NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 277-291
SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES
C. V i c t o r i a (Received May 1996)
Abstract. A ^-graded algebra of ^-symmetric Z2-tensors associated with a free supermodule of finite rank is introduced as a suitable generalization of the usual symmetric and antisymmetric tensors.
1. Introduction The concepts of multilinearity, symmetric and antisymmetric tensor products in superalgebra are treated in [2], [3], [4], [5] and [6]. We introduce here the Z 2-graded algebra of Z 2 -symmetric Z2-tensors associated with a free supermodule of finite rank (Definition 3.1). For any (po,pi) 6 Z2, the supermodule of Z 2 -symmetric (Po,Pi)-tensors is proved to be free of finite rank and we give a basis for it. This basis has two descriptions, one in terms of tensor products (Theorem 3.6) and the other in terms of differential operators (Theorem 3.10). This first description needs ‘sign functions’ (Definition 3.4) which although in some cases coincide with the usual symmetric or trivial and antisymmetric representations of the group of permuta tions of po +p\ objects (Proposition 3.13), they are not, in general, representations of it (Proposition 3.12). As a consequence, the tensor product does not induce any product of Z2 -symmetric Z 2-tensors, as it does in the case of modules over commutative algebras. Thanks to the description of Z 2 -symmetric Z2-tensors in terms of certain differential operators (Theorem 3.10), however, their product may be defined as the composition of their corresponding differential operators (Defini tion 3.17). The usual Z-graded algebras of symmetric and antisymmetric tensors of free modules over commutative algebras are proved to be natural subalgebras of the Z2-graded algebra of Z 2 -symmetric Z2-tensors (Propositions 3.19 and 3.20). Then, although the usual symmetric and antisymmetric products have independent definitions, our constructions make them merge both as particular cases of the same composition law (i.e. composition of differential operators). The Z-graded algebra of symmetric tensors of free supermodules introduced in [2], [3] and [4], is in a natural way, a subalgebra of the Z 2-graded algebra of Z 2 -symmetric Z2-tensors (Propositions 3.21). Since the functoriality of all of our constructions with respect to the general linear supergroup is proved in Proposition 3.22, there is no loss of generality by stating the
1991 AMS Mathematics Subject Classification: 15A69, 15A75, 16S32, 16W50, 16W55, 17A70, 17C70. Key words and phrases: Multilinear algebras, tensor products, exterior algebra, grassmann al gebras, rings of differential operators, graded rings and modules, “super” (or “skew” ) structure, super algebras, super structures . 278 C. VICTORIA definitions and results over which is the model of free A-supermodules of rank (mo, m i) (see [4]). In relation with forthcoming articles (see [8]), let us say that the functoriality of our constructions allows us to consider the Z2-symmetric Z 2-tensors on super manifolds, as the analogues of differential forms (that is, antisymmetric Z-graded tensors) in manifolds. Once the exterior differential is suitably generalized to these Z2-forms, a Z 2-graded algebra cohomology is obtained. This cohomology ring the ory is proved to be functorial with respect to all morphisms of supermanifolds (in contrast with the cohomology group of [1] and [9]) and is different from the coho mology ring of the underlying manifold (in contrast with the cohomology defined in [2], [3] and [4]). The contents of this paper are part of the author’s Ph.D. dissertation [7].
