(cid:127) i (cid:47)(cid:47) Y(cid:31)(cid:127) u (cid:47)(cid:47) S(cid:48)
π
p q
(cid:15)(cid:15) (cid:31)(cid:127) j (cid:31)(cid:31)(cid:47)(cid:47) (cid:15)(cid:15)
X S
Theorem 2.2 (Orlov). The functor (cid:82)i∗p∗: Db(X) → Db(Y) descends to an
equivalenceofcategoriesstmod(C)=D (X)→D (Y). Therightadjoint
sg sg
(cid:82)p∗i(cid:91)=(cid:82)p∗(cid:76)i∗(−⊗ωZ/Y)[−r+1] (2.3)
givestheinverseequivalence.
Our goal is to prove the following lift of the Orlov correspondence to the
derivedcategory.
Theorem2.4. ThereisanequivalenceofcategoriesDb(C)(cid:39)D (A)liftingthe
sg
equivalenceD (X)(cid:39)D (Y)ofOrlov.
sg sg
Remark2.5. InOrlov’sversion,hemakesuseoflocaldualityasdescribedin
ChapterIII,Corollary7.3of[Hartshorne1966]toidentifytherightadjoint(2.3).
Inourversion,thisisreplacedbytheself-duality(6.1)ofacomplex (cid:49)ˆ giving
A A
aTateresolutionof A asan A-A-bimodule. Thediscrepancybetweentheshift
of−r+1in(2.3)andtheshiftof−r in(6.1)isexplainedbythefactthatinour
situationthesheafωZ/Y isjustO(−1).
In the following corollary, we spell out the consequences for the modular
representationtheoryofelementaryabelian p-groups.
Corollary 2.6. Let E be an elementary abelian p-group of rank r. Then the
followingtriangulatedcategoriesareequivalent:
(1) thederivedcategoryDb(kE);
26 DAVIDJ.BENSON
(2) thesingularitycategoryofgraded A-modulesD (A)where
sg
A= R/(f), R=k[y ,...,y ,X ,...,X ], f = y Xp+···+y Xp,
1 r 1 r 1 1 r r
the X havedegreezeroandthe y havedegreeone;
i i
(3) thestablecategoryofmaximalCohen–Macaulaygraded A-modules;
(4) thecategoryofreducedgradedmatrixfactorisationsof f over R.
WeshallseethatunderthecorrespondencegivenbythisCorollary,theimage
ofthetrivialkE-modulek isa2r−1×2r−1 matrixfactorisationgivenbytaking
theevenandoddtermsinabidirectionalKoszulcomplex. Theperfectcomplexes
correspondto themaximal Cohen–Macaulayapproximationsto the A-modules
whicharenonzeroonlyinfinitelymanydegrees,sothattheequivalencedescends
toOrlov’sequivalence
stmod(kE)(cid:39)D (Proj(A)).
sg
Theelements y correspondtothebasisfortheprimitiveelementsin H2(E,k)
i
obtained by applying the Bockstein map to the basis of H1(E,k) dual to
X ,...,X .
1 r
3. Thefunctors
Firstwedescribethefunctor(cid:56): Db(C)→Dsg(A). If M∗ isaboundedcomplex
offinitelygeneratedC-modulesthenthetensorproduct
k[y ,...,y ]⊗ M
1 r k ∗
is a bounded complex of finitely generated B-modules, which may then be
regardedasaboundedcomplexoffinitelygenerated A-modules. Passingdown
to the singularity category Dsg(A), we obtain (cid:56)(M∗). Thinking of Dsg(A) as
equivalent to MCM(A), we can view (cid:56)(M∗) as a maximal Cohen–Macaulay
approximationtok[y1,...,yr]⊗k M∗.
Next we describe the functor (cid:57): D (A)→Db(C). An object N in D (A)
sg sg
can be thought of as a maximal Cohen–Macaulay A-module. It is therefore
representedbyareducedgradedmatrixfactorisationofthepolynomial f over
R. Namely,wehaveapairoffinitelygeneratedfree R-modules F and F(cid:48) and
maps
F −→U F(cid:48)−→V F(1),
MODULESFORELEMENTARYABELIANGROUPS 27
suchthatUV and VU arebothequalto f timestheidentitymap. Thenwehave
thefollowingsequenceoffreeC-modules:
... ...
V
(cid:119)(cid:119)
C⊗ F U (cid:47)(cid:47) C⊗ F(cid:48)
R0 0 R0 0
V
(cid:120)(cid:120)
C⊗ F U (cid:47)(cid:47) C⊗ F(cid:48)
R0 1 R0 1
V
(cid:119)(cid:119)
... ...
