THE GRASSMANN ALGEBRA OVER ARBITRARY RINGS AND MINUS SIGN IN ARBITRARY CHARACTERISTIC

An analog in characteristic 2 for the Grassmann algebra G was essential in a counterexample to the long standing Specht conjecture. We define a generalization G of the Grassmann algebra, which is well-behaved over arbitrary commutative rings C, even when 2 is not invertible. This lays the foundation for a supertheory over arbitrary base ring, allowing one to consider general deformations of superalgebras. The construction is based on a generalized sign function. It enables us to provide a basis of the non-graded multilinear identities of the free superalgebra with supertrace, valid over any ring. We also show that all identities of G follow from the Grassmann identity, and explicitly give its co-modules, which turn out to be generalizations of the sign representation. In particular, we show that the nth co-module is a free C-module of rank 2n−1. 1. The Specht problem, sign, and Grassmann algebra Supertheory, based on the partition of objects into even and odd parts, is ubiquitous in mathematics. It features the Grassmann algebra, whose definition requires the sign, and thus avoids characteristic 2, hampering the development of a complete supertheory. Indeed, in order to allow arbitrary reductions and quotients, it is important to consider such objects over arbitrary base rings and fields, in arbitrary characteristic. Similarly, one would like to investigate noncommutativity under deformations of the base ring, similarly to Poisson brackets in quantum mechanics. Our starting point is a Grassmann algebra in characteristic 2, which was constructed in connection with the famous Specht problem. We present a characteristic-free construction leading to generalized notions of superalgebras and superidentities, valid over an arbitrary commutative ring, thus supporting the possibility of having a supertheory in arbitrary characteristic. Let C be an arbitrary commutative unital ring, and let C〈X〉 be the free (associative) algebra over a countable infinite alphabet X. Recall that an ideal of C〈X〉 is a T-ideal if it is closed under transformations. Received by the editors July 9, 2015, and, in revised form, February 27, 2018, and August 7, 2019. 2010 Mathematics Subject Classification. Primary 16R10; Secondary 17A70, 16R30, 16R50.


The Specht problem, sign, and Grassmann algebra
Supertheory, based on the partition of objects into even and odd parts, is ubiquitous in mathematics. It features the Grassmann algebra, whose definition requires the sign, and thus avoids characteristic 2, hampering the development of a complete supertheory. Indeed, in order to allow arbitrary reductions and quotients, it is important to consider such objects over arbitrary base rings and fields, in arbitrary characteristic. Similarly, one would like to investigate noncommutativity under deformations of the base ring, similarly to Poisson brackets in quantum mechanics. Our starting point is a Grassmann algebra in characteristic 2, which was constructed in connection with the famous Specht problem. We present a characteristic-free construction leading to generalized notions of superalgebras and superidentities, valid over an arbitrary commutative ring, thus supporting the possibility of having a supertheory in arbitrary characteristic.
Let C be an arbitrary commutative unital ring, and let C X be the free (associative) algebra over a countable infinite alphabet X. Recall that an ideal of C X is a T-ideal if it is closed under transformations.
The main disadvantage of G + is that it degenerates in any characteristic not equal to 2, and superficially looks very different from the ordinary Grassmann algebra G. Therefore, our goal in this paper is to present and study an algebra G, unifying both constructions in a way that is well-behaved over arbitrary commutative rings. We show that G possesses properties similar to the ordinary Grassmann algebra G, and generalize various theorems regarding G over fields of characteristics not 2 to theorems on G which hold in general.
In particular, we prove that id(G) is generated as a T-ideal by the Grassmann identity, [x, [y, z]] = 0 (Theorem 3.5). Moreover, when 2 is invertible in C, G is strongly PI-equivalent to the free supercommutative algebra S, in the sense that id(A ⊗ C S) = id(A ⊗ C G) for every C-algebra A (Theorem 3.14).