2. Previous Results 2.1. Generalities and Notation. Let S = S[o] + S[i] be a Z2-graded algebra or a Z2-graded module. A homo geneous element s G S is called even if s G S[o] and odd if s G S[ij. The value of the parity function Xs on a homogeneous element s G S, Xs(s), is defined to be zero if s is even and 1 if s is odd. As it is sometimes used in the literature, s denotes the parity, A^(s), of s G S. All superalgebras (that is, all Z 2-graded associative algebras with unit 1 G A) A are assumed to be supercommutative; that is x y — (—1 )xVyx, \/x,y homogeneous elements in A (see [4]). A Z2-graded module M on a superalgebra A is a supermodule if A^jMfj] C M[i+J-], i, j G Z2 = {0,1}. Supermodule isomorphisms are always assumed to be homogeneous of degree zero. Besides the parity function As associated with a superalgebra or with a super module, we will also need to consider the parity functions defined on Z*, resp. on N, as
Jo if A > 0 fo i f F < m 0 X*'iA) = \l if A < 0 reSP' = \ l ifF>m„ where (mo, m i) G N2 has been fixed. In expressions labeled by superindices and subindices, like X £ , the first group of capital letters are used to denote generic superindices in Z* and the second one F,G , . . . to denote generic subindices in {1 ,... , mo + m i}. So, A,B,... denote A z* (A), X (B ) ,... and F,G,... denote V o,n »i)(^ )i ^(m0,mi)(Gr), .. . , and so on. Moreover, the parity j of small latin letters j G N means \ Po,Pl){j) for the pair (po,pi) fixed in the text. We warn the reader about the fact that the Einstein convention about summation over repeated indices will be assumed throughout this work, unless stated otherwise. The parity change functor assigns to a supermodule M = M[0] + M[i] the super module IIM defined as follows:
(1) (IIM)^] = M [i+i], Vi G {0,1} = Z2.
(2) If x ,y are homogeneous elements in M and lire, IIy are the same elements but with reversed parity in IIM, then by definition, IIx + II y = II(x 4- y).
(3) If a G A[i], IIx G IIM, then by definition, allx = (—l) aIIax. SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 279
For m = (m o,m i) G N2 and M an A-supermodule, M m is defined as M®m° © (IIM)®mi. An A-supermodule is free of rank m = (mo,mi) G N2 if it is isomorphic to Am. We refer to [4] for more details. 2.2. Symmetric and Antisymmetric Z-Graded Tensor Algebras. In this section we recall the definition of the Z-graded algebras of symmetric and antisymmetric tensors of a supermodule introduced in [2], [3] and [4] as a general ization to supermodules of the Z-graded algebra of symmetric and antisymmetric tensors of a module. The tensor product of two A-supermodules M and N is the A-supermodule defined as follows: as A-module, the tensor product of M and N is M 3. Z 2 —Symmetric Z2—Tensors 3.1. Definitions. Fix p — (po,Pi) G N2 and let © = { A i,... ,A Po+Pl} C Z* be a subset with A i , ... , Ap0 even superindices and with A Po+1,... , ^4Po+Pl odd ones. Recall from Section 2.1 that the parity A = Xz*{A) of A G Z* is to be 0, resp. 1, if A > 0, resp. A < 0. When the parity of A is 0, resp. 1, A is called an even, resp. odd, superindex. The elements of the canonical basis X\,... , X mo+mi of Am, m — (m o,m i) G N2, have parities X p — Am(F) = F. S(Am) is by definition the 280 C. VICTORIA A-superalgebra generated by the homogeneous elements X £ , 1 ^ F ^ mo + mi, A G Z*, X^ = A + P, modulo the relations spanded by X £ X £ = 0, X £ X £ = (_ 1)(f+a)(g+b)X bX ^ for i ^ F G ^ m0 + mi, A ± B G Z*. Se (Am) denotes the A-subsupermodule of S(Am) spanded by X^, 1 ^ F ^ mo + m i, A G 0. S0 (ATO) = A by definition, is the A-superderivation on S(Am) defined by jgytK-) =«««;. Definition 3.1. For p = (pojPi) € N2, the A-supermodule of Z2 -symmetric p - tensors on Am is the subsupermodule of 5 e (Am), 0 = { — pi,... , —1,1 ,... ,po} (no summation is assumed) ^ = + ( + } d X B d X * ) F,G where A,B G © and 1 ^ P, G ^ mo + mi. The A-supermodule of Z 2 -symmetric Z 2-tensors, is ® peZ2 where = 0 if p0 < 0 or p1 < 0. Notation 3.2. We use the following notation when it is explicitly said in the text: if p = (po,Pi) G N2 is fixed, we consider the bijection ip: {1,... ,Po + Pi} —> © = {~P i, • • • , -1 ,1 , • • • ,Po}, v (j) = j if 1 ^ j ^ Po, v{j) = P o-j ifpo < j ^ Po+Pi- V is homogeneous of degree 0. Then the generators X k ^ \ Lemma 3.3. With the above notations: (1) f is a Z 2-symmetric p-tensor if and only if so are f ^], i = 0,1. (2) /[j] is a Ij^-symmetric p-tensor if and only if = Vj < f where SgnpF(j,j') = FjF? + IF, + Fy j(]' + £ j Proof. Expand in coordinates the equation ( /) = 0 . □ Definition 3.4. Assume that p = (.Po,Pi) G N2 has been fixed. For a given se quence F = (Pi,... ,PPo,PPo+i,... ,PPo+Pl), 1 ^ Fi ^ m0 + mi the value of the sign function Sgn£: &p —> {0 ,1 } on a permutation of the symmetric group of SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 281 P = Po + Pi elements, a € 6 P, is defined by induction on p as if ap ^ p and moreover, whenpo+Pi = 1> SgnpF(id) = 1, id G ©i, 1 < F ^ m o + m i. One of the basic properties of these sign functions is the following: Theorem 3.5. Let F be the sequence (F\,... , FPo, FPo+i, ... , FP0+Pl), 1 ^ Fi < mo + m\. For a, G 6 P with j < j ' , one has Proof. For short, let us fix the pair p and the sequence F and denote Sgn£ simply by Sgn. The theorem can be proved by induction on p = p0 + p i and distinguishing several cases: 1. p = ap: (a) j ' < p: one can apply induction. (b) j' = p: Sgn (a o (j,j')) = Sgn(cr) + FpFaj + {Fp + Fa.) 2. p = ai,i^p: (a) j' < i : Sgn ((7 o (j,f)) = FpF„r + (Fp + F„p) p + £ (F0a + 5) i + Sgn (a o ( j j ' ) o (i,p)) = Sgn(<7 o (i,p)) + FpFap 282 C. VICTORIA (b) i = j ' : ( \ Sgn(cr o (j , j ')) = FpFap + (Fp + Fap) P + X {F<7a + a) + (F<7j + f ) \ I + F(TpFaj + (Fap + Faj) ( / + X (FCTa + a) ] + Sgn(cr o (j',p)) j < a < j ' — FpFap + (Fp + F(Tp) i j + FapFaj + (Faj> + FCTj.) ( j ' + X (FCTa+a) j < a < j ' + FpFGp + (Fp + Fap) [ p ^ (FtJa + a) ] + Sgn (a) j'< a < p = FPF„, + (Fp + F^.) J'+ Y , (F° . + “) + SSnM- \ j' Theorem 3.6. Let e(a) denote the parity of a permutation a € &p. The non vanishing elements of the system { - i y {(j)+S9n W x 1Fai. . . x pFo erEGp j p form an A-basis of where F = (F i,... , Fp) is any ordered sequence 1 ^ Fi <•...< ^ m0 + mi . P roof. It is an immediate consequence of Lemma 3.3. □ The following examples show that in fact some of the elements of the above system of generators can be zero. Exam ple 3.7. Let m = (1,1 ),p = (1,1) and Fi = F2 = 1. Then ^ ( _ l) £(ff)+Sgnj;;lj(a)x l^ x 2^ = -j- (—l)1"l’°X iX i = 0. 