a r X i v :m a t h /0202227v 1  [m a t h .A G ]  22 F e
b  2002A Fitting Lemma for Z /2-graded modules
by
David Eisenbud and Jerzy Weyman*
Abstract.We study the annihilator of the cokernel of a map of free Z /2-graded modules over a Z /2-graded skew-commutative algebra in characteristic 0and define analogues of its Fitting ideals.We show that in the “generic”case the annihilator is given by a Fitting ideal,and explain relations between the Fitting ideal and the annihilator that hold in general.Our results generalize the classical Fitting Lemma,and extend the key result of Green [1999].They depend on the Berele-Regev theory of representations of general linear Lie super-algebras.
Introduction.The classical Fitting Lemma (Fitting [1936])gives information about the annihilator of a module over a commutative ring in terms of a presentation of the madule by generators and relations.More precisely,let φ:R m →R d be a map of finitely generated free modules over a commutative ring R ,and for any integer t ≥0let I t (φ)denote the ideal in R generated by the t ×t minor
s of φ.Fitting’s result says that the module coker φis annihilated by I d (φ),and that if φis the generic map—represented by a matrix whose entries are distinct indeterminates—then the annihilator is equal to I d (φ).Thus I d (φ)is the best approximation to the annihilator that is compatible with base change.Morever,I d (φ)is not too bad an approximation to ann coker φin the sense that I d (φ)⊃(ann coker φ)d ,or more precisely (ann coker φ)I t (φ)⊂I t +1(φ)for all 0≤t <d .In this paper we will prove a corresponding result in the case of Z /2-graded modules over a skew-commutative Z /2-graded algebra containing a field K of characteristic 0.Let R be a Z /2-graded skew-commutative K -algebra:that is,R =R 0⊕R 1as vector spaces,R 0is a commutative central subalgebra,R i R j ⊂R i +j (mod 2),and every element of R 1squares to 0.Any homogeneous map φof Z /2-graded free R -modules may be written in the form
φ:R m ⊕R n (1)
*The second named author is grateful to the Mathematical Sciences Research Institute for support in the period this work was completed.Both authors are grateful for the partial support of the National Science Foundation.Version of 2/18/02
Now let K be afield,and let U=U0⊕U1and V=V0⊕V1be Z/2-graded vector spaces of dimensions(d,e)and(m,n)respectively.We consider the generic ring
S=S(V⊗U):=S(V0⊗U0)⊗S(V1⊗U1)⊗∧(V0⊗U1)⊗∧(V1⊗U0),
where S denotes the symmetric algebra and∧denotes the exterior algebra,and the generic,or tautological map
Φ:S⊗V
X A B Y
-R d⊕R e(1),
be a Z/2-graded map of free modules over a Z/2-graded skew-commutative K-algebra R.
a)When R=S andφ=Φ,the generic map defined above,the annihilator of the cokernel ofΦis IΛ(d,e)(Φ),whereΛ(d,e)is the partition(d+1,d+1,...,d+1,d)of(d+1)(e+1)−1into e+1 parts.In general we have IΛ(d,e)(φ)⊂ann coker(φ).
b)If x1,...,x e∈ann coker(φ), e∈IΛ(0,e)(φ).Moreover,if0≤s≤d−1,and x1,...,x e+1∈ann coker(φ), e+1IΛ(s,e)(φ)⊂IΛ(s+1,e)(φ).
The proof is given in sections2and3below.
In the classical case(e=n=0)we can also describe the annihilator of cokerΦby saying that it is nonzero only if m≥d,and then it is generated,as a gl(V)×gl(U)-ideal,by an m×m minor ofΦ.To simplify the general statement we note that a shift of degree by1does not change the annihilator of the cokernel ofΦ,but has the effect of interchanging m with n and d with e. Corollary2.With notation as above,the annihilator of the cokernel ofΦis nonzero only if
a)m>d(or symmetrically n>e)or
b)m=d and n=e.