Next, we present a generalization of the notion of signs of permutations, that is associated with G in much the same way that ordinary signs are associated with the ordinary Grassmann algebra G. We refer to this generalization as the generalized sign representation. We show that over any commutative ring C, the generalized sign representation is actually the full co-module of G: The S n -module of generalized signs C[ε] n over a ring C is the n-th co-module of G (Theorem 3.28). Furthermore, the n-th co-module of G is a free C-module, of rank 2 n−1 (Theorem 3.31). This generalizes the well known fact that the co-dimension sequence of G (in characteristic not 2) is c n (G) = 2 n−1 .
The ring C[ε] is defined in Subsection 3.1. In Section 4 we define generalized superalgebras (henceforth called Σ-superalgebras for brevity), as algebras over C [ε] which are graded by the group (Z/2Z) N . The free Σ-superalgebra S is defined in Example 4.5. Using this object we define the generalized Grassmann hull of an arbitrary Σ-superalgebra A, and determine its Σ-superidentities in terms of the Σ-superidentities of A (Theorem 4.14).
For the reader's convenience, let us collect here the notation used for the four objects studied and compared in this paper: superalgebra Σ-superalgebra Grassmann G G free commutative S S In Section 5 we define generalized supertraces (Σ-supertraces), and show that when 2 is invertible, these notions coincide with the notions of ordinary supertheory: Theorem (Theorem 5.8). Suppose that 2 is invertible in C. Let A be some Calgebra with trace tr. Let str be the associated Σ-supertrace of A⊗ C S, and associate a supertrace str to A ⊗ C S. Then the supertrace identities of A ⊗ C S are the same as the Σ-supertrace identities of A ⊗ C S, with sTr replaced by sTr.
The next question is what properties do supertraces (and more generally, Σsupertraces) satisfy. Thus we turn our attention to the question of ungraded identities satisfied by supertraces. In Theorem 5.12 we give a complete basis of identities for the multilinear part of the ideal of identities of the free Σ-superalgebra with Σ-supertrace (over any ring).
Another approach to superalgebras in characteristic 2, based on [Ven16], was recently presented in [Kau18].

Preliminaries
Throughout the paper, algebras are associative, but not necessarily unital. The base ring C will always be commutative and unital. We will assume nothing about the characteristic of C, except where explicitly stated.
Let A be an algebra over C, and let C X be the free (associative) algebra over a countable infinite alphabet X. A polynomial f (x 1 , . . . , x n ) ∈ C X is an identity of A if for all substitutions a 1 , . . . , a n ∈ A, we have that f (a 1 , . . . , a n ) = 0. We let id(A) = {f ∈ C X | f is an identity of A}. denote the ideal of identities of A. An algebra satisfying some non-zero identity with at least one invertible coefficient is called a PI-algebra.
Obviously, id(A) is an ideal of C X , which is invariant under substitutions. For any ring R, a T-ideal is an ideal I R such that τ (I) ⊆ I for every endomorphism τ of R. We will implicitly assume throughout that all T-ideals are T-ideals of C X . With this terminology, id(A) is a T-ideal for every algebra A.
Given that an algebra A over an infinite field C = F satisfies an identity f , it is always possible to break f down into its multi-homogenous components, by multiplying each variable by suitable scalars, and using a standard Vandermondetype argument. Furthermore, in characteristic 0, one can multilinearize any identity to an equivalent multilinear identity. Thus, in characteristic 0 over a field, any Tideal is generated by its multilinear part.
Because of this, one considers the spaces of multilinear polynomials in the variables x 1 , . . . , x n . This space has the structure of an S n -module by defining: With the above definition, C[S n ] ∼ = P n as S n -modules, with an isomorphism given by The multilinear part of degree n of a T-ideal Δ is given by Δ ∩ P n , which is an S n -submodule of P n . The quotient P n /(Δ ∩ P n ) is called the n-th co-module of Δ, and (in case C = F is a field) c n = dim P n /(Δ ∩ P n ) is the n-th co-dimension.
Remark 2.1. In addition to Remark 1.2, it is known that the co-dimension sequence of G is exactly c n = 2 n−1 . This result is obtained by first applying a combinatoric argument showing that the identity [x, [y, z]] = 0 has enough consequences to reduce the co-dimension to be c n ≤ 2 n−1 , and then using the representation theory of S n to show that it is bounded from below by the same quantity.