2 To state which generators in Theorem 3.6 are non zero, we describe the Z 2 -symmetric Z2-tensors in terms of the following differential operators. SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 283 Definition 3.8. [Notation 3.9 not assumed] Fix A E Z*. For F e {1 ,... , m o+m i} we define the differential operator d£: S(Am) —> S(Am) by k - E (-i)sa+'(i+a)^ ^ ; a | i - • B ,G G N otation 3.9. When Notation 3.2 is assumed, d£, A G {1,... ,p = po +f>i} denotes dj^A\ Notation 3.9 is assumed unless stated otherwise. Theorem 3.10. Fix the pair p = (po,pi) € N2 and let p = po + p \ . (1) For any F = (Fi,, Fp): dPFp 0...0 4 ^ 1) = (-if i (2) It is satisfied . . . o dA o dB o . . . = ( — 1 )(^+F)(^+6) ...od®o^o.... (3) The expression (fFp o . . . o d}Fi (1) is zero if and only if there exist j and j' such that j ^ j ' with Fj = F'- and Fj = Fj> = 0 (in other words, if there is some repeated even subindex). (4) dAFdBG = (-1 )1+ABdBdA if A = B. P roof. Statement (1) can be proved by induction on p = p0 + p i- Parts (2) and (4) can be straightforwardly verified by an explicit computation. Let us demonstrate part (3). If an even subindex F appears twice: dfp o • • • o dp o... o d® o ... o d^ (1) = ±d ppo...od p 0...o d p 0 ...o d p i odpod® (1), and therefore dJod?(l) = d t ( X ? ) = XI __ \rA \TB VA y B __ F F F To see the reciprocal statement we will use induction on p = p$ + p\. For p = 2 the equality _ ^ d£ o d® (1) = X £ X ‘ - ( - 1 ) sl* W * d} X g X ? = 0 implies F = G and F = 0. Now consider the pair (po + l,p i) and 0 = po + 1 or the pair (po>Pi + 1) and 6 = pi + 1, and suppose that dp o dp'- o ... o d* (1) = 0. Then either dpp o. . .o d ^ (1) = 0, and we conclude by induction, or dpp o .. .o d ^ (1) / 0. 284 C. VICTORIA In that case 0 = ± d F o dpF°p+Pl o ... o dFl (1) = (x eF-Y ^ (-^ fi+Hk+i)x eLx ‘ j V b ,l L / a = ^ ( - l ) £^ +s^nF^ [ x eFX l ai ... X jFoj . . . X pap G + (_ l) B ^ +F(B+^ )+(f+Fff^ a If F ^ {F i,... , Fp}, then X FX Fi ... X Fp will be cancelled out by an expression on which the sum runs over some permuta tions which only move some repeated odd subindices. For such a permutation a, Sgnjr(a) +£{&) = 0. Therefore the above expression 1)£(CT)+Sgnj:’^ X FX F (1) if $i : Fi = 0 and writting Fp+i instead of F, then ± dj o d?p o ... o dj., (1) = ... X * ' t X l w = E x k ■ ■ ■ x k , n . v+l = Sym ( x j , ... X ’ X°F^ ) + 0, which is a contradiction. (2) if 3* : Fi = 0 then, ±d£ o dpFp o ... o d*F. o ... o d ^ (1) = d*F.d* o ... o dpp o ... o dF. o ... o d^ (1) = 0. In this case there are two possibilities: dFo . .. odF. o . . .odFp o . . .od Fl (1) = 0, and we finish by induction, or dF o ... o dF. o ... o dpFp o ... o dFl (1) ^ 0. In that case, the preceding arguments show that it has to exist h such that h ^ i and Fi — Fh, and in particular Fj = F^ = 0. □ Corollary 3.11. ,E(re6p(_ ^ £^ +Ssnf^^F SgnfUJ') = PvjFoy + (Fvj + F) ( J1 + ^ 2 + a ) ) ’ mod2 j< a < j' but by definition Sgn,a,/) = FjFji + ( Fj + Fy)W + £ (F< + 1 ) ) . This shows that for some F, Sgn^r is a representation of the symmetric group, while for others it is not. Proposition 3.12. Let F = (Fi,... ,FP) denote any sequence 1 ^ Fi,. . . ,FP ^ ra0 + mi (1) Sgnp is a representation if and only if for all a G 6 P FCiFCj, + (F0) + ) [ ? + E ( & . + « ) — F jF j' + {Fj + F j') { j ' + ^2 (^a + a))> mod 2 . j< a < j' (2) For 7 G 6 p; let us denote by 7 F the sequence (F7l,. . . , F7p). If SgnF is a representation then Sgnp = Sgn^p, V7 . Proof. (1) Prom Theorem 3.5 Sgn?(cr o ( j , / ) ) = Sgnf(a) + F^Fe^-H (FCTi + ^ , ) ( ? + E j< a < j'(^ a + <*))• If SgnF is a representation, then SgnF(j, j') must be equal (mod 2) to FajFa., + (F SgnF (a o (j , j ')) = Sgnir(a) + SgnF {j,j'), Yj < j ' . From that and the fact that adjacent transpositions generate the symmetric group, statement (1) follows. 