In each of these cases the annihilator is generated as a g-ideal by one element Z of degree de+d+e defined as follows:
In case a)when m>d
Z=Z1·X(1,...,d|1,...,d)
where X(1,...,d|1,...,d)is the d×d minor of X corresponding to thefirst d columns and Z1= j≤e,k≤d+1b j,k is the product of all the elements in thefirst d+1columns of B(and symmetrically if n>e);
In case b)
Z=W1···W e·det(X)
where W s is the(d+1)×(d+1)minor ofΦcontaining X and the entry y s,s,that is,
W s=det(X)y s,s+ 1≤i,≤d±det(X(ˆi,ˆj))a i,s b s,j.
In this case our result shows that the cokernel has annihilator equal to the product
(x1,1,x1,2)(b1,1b1,2,x1,1b1,2−x1,2b1,1),
which is minimally generated by4elements.The element Z is x1,1b1,1b1,2.
Example3.As afinal2×2example,consider the case m=n=d=e=1which for simplicity we write as x a
b y .
Here the annihilator of the cokernel is again minimally generated by4elements,namely
axy,bxy,(xy−ab)x,(xy+ab)y.
The element Z is(xy+ab)x.In an Appendix we will explain the g action on these elements. Positive ch
aracteristics.Already with m=n=d=e=1as in Example3the annihilator is different in characteristic2:in characteristic zero the annihilator is generated by forms of degree3, but in characteristic2the algebra R is commutative,so the determinant xy−ab is in the annihilator as well.
The annihilator can differ in other characteristics as well.Macaulay2computations show that the case d=1,e=p−1,m=2,n=0is exceptional in characteristic p for p=3,5and7. Perhaps the same holds for all primes p.
The cokernel of the generic matrix over the integers can also have Z-torsion.For example Macaulay2computation shows that if d=1,e=2,m=3,n=1then the cokernel ofΦZ has 2-torsion.
Our interest in extending the Fitting Lemma was inspired by Mark Green’s paper[1999]where he shows that the exterior minors are in the annihilator.Green’s striking use of his result to prove one of the Eisenbud,Koh,Stillman conjectures on linear syzygies turns on the fact that if N is a module over a polynomial ring S=K[X1,...,X m]then T:=Tor S∗(K,M)is a module over the ring R=Ext∗S(K,K),which is an exterior algebra.Green in effect translated the hypothesis of the linear syzygy conjecture into a statement about the degree1part of the R-free presentation matrix of the submodule of T representing the linear part of the resolution of N,and then showed that the exterior minors generated
a certain power of the maximal ideal of the exterior algebra,which was sufficient to prove the Conjecture.Green’s result only gives information on the annihilator in the case where the elements of the presentation matrix are all odd.Elements of even degree in an exterior algebra can behave(if the number of variables is large)very much like variables in a polynomial ring,at least as far as expressions of bounded degree are concerned.Thus to extend Green’s work it seemed natural to deal with the case of Z/2-graded algebras.
This work is part of a program to study modules and resolutions over exterior algebras;see Eisenbud-Fløystad-Schreyer[2001],and Eisenbud-Popescu-Yuzvinsky[2000]for further informa-tion.
generatedWe would never have undertaken the project reported in this paper if we had not had the program Macaulay2(www.math.uiuc.edu/Macaulay2)of Grayson and Stillman as a tool;its ability to compute in skew commutative algebras was invaluable infiguring out the pattern that the results should have and in assuring us that we were on the right track.
1.Berele-Regev Theory
For the proof of Theorem1we will use the beautiful results of Berele and Regev[1987]giving the structure of R as a module over g.For the convenience of the reader we give a brief sketch of what is
needed.We make use of the notation introduced above:U=U0⊕U1and V=V0⊕V1are Z/2 graded vector spaces over thefield K of characteristic0with dim U=(d,e)and dim V=(m,n).
The Z/2-graded Lie algebra gl(V)is the vector space of Z/2-graded endomorphisms of V= V0⊕V1.Thus
gl(V)=gl(V)0⊕gl(V)1,
where gl(V)0is the set of endomorphisms preserving the grading of V and gl(V)1is the set of endomorphisms of V shifting the grading by1.Additively
gl(V)0=End K(V0)⊕End K(V1),
gl(V)1=Hom K(V0,V1)⊕Hom K(V1,V0)
The commutator of the pair of homogeneous elements x,y∈gl(V)is defined by the formula
[x,y]=xy−(−1)deg(x)deg(y)yx.