The generalized Grassmann algebra
The standard Grassmann algebra G is well behaved in characteristic not 2, while the generalized Grassmann algebra G + is defined in characteristic 2. Our first objective is to combine the two objects into an algebra defined over an arbitrary (commutative) ring, in a way which is amenable to reductions and inverse limits.
3.1. The generalized Grassmann algebra. Starting from the relations [e i , e j ] = ε i ε j e i e j of Definition 1.6, we immediately obtain for every i, j (in particular θε i e 2 i = ε 2 i e 2 i = 0). The following version of (3) will be frequently used: Remark 3.3. The elements e 2 j are central in G, as Modulo θ we recover the extended Grassmann algebra. More precisely: Remark 3.4. The quotient G/θG is the extended Grassmann algebra G + over C/2C.
The terminology attached to G is justified by the following theorem.  We need to prove that this is zero. We will do so by induction. If w 2 = e k v, and if we assume that the expression is zero for all shorter words, then We are now left with proving the other direction of Theorem 3.5.  Proof. We would first like to reduce to the multi-homogenous case. So, note that Thus, if f (x 1 , . . . , x n ) is an identity, then f (λ 1 ⊗ x 1 , λ 2 ⊗ x 2 , . . . , λ n ⊗ x n ) is also an identity. If we let f d 1 ,...,d n (λ 1 ⊗ x 1 , . . . , λ n ⊗ x n ) be the component of f (λ 1 ⊗ x 1 , . . . , λ n ⊗ x n ) of degree d i in λ i , we see that f d 1 ,...,d n (λ 1 ⊗ x 1 , . . . , λ n ⊗ x n ) = λ d 1 1 · · · λ d n n ⊗ f d 1 ,...,d n (x 1 , . . . , x n ) are the multihomogenous components of f , and must be equal to zero separately. Thus, we can assume that f is multi-homogenous.
So, let f be a multi-homogenous identity of G. We need to prove that it is a consequence of the Grassmann identity. Since commutators are central, f can be rewritten as a sum of terms of the form where k 1 ≤ · · · ≤ k m . Using (5), we may assume that k m+1 < · · · < k n .
Substitution of 1 for all of x 1 , . . . , x n sends f to the coefficient of the term x 1 · · · x n , and since f is an identity, this coefficient is zero. For every pair of variables x i , x j , substitute 1 for the other variables and e 1 , e 2 for x i , x j ; the only nonzero term is the one in which exactly these two variables are in the commutator, which again proves that the coefficient of this term is zero. Repeating this argument for all subsets of four variables, then six, and so on, we see that f is zero modulo the Grassmann identity.
3.2. The ring C[ε] and the connection to the Grassmann algebra. Our next goal is to show that when 2 is invertible, C[ε] has enough idempotents to break G into a sum of supercommutative pieces. The basic observation is that the expressions 1 2 θε i (if defined) are idempotents. Definition 3.11. For any subset X ⊆ N, let G X = C e j , ε j , θ | j ∈ X ⊂ G be the subalgebra generated by all generators ε j and e j whose indices are in X. (1 − 1 2 θε b ).
Proposition 3.13. Assume that 2 is invertible in C. Let X ⊆ N be a finite subset.
(2) For every s : X → {±1}, the algebra Λ s G X is a free supercommutative algebra, with even generators θ and Λ s e b for s(b) = +1, and odd generators Λ s e a for s(a) = −1.
Proof. The defining relations imply that the elements 1 2 θε i are idempotents, from which it follows that every Λ s is an idempotent. Furthermore so e a Λ s and e a Λ s anticommute. The proof that Λ s e b are central is analogous. Freeness then easily follows.