2 8 6 C. VICTORIA By part (1), Sgn7j? is a representation if and only if F(Jlj Ffjry., + (F — F7iFljf + (F7j. + F^.,) J jf + ^ (F7q 4- a) j< a < j' = F jF j> + (F; + fj,) I ? + £ ( £ + a) I . \ j< a < j' J Therefore, if Sgnjp is a representation, then Sgn7jp is also a representation, and they both have the same values on adjacent transpositions. Statement (2) follows because adjacent transpositions generate the symmetric group. □ We conclude this section with a result that will be useful in Section 2.6: Proposition 3.13. Let F = (Fi,... , Fp), 1 ^ Fi ^ mo + mi and fix p € N2. (1) If Fi = 1, VFj, then for any cr £ &p one has SgnF (o) = £(<7 ), w/iere £(<7) is i/ie usua/ parity of a. (2) If Fi = 0, VFj, then for any a £ 6 P one has Sgn?(a) = 0. (3) Suppose that { j i , . . . ,jp } C {1 ,... ,p } and that Fjb = 1, V6 = 1, For 7 € ,jp} C 6 {i,...,p} and a £ 6 {i,...,p} one /ias SgnF (a 0 7 ) = SgnF {a) + £(7 ) where £(7 ) is £/ie itswa/ parity of 7 . (4) / / £/ie sequence F satisfies that Fi = Ap(i), then the restriction of the function ®Po+Pi y ^2 CT ___> + on ©Po x ©Pl is a representation. P roof. According with Definition 3.4 to compute Sgnj-(o-) one has to decompose a as cr = (.(<7(p~2 ))2 \ 2 ) 0...0 ((a<1)) ^ 1, p - 1) o ((ct(0)>“‘1,p), where cr^ = a and a^t+1^ = o ((0-^ )“^ ,^ — i). If Fj = Fj, Wi,j, then p-2 _ SgnF(cr) = Y l F^ ) ^ 1_jFp -3- j = 0 In case Fi = Fj = 1, V i,j, then p—2 p—2 p—2 SgnF(cr) = X^ F {aU))-i_FF p-j-p-i = 1 = X !e ((( and in case F{ — Fj = 0, \/i,j then P -2 _ SgnF(a) = = °' j = 0 To prove (3) one can write 7 as a product of transpositions 7 = 7 , o ... o 7 X and then one applies Theorem 1 in Section 3.2 Sgn^(a o 7 , o ... o 7 i) = SgnF (a o o ... o 72) + £(7 ^ = ... = Sgnf (cr) + ^ £ (7 j) = SgnF((r) + 5(7 ). j = 1 Part (4) follows from parts (1) and (2). □ 3.4. Multiplicative Properties. Notations 3.2 and 3.9 are assumed in this section. Before of dealing with the general problem of defining a product of Z 2 -symmetric Z 2 -tensors, let us examine a particular case: Example 3.14 (Even case). Consider an even superalgebra; that is a commutative algebra A = A[0]. Then for all F and a, Sgnp(cr) = 0, and we have an A-module isomorphism < i o > — A i - a g When one composes the usual antisymmetrization morphism A^m (A771)®25 with the projection (Am)®p — ♦ (Am)®*7{XFCTi n ( ° ’P) ~ \ p ( &m \®p 14A(m,0) — ilAm J — * (Amf»/{xli...x^p-(-i)‘^ x 1ri...xit}F,a~ n ^ 0). From this isomorphism one usually defines the product of antisymmetric tensors as the induced tensor product. The following example shows that the analogous morphism <-> (Am)®p — c * m = (A m)®^po+pl^ / { X Foi ® . . . ® xr,r - (-lfW +SgnrM x^ ® ... ® which sends ^ ( - I J 'W + S b t W x i , .. , X PF to [JJ-l)sWtSm(»)l, ®...® is not an isomorphism for general m and p. Example 3.15. Consider m — (2,1), p = (0,3) G N2. Let us see that X\ / = {XFCI ® X f^ ® X f ,3 - which defines the quotient c ! ^ . In fact, for the permutation a = ( 3 1 2 ) one has X\ X 2 — (—l)°X i (g) X 2 g> X 3 G / and their difference 2X i However, there are some important cases in which — > Cpm is an isomor phism: Proposition 3.16. If for any F the function SgnF is a representation, then: (1) The function &p — » EndA(Sp(Am)) defined by o(X l...X lr) = is a representation. (2) The