By a gl(V)-module we mean a Z/2-graded vector space M=M0⊕M1with a bilinear map of Z/2-graded vector spaces◦:gl(V)×M→M satisfying the identity
[x,y]◦m=x◦(y◦m)−(−1)deg(x)deg(y)y◦(x◦m))
for homogeneous elements x,y∈gl(V),m∈M.
In contrast to the classical theory,not every representation of the Z/2-graded Lie algebra gl(V)is semisimple.For example its natural action on mixed tensors V⊗k⊗V∗⊗l is in general not completely reducible.However,its action on V⊗t decomposes just as in the ungraded case: Proposition1.1.The action of gl(V)on V⊗t is completely reducible for each t.More precisely,the analogue of Schur’s double centralizer theorem holds and the irreducible gl(V)-modules occurring in the decomposition of V⊗t are in1-1correspondance with irreducible representations of the symmetric groupΣt on t letters.These irreducibles are the Schur functors
Sλ(V)=e(λ)V⊗t
where e(λ)is a Young idempotent corresponding to a partitionλin the group ring of the symmetric groupΣt.
Proposition1.1implies that the parts of the representation theory of gl(V)×gl(U)that involve only tensor products of V and U and their summands are parallel to the representation theory in the case V1=U1
=0,which is the classical representation theory of product of the two general linear Lie algebras gl(V0)×gl(U0).
The Proposition also implies that the decompositions into irreducible representations of tensor products of the Sλ(V),as well as the decompositions of their symmetric and exterior powers, correspond to the decompositions in the even case:we just have to replace the ordinary Schur functors S,∧by their Z/2-graded analogues S, .
The formulas giving equivariant embeddings or equivariant projections may also be derived from the corresponding formulas in the even case by applying the principle of signs:The formulas in the even case involve many terms where the basis elements are permuted in a prescribed way. The basis elements have degree0.To write down a Z/2-graded analogue of such formula we simply allow the basis elements to have even or odd degree and we adjust the signs of terms in such way that changing the order of two homogeneous elements x and y of V in the Z/2-graded analogue of the formula will cost the additional factor(−1)deg(x)deg(y).
There is a Z/2-graded analogue of the Cauchy decomposition,which follows as just described from Proposition1.1together with the corresponding result in the even case(proven in[MD],ch.1 and[DC-E-P]).Recall that g=gl(V)×gl(U).
Corollary1.2.The t th component S t(V⊗U)of S(V⊗U)decomposes as a g-module as
S t(V⊗U)=⊕λ,|λ|=t Sλ(V)⊗Sλ(U).
Although it is not so simple to describe the vectors in S t(V⊗U)that lie in a given irreducible summand,we can,as in the commutative case,define afiltration that has these irreducible repre-sentations as successive factors.We start by defining a mapρt: t V⊗ t U֒→S t(V⊗U)as the composite
t V⊗t U-S t(V⊗U)
where thefirst map is the tensor product of the two diagonal maps(here we use the sign conventions for Z/2-graded vector spaces)and the second map simply pairs corresponding factors.Thus
ρt(v1∧...∧v t⊗u1∧...∧u t)= σ∈Σt±(v1⊗uσ(1))·...·(v t⊗uσ(t))
where the sign±is the sign of the permutationσadjusted by the rule that switching homogeneous elements x,y from either V or U means we multiply by(−1)deg(x)deg(y).For example,if V and U were both even,the image of this map would be the span of the t×t minors of the generic matrix; when V is even and U is odd,the image is the span of the space of“exterior minors”as in Green [1999].
For any partitionλ=(λ1,...,λs)we define Fλto be the image of the composite map
m◦(ρλ
1⊗...⊗ρλ
s
):
λ1
V⊗λ1 U⊗...⊗λs V⊗λs U→S|λ|(V⊗U)

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