Multiplying by a suitable idempotent, we may thus declare finitely many of the e 1 , e 2 , . . . even, and finitely many others, odd. With this new understanding, we can prove a much stronger correspondence between G and S: Proof. We first show that any identity of A ⊗ G is an identity of A ⊗ S. Indeed, define a homomorphism of C-algebras, φ : S → G, by φ(e a ) = 1 2 θε a e a ∈ G for odd generators e a , and φ(e b ) = (1− 1 2 θε b )e b ∈ G for even generators e b (note that the e i on the left hand side of this equation are elements from S, and on the right hand side from G). This homomorphism is clearly injective. Since S, G and the image of φ are all free C-modules, and the image of a base of S under φ can be completed to a base of G (by considering the base of words in S, and the base of words multiplied by all idempotents associated to generators in the word, possibly times θ), we see that the map 1 A ⊗φ : In the other direction, let f ∈ id(A ⊗ C S), and let x i →x i ∈ G be a substitution of elements from G in the variables appearing in f . Let X be the (finite) collection of all the indices j of all e j or ε j appearing in some of thex i . Recall the definition of the subalgebra G X = C e j , ε j , θ | j ∈ X ⊂ G. By Proposition 3.13, the idempotents Λ s , with s : X → {±1}, form a complete set of idempotents for G (and thus G X ). Then it is sufficient to consider substitutions x i → Λ sxi ∈ Λ s G X for some fixed s : X → {±1}. But now, Proposition 3.13 shows that Λ s G X is a free supercommutative algebra, so we can fix a canonical embedding ψ : Λ s G X → S of Λ s G X in S. Again, we see that it maps the base of Λ s G X into a set that can be completed to a base of S (take the base generated by Λ s times words in G X , and the base of words in S). Hence, the map is an injective homomorphism. Thus, f is zero on substitutions from Λ s G X and is therefore zero on the substitution Corollary 3.15. Suppose that 2 is invertible in C. Then the ideal of identities of the free supercommutative algebra, id(S), is generated as a T-ideal by the Grassmann identity.
Remark 3.17. Over a finite field, id(G) strictly contains id(S). For example over C = F 3 , the polynomial x 9 y 3 − x 3 y 9 is an identity of G, which does not follow from the Grassmann identity.
Indeed, working modulo 3, if x = x 0 +x 1 is the decomposition of x to homogenous parts, then x 3 = x 3 0 + x 3 1 = x 3 0 . But, the even part of G is spanned by 1 and words of positive even length, so writing x 0 = λ + w, where λ ∈ F 3 , we have that Thus, the identity becomes λμ(λ 2 − μ 2 ), which is an identity of F 3 . A similar construction works over any finite field.
As an immediate corollary, we now have a proof of the following theorem, proved by Regev and Krakowsky in characteristic 0 [KR73], and by Giambruno and Koshlukov in characteristic p = 2 [GK01].
Corollary 3.18. Suppose that 2 is invertible in C. Then the ideal of identities of the free supercommutative algebra, id(S), is generated as a T-ideal by the Grassmann identity.
Proof. According to Theorem 3.14, in this case id(S) = id(G). But we have already seen that id(G) is generated by the Grassmann identity (see Theorem 3.5).
3.3. Generalized signs. Now that we have a clear understanding of the role taken by the ε i 's, we can introduce some helpful notation. If w ∈ G is a word in the generators, w = e i 1 · · · e i n , then define: ε w = ε i 1 + · · · + ε i n . Clearly, for any two such words w and w , we have ε w ε w ∈ span Z/2Z {ε i ε j }.
Remark 3.20. The exponent, a-priori defined on span for any a, b ∈ span Z/2Z {ε i ε j }. For the same reason, exp(a) 2 = exp(2a) = 1 for every a.
Proposition 3.21. For any two monomials u, w ∈ G in the generators e i , Proof. Equation (4) proves the case u = e i , w = e j . Let us verify the claim for u = e i , w = e j 1 · · · e l m . Indeed, we see that Now, let u = e i 1 · · · e i n , w = e j 1 · · · e j m . Then: Let us introduce a further generalization of the exponent map, which we call a generalized sign. We use the natural action of the infinite symmetric group S N on Definition 3.22. Let w = (w 1 , . . . , w n ) be an n-tuple of words in the generators e i . For σ ∈ S n , a permutation on the set {1, . . . , n}, we define the generalized sign to be: Proposition 3.23. Let w = (w 1 , . . . , w n ) be an n-tuple of words in the generators e i .