morphism — > Cpm is an isomorphism. Proof. Part (1) is obvious. To prove (2), it is enough to observe that the morphisms 4,: - C l and 0: Cfc defined by ■ ■ ■ X £ J = [Xkl ® ... ® X kp] and The above example shows that in the tensor category of supermodules the tensor product does not induce any product of Z2 -symmetric Z2-tensors as naturally as it does in the tensor category of modules. Let us now give the definition of the product of Z 2 -symmetric Z2-tensors by using their description in terms of differential operators given in Theorem 3.10. Definition 3.17. The product of ip o d^f^ o ... o d“^ o dp°o o ... o d^ (1) G times ^ o d ^ 91 o .. .od^11 odq o. ..o d ^ (1) G is the composition of differential operators o . . -od-l'-'odf+too. . .o d ^ o ^ o d ^ O.. '.odg^ odgQ0 o. . .od^ (1) in where ip, ip € A. Proposition 3.18. &Am is a 1? -graded A-algebra which satisfies a(3 = { - l f P +Poqo(3a for any homogeneous a G and (3 G Proof. It is a direct consequence of the definition and part (2) and (4) of Theorem 3.10. > * □ We call the Z2-graded algebra of Z 2 -symmetric Z2-tensors of the free A-supermodule Am. Its relations with the symmetric and antisymmetric algebras of a free module, resp. supermodule, over a commutative algebra, resp. superalgebra, are stated in Section 3.5. SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 289 3.5. Symmetric and Antisymmetric Tensors. In this section p = (po,pi) G N2 is fixed, p denotes p o + p i and Notation 3.2 and 3.9 are assumed unless stated otherwise. The goal of this section is to show that the Z-graded algebras of symmetric and antisymmetric tensors of a free module and the Z-graded algebras of symmetric and antisymmetric tensors of a free supermodule (see [2], [3] and [4] and Section 2.2) are all particular cases of the Z2-graded algebra of Z2 -symmetric Z 2-tensors. Then, since it is clear from the definition that the algebra can be naturally considered as a ring of differential operators, we have a new description, as far as we know, of the algebras of symmetric and antisymmetric tensors of a free module or supermodule as algebras of differential operators. Proposition 3.19. Assume A = A[0j; that is, A is a commutative algebra. There exist natural group isomorphisms aPo+Pi ^ q(P0,Pi) 7 q,rr)Po+Pl ~ o(p°,Pi) Proof. From Theorems 3.6 and 3.10 we know that the elements of the system I ... x pap \ ^(7G6p ) 1 cr€& p The second isomorphism can be deduced in a similar way from part (1) of the same Proposition 3.13, by realizing that in this case = xFle...exFr. cre& p □ Proposition 3.20. As in Proposition 3.19, assume A = A[0j. There exist natural algebra isomorphisms AAmo ~ O^0(’Jo,0) and SymAmi ~ Proof. From Proposition 3.18 it is easy to check that the group isomorphisms in Proposition 3.19 induce morphisms of algebras. □ Therefore, in case A = A[0] we can consider the subalgebras AA(mo 0) c and SymA(mi 0) C mi) ’ These inclusions give a new description, as far as we know, of both the usual symmetric and antisymmetric products of free modules over commutative algebras as particular cases of a general product formula as com position of differential operators. Now, let us see that the Z-graded algebras of symmetric and antisymmetric ten sors of free supermodules ([2], [3] and [4], see Section 2.2), can be both considered as subalgebras of the Z 2-graded algebra of Z 2 -symmetric Z2-tensors. Proposition 3.21. There exists natural A-algebra isomorphism — Sym ^m . 