Sincethefreemodulesarefinitelygenerated,thisiszerofarenoughupthepage.
WeshallseeinLemma3.1below,thattheresultingcomplexonlyhashomology
in a finite number of degrees. It is therefore a complex of free C-modules,
bounded to the left, and whose homology is totally bounded. It is therefore a
semi-injectiveresolutionofawelldefinedobjectinDb(C). Weshiftindegree
sothat theterm F ⊗ C appearsin degree−r, andthis isthe object(cid:57)(N)in
0 R0
Db(C).
Anotherwayofviewingtheobject(cid:57)(N)istotakeacompleteresolutionof
N asan A-module:
··· (cid:47)(cid:47) P (cid:47)(cid:47) P (cid:47)(cid:47) P (cid:47)(cid:47) ···
1 0 (cid:61)(cid:61) −1
(cid:33)(cid:33)
(cid:61)(cid:61) N
(cid:34)(cid:34)
0 0
andthen(cid:57)(N)isthecomplex(B⊗A P∗)0[−r],whosedegreen termis(B⊗A
P∗)(n−r,0).
Lemma3.1. If M isamaximalCohen–Macaulay A-modulethenforall j ≥0,
TorA(B,M)isnonzeroonlyinfinitelymanydegrees.
j
Proof. Thisfollowsfromthefactthat B islocally(butnotglobally)acomplete
intersectionasan A-module. Moreexplicitly,for1≤i ≤r,inthering R[y−1]
i
wehavethefollowingequation:
i
Xp = y−1(f −y f −···↑···−y f ).
i i 1 1 r r
Itfollowsthat
i
B[y−1]= A[y−1]/(f ,···↑··· , f )
i i 1 r
28 DAVIDJ.BENSON
is a complete intersection of codimensionr −1 over A[y−1]. Using a Koszul
i
complex,itfollowsthatforall A-modules M andall j ≥r wehave
TorA(B,M)[y−1]=TorA[yi−1](B[y−1],M[y−1])=0.
j i j i i
If M isamaximalCohen–Macaulaymodulethentheminimalresolutionof M is
periodicandsoTorA(B,M)[y−1]=0forall j ≥0. SinceTorA(B,M)isfinitely
j i j
generated,itisannihilatedbyahighenoughpowerofeach y andishenceitis
i
nonzeroonlyinfinitelymanydegrees. (cid:3)
4. Anexample
Before delving into proofs, let us examine an example in detail. Let E =
((cid:90)/p)2=(cid:104)g ,g (cid:105),anelementaryabeliangroupofranktwo,andletk beafield
1 2
ofcharacteristic p. Thensetting X =g −1, X =g −1,wehave
1 1 2 2
C =kE =k[X ,X ]/(Xp,Xp),
1 2 1 2
A=k[y ,y ,X ,X ]/(y Xp+y Xp).
1 2 1 2 1 1 2 2
Letuscompute(cid:56)(k),wherek isthetrivialkE-module. Thismeansweshould
resolvek[y ,y ]asan A-module,andlookatthecorrespondingmatrixfactori-
1 2
sation. Thisminimalresolutionhastheform
(cid:18)y Xp−1 −y Xp−1(cid:19) (cid:18) X y Xp−1(cid:19)
2 2 1 1 2 1 1
X X −X y Xp−1
A(−1)⊕A(−1)−−−−−1−−−−−−−2−−→ A⊕A(−1)−−−−−1−−2−−2−−→ A⊕A
(cid:0) (cid:1)
X X
−−−1−−−2→ A→k[y ,y ].
1 2
This pair of 2 × 2 matrices gives a matrix factorisation of the polynomial
y Xp + y Xp, and it is the matrix factorisation corresponding to the trivial
1 1 2 2
kE-module.
Applying the functor (cid:57) to this matrix factorisation, we obtain the minimal
injective resolution of the trivial kE-module, shifted in degree by two. The
elements y and y givetheactionofthedegreetwopolynomialgeneratorsof
1 2
H∗(E,k)ontheminimalresolution.
Similarly,wecompute(cid:56)(kE)usingthefollowingresolution:
(cid:18)y −y (cid:19) (cid:18) Xp y (cid:19)
2 1 2 1
Xp Xp −Xp y
A(−1)⊕A(−1)−−−−1−−−2−→ A⊕A(−1)−−−−−1−−−2→ A⊕A
(cid:0)Xp Xp(cid:1)
−−−1−−−2→ A→kE⊗ k[y ,y ].
k 1 2