(3) In particular, when w = (e 1 , . . . , e n ), Proof. Write σ = s 1 · · · s m where s j = (k j , k j + 1) are Coxeter generators of S n . We prove (1) by induction on m. For m = 0, the claim is trivial. Assume the claim holds for π = s 1 · · · s m−1 . Then according to Proposition 3.21 and since s m transposes w π(k m ) and w π(k m +1) , we have: where the last equality follows from the induction hypothesis. Acting by s m = (k m , k m + 1) does not affect the order of any of the pairs i < j, except for flipping the order of the pair k m , k m + 1. Thus, as claimed.
To prove (2), we compute But since each pair i < j whose order is inverted by στ is inverted by σ or by τ , and not both, we have that the latter sum is equal to Since in the Grassmann algebra G we have that e σ(1) · · · e σ(n) = sgn (σ)e 1 · · · e n , Proposition 3.23.(1) shows that the generalized sign sgn w (·) plays in G the same role that the usual sign plays in G. Furthermore, the idempotent corresponding to the constant function s(i) = −1 (i = 1, . . . , n) satisfies since the e i anticommute in the presence of Λ s .
Remark 3.24. The generalized sign can be given a cohomological interpretation. Let W denote the set of n-tuples of words, on which S n acts by σ : w → σ −1 (w). Let M be the multiplicative group of functions W → C[ε] × , on which S n acts by s → s • σ −1 . Proposition 3.23.(1) is the claim that sgn(·) : S n → M , defined by sgn(σ) : w → sgn w (σ), defines a cohomology class in H 1 (S n , M).
3.4. The co-module sequence of G. We now turn our attention to the comodules and co-dimensions of G. We begin by defining an S n -representation analogous to the usual sign representation.  Fix w = (e 1 , . . . , e n ). We consider the natural action of S n on C[ε] twisted by signs: For each σ ∈ S n and λ ∈ C[ε], Also let C[ε] n denote the S n -submodule of C[ε] generated as a module by 1 ∈ C[ε].
We can now state the main result of this section.
Theorem 3.28. The n-th co-module of G is isomorphic, as an S n -module, to C[ε] n .
To prove the theorem, we will first establish that a multilinear polynomial that vanishes on e 1 , . . . , e n vanishes on any other substitution. Since S n acts on the space P n defined in (2) by reordering variables, and since reordering variables multiplies by the generalized sign, Theorem 3.28 follows (as will be explained below).
We first observe that G has plenty of endomorphisms.
Lemma 3.29. For any n-tuple of words w = (w 1 , . . . , w n ) in the generators e i , there is a morphism η w : G → G such that for all 1 ≤ i ≤ n: Proof. First we show that for every and for every word w of length 1 or 2, there is a homomorphism G → G of C-algebras such that e i → e i (i = ) and e → w.
Indeed, when w = e j , define the map on C[ε] by θ → θ, ε i → ε i for every i = , and ε l → ε j . This is easily seen to be well defined.
Likewise when w = e j e k , define the map η j,k, by θ → θ, ε i → ε i , e i → e i for i = , and ε → ε j +ε k −θε j ε k , e → e j e k . In order to show that this homomorphism is well defined, it suffices to check that η j,k, respects the relations ε 2 = θε l and [e , e i ] = ε ε i e e i for all i. For the first relation we have As for the second relation, for i = we have where v = e j e k . But by Proposition 3.21, we know that  Proof. If f is an identity then obviously f (e 1 , . . . , e n ) = 0. On the other hand assume f (e 1 , . . . , e n ) = 0. For every w 1 , . . . , w n we have that f (w 1 , . . . , w n ) = η w (f (e 1 , . . . , e n )) = 0, so we are done by multilinearity.
Let ν : C[ε] n e 1 · · · e n → C[ε] n denote the isomorphism of C-modules defined by setting ν(λe 1 · · · e n ) = λ. Let ψ = ν • μ : M n → C[ε] n . We will prove that ψ is an isomorphism of S n -modules. Indeed, ψ( σ∈S n a σ x σ(1) · · · x σ(n) ) = σ∈S n a σ sgn w (σ). But, for every π ∈ S n , showing that ψ is a homomorphism of S n -modules. Since 1 = ψ(x 1 · · · x n ) generates C[ε] n , ψ is surjective. Injectivity follows once we show that if f ∈ P n becomes zero under the substitution x i → e i then f is an identity, which is the content of Lemma 3.30.