290 C. VICTORIA Proof. Notice that £ (Faj + 1 )(FV + 1) = Sgn^'V), i 3.6. Functoriality with Respect to the Action of the General Linear Supergroup. Let (f) : Am —► Am be an isomorphism of A-supermodules. For any p = (po,Pi), it induces an isomorphism of A-supermodules 0®p : (Am)®p —> (Am)®p, p = po+pi by the usual formula (j)®p(aXFl © ... © X Fpo+pi) = a(f)(XFl) © ... © (f)(XFpo+pi), a G A (this formula makes sense because supermodule isomorphisms are always even, as it was recalled in Section 2.1). The following proposition shows that all of our constructions are functorial with respect to supermodule isomorphisms. Proposition 3.22. Any isomorphism of A-supermodules P roof. If we write 4>{X^k) = ^ k, X j and use the description of given in Theorem 3.10, then an explicit computation shows that m (< £ o... o dj. (1)) = o... o d * (1). From that, it is clear that f)(0) and J7(0_1) are inverses to each other. □ References 1. M.A. Baranov and A.S. Shvarts, Cohomology of supermanifolds, Funkts. Analiz. i.ego Pril. 18 no. 3 (1984), 69-70. 2. K. Gawedzki, Supersymmetries-mathematics of supergeometry, Ann. Inst. Henri Poincare, X X V II no. 4 (1977), 335-366. 3. B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Maths. 570 (1977), 177-306. 4. Yu.I. Manin, Gauge Field Theory and Complex Geometry, Grundle-hren der mathematischen Wissenschaften, 289, Springer-Verlag 1988. 5. O.A. Sanchez, Linear supergroup actions. I: On the defining properties, Trans actions of the American Mathematical Society, 307 no. 2 (1988), 569-595. 6. O.A. Sanchez, Matrix computations in linear superalgebra, Linear Algebra and its Applications, 111 (1988), 151-181. 7. C. Victoria, Cohomologia de Supervariedades, Ph.D. Dissertation, Directed by S. Xambo-Descamps, Dep. de Algebra, Universidad Complutense de Madrid, 1993. 8. C. Victoria, Cohomology ring of supermanifolds, New Zealand J. Math. 27 (1998),123-144. SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 291 9. T. Voronov, Geometric Integration Theory on Supermanifolds, Mathematical Physics Reviews, Vol. 9, Part 1, Harwood Academic Publishers, 1991. C. Victoria Departament de Matematica Aplicada i Telmatica Universitat Politecnica de Catalunya 08034 Barcelona SPAIN [email protected] a N ; the Z 2 ~graduation is obtained by defining as homogeneous elements the product of homogeneous elements of M and N, with parity m <8> n — m + n; the product is defined as a(m ® n ) = (am) ® n = (—l)amm 0 (an). Similarly the tensor product of a number p G N of A-supermodules can be defined and one can check that the A-supermodules constitute a tensor category (see [4]). Let M be an A-supermodule. The A-supermodule ® n^0 M ® n, where M®° = A, has a natural structure of Z-graded algebra with the tensor product ® M ® b —> M ® (a+b\ This is the Z-graded tensor algebra of M. In this alge bra the ideal Is, resp. I a, is the ideal generated by mi ® m2 + (—l)^*^m2 <8> mi, Vm i,m 2 homogeneous elements in M, where (*) = 1 + m im 2, resp. (*) = m im 2. The Z-graded algebra of symmetric, resp. antisymmetric, tensors is defined as the quotient ( 0 n>o M ® n) /I , where I = Is, resp. I = 7a- They are denoted by SymM and A m respectively, and as usually Sym?,, resp. A?,, denotes the A-supermodule M ® P /Is n M®P, resp. M ® p /Ia n M®*>. When a transposition acts on a symmetric, resp. antisymmetric, Z-graded tensor one has mi