In addition to having the co-modules of G, we can already calculate its codimensions: Theorem 3.31. The S n -module C[ε] n is a free C-module of rank 2 n−1 .
Proof. In the proof of Lemma 3.10 we have seen that modulo consequences of the Grassmann identity, every non-commutative polynomial f of degree n, and in particular every multilinear polynomial f of degree n, is a sum of elements of the form where i 1 ≤ · · · ≤ i m and we can assume that i m+1 < · · · < i n . Therefore, those elements generate the n-th co-module of G as a C-module. Thus, if we let then N/(N ∩ id(G)) is the n-th co-module of G, which is (by Theorem 3.28) isomorphic to C[ε] n . Hence, C[ε] n is the quotient of N by all identities of G. But, we have seen in the proof of Lemma 3.10 that all identities of G in N are zero, and hence N is isomorphic to C[ε] n .
However, there are exactly 2 n−1 polynomials in the set spanning N , and we have already seen that they are linearly independent: indeed, in the proof of Lemma 3.10, we have shown that if is an identity (in particular, a linear relation among the generators of N ), then the coefficients a i are zero. Hence, they are linearly independent.
Corollary 3.32. For any field C = F of any characteristic, the co-dimension sequence of G is c n (G) = 2 n−1 .
An immediate consequence is that we know the co-dimension of G, the usual Grassmann algebra, for any field of characteristic different than 2, generalizing the well known classical result in characteristic 0 (see also [LPT05] for a purely combinatoric proof).
Corollary 3.33. For any field F with char F = 2, we have c n (G) = 2 n−1 .
Proof. We have shown that when 2 is invertible, id(S) = id(G) (see Theorem 3.14), and since S is an extension by scalars of G they have the same co-dimension.

Generalized superalgebras
4.1. Generalized superalgebras. Now that we have the basic machinery of the generalized Grassmann algebra, we use it to replicate the success of the standard Grassmann algebra in characteristic 0. The first problem is that while the Grassmann algebra G has a natural superalgebra structure, given by the words of even and odd length, the even-odd grading on G is uninteresting, as exemplified by Lemma 3.29.
Recall the definition of C[ε] in Definition 3.1. Taking advantage of the many idempotents of C[ε], we choose the following grading.

Definition 4.1. A C[ε]-algebra is called a Σ-superalgebra over C if it is graded by 2 <ω .
Our first example is the algebra G itself: The zero component is thus G 0 = C[ε][e 2 1 , e 2 2 , . . . ], which is contained in the center of G. For every g = (g 1 , g 2 , . . . ) ∈ 2 <ω , which is eventually zero by definition, let e g = e g i i and ε g = g 1 ε 1 + g 2 ε 2 + · · · ∈ span Z/2Z {ε i }. The corresponding component G g = G 0 e g is a rank 1 module over G 0 , so the grading is "thin".

Definition 4.3. Let A =
g∈2 <ω A g be any Σ-superalgebra over C. We define the Σ-supercommutator {a, b} ∈ A for homogenous elements a ∈ A g and b ∈ A h by setting We say that A is Σ-supercommutative if {a, b} = 0 for all a, b ∈ A.
Example 4.4. The extended Grassmann algebra G is Σ-supercommutative. Indeed, by Proposition 3.21, for any pair of words u ∈ G g and v ∈ G h we have that uv = exp(ε g ε h )vu = exp(ε g ε h )vu, or in other words, {u, v} = 0.
We will use regular font for the standard supertheoretic notions, such as sgn (·), sCent, str, A, B, C, G, and the Fraktur font for the corresponding Σ-supertheory notions, sgn(·), sCent, str, A, B, C, G, etc. Example 4.5. As another example, one can consider S, the free Σ-supercommutative Σ-superalgebra on the generators e (n) g (n = 1, 2, . . . ) where e (n) g ∈ S g is a homogenous generator of the component with degree g. As a result, S is generated by the generators e (n) g under the relations: h . Note that id(G) = id(S), because G ⊂ S and S satisfies the Grassmann identity.
4.2. The generalized Grassmann hull. Now that we have an appropriate grading, we can generalize the Grassmann hull of an algebra (see Theorem 1.5 for the notion of the Grassmann hull for superalgebras). Similarly to the standard Grassmann hull, one can use either the Grassmann algebra or the free Σ-supercommutative algebra to define it (for an example in the case of char = 0, see [GZ05,). For our purposes, it will be more convenient to use the free Σ-supercommutative algebra.
Definition 4.6. Let A = g∈2 <ω A g be a Σ-superalgebra. The generalized Grassmann hull of A is by definition

Example 4.7. Let A be any C-algebra. Tensoring with the C[ε]-group algebra C[ε][2 <ω ], which is naturally a Σ-superalgebra over C, gives A ⊗ C C[ε][2 <ω
] a natural Σ-superalgebra grading, where the homogeneous components are We will now define the notion of a Σ-superidentity: , denoted by C[ε] X (g) for brevity, to be the free Σ-superalgebra on countably many generators in each degree. The elements of this algebra are called Σ-superpolynomials.
We define the set of Σ-superidentities of any Σ-superalgebra A as the intersection of all kernels of all grading-preserving C[ε]-homomorphisms φ : C[ε] X (g) → A, and denote it by id Σ (A).  i } 1≤i≤n (g) ,g∈2 <ω . We will refer ton as the associated multidegree. We will also write n = n (g) , the total degree of identities in Pn [ε].
Again, keeping the analogy to the case of characteristic 0, we can define the operation of the generalized Grassmann hull on an identity. Definition 4.10. We define the Grassmann involution on Σ-superpolynomials as follows. Let f = σ∈S n a σ x σ(1) · · · x σ(n) ∈ Pn[ε] be a multilinear 2 <ω -graded identity of multidegreen, such that each variable x j is in the homogenous component of C[ε] X (g) corresponding to some g j . Then where w = (e g 1 , . . . , e g n ).
This is indeed an involution: Lemma 4.11. The map f → f * is an involution.
Also, in this case, we let Δ * be the T Σ -ideal generated as a T Σ -ideal by the images of all multilinear identities in Δ under the involution * .
In other words, using * on all multilinear identities of a T Σ -ideal Δ gives all multilinear identities of Δ * .
Recall that id Σ (A) is the set of Σ-superidentities of A, Definition 4.8. Proof. Let f = σ∈S n α σ x σ(1) · · · x σ(n) ∈ Pn [ε]. Let x i → a i ⊗ w i be any substitution where w i ∈ S g i is a word in the generators e (n) j of S, in the component corresponding to g i , and a i ∈ A g i . Then, under the substitution: as we wanted to show.
We say that Σ-superalgebras A and B are multilinearly equivalent if id Σ (A) and id Σ (B) share the same multiliner identities.

Corollary 4.15. For every Σ-superalgebra A, S[S[A]] is multilinearly equivalent to A.
Proof. Use the result of Theorem 4.14 twice, and then apply Lemma 4.11.
Remark 4.16. We have not proved that id Σ (S[A]) = id Σ (A) * . In characteristic 0, having the same multilinear identities would have implied that they are the same. However, this is not the case in positive characteristic: id Σ (A) is not necessarily generated as a T Σ -ideal by its multilinear component.
We see that even though the language of generalized Grassmann hulls generalizes the ordinary notion of Grassmann hull, its formulation could be considered more elegant; rather than defining the involution on a multilinear identity by multiplying by the sign of only the odd variables, we simply multiply by the generalized sign of all variables. This is mainly because all words in the generators e i of G are, in a way, generic, so no special treatment is needed for any specific component of the grading.

Generalized supertraces
The superization of basic concepts in linear algebra, such as the supertrace and supercommutator, is defined in characteristic zero. We now begin the development of a supertheory based upon G and the concept of the generalized superalgebra. Such a Σ-supertheory will have the advantage of being characteristic free, valid over any ring.
We will begin by defining the notion of Σ-supertraces. Recall that an (abstract) trace function on a C-algebra A is a function tr : The concepts of Σ-supertrace Σ-superidentities naturally follows (see [BR05, Chapter 12]).

Definition 5.3.
Define the algebra C[ε] X (g) , sTr to be the free Σ-superalgebra with Σ-supertrace sTr. This algebra is spanned over C[ε] by words of the form w 0 sTr(w 1 ) · · · sTr(w ) where w i ∈ X (g) , and the grading is such that the grade of sTr(w) is the same as that of w. The defining relations are the axioms of Definition 5.2.
Remark 5.4. We use different capitalization to differentiate between formal traces (traces in the free algebra) and traces of the object under discussion. That is, Tr, sTr and sTr are formal traces, formal supertraces and formal Σ-supertraces in the algebras C X, Tr , C X (0) , X (1) , sTr and C[ε] X (g) , sTr , respectively. At the same time, tr, str and str are arbitrary trace functions, in any algebra we happen to be currently working with.
For example, the equality str(a p ) = str(a) p holds in the algebra A for all a, if and only if A satisfies the Σ-supertrace Σ-superidentity sTr(x p ) = sTr(x) p . In other words, sTr(x p ) = sTr(x) p is an identity, while str(a p ) = str(a) p is the value of that identity after substituting the function str to the variable sTr.
We come to our most important example.
Definition 5.5. Let A be a Σ-superalgebra with a grading preserving trace function tr : A → C. Define the associated Σ-supertrace function str = tr * on S[A] by str(a ⊗ w) = tr(a) ⊗ w. Conversely, if A has a Σ-supertrace str, define its associated trace function tr = str * on S[A] by tr(a ⊗ w) = str(a) ⊗ w. Note that str * preserves the grading.
Lemma 5.6. The above definitions of the associated trace function str * and the associated Σ-supertrace function tr * indeed give a trace function and a Σ-supertrace function, respectively. This construction generalizes Definition 5.5 to the case of (non-graded) C-algebras.

Proof. This follows since for all
Now, analogously to Theorem 3.14, we show the equivalence of supertrace and Σ-supertrace identities (the identities are not graded, so these are not Σ-superidentities).
Theorem 5.8. Suppose that 2 is invertible in C. Let A be some C-algebra with trace tr. Let str be the associated Σ-supertrace of A ⊗ C S, and in a similar manner, associate a supertrace str to A ⊗ C S, where S is the free supercommutative algebra. Then the supertrace identities of A ⊗ C S are the same as the Σsupertrace identities of A ⊗ C S, with sTr replaced by sTr.
Proof. The proof is virtually identical, word for word, to the proof of Theorem 3.14.
A key result in PI-theory is the "Kemer supertrick" (see e.g. [Zel91]), which heavily relies on representation theory, which fails to deliver in positive characteristic. The Kemer supertrick can be reformulated as the claim that for every algebra A there is some n such that id(A) ⊇ id(M n (G)). In this sense, the Kemer supertrick has already been proven in characteristic p (by Kemer, [Kem95]), but with very bad bounds.
Eventually, one might hope to bypass this difficulty by directly adding formal supertraces to algebras (and then show that their identities imply all identities of M n (G)), just like Zubrilin's theory enables the introduction of traces to an algebra and showing that affine PI-algebras satisfy all identities of a matrix algebra (see [AB10] for an overview of Zubrillin traces).
This motivates the following question about Σ-supertraces: Question 5.9. Let A be an (ordinary) algebra on which a linear function f is defined. What identities on A and f allow us to introduce a grading to the algebra such that f becomes a Σ-supertrace?
More formally, we define Definition 5.10. Let C X, F be the free algebra over C with a C-linear function F acting freely on it. Let A be any C-algebra with a linear function f : A → A.

GAL DOR, ALEXEI KANEL-BELOV, AND UZI VISHNE
We define the identities of A with linear function f to be the intersection of all kernels of all homomorphisms φ : C X, F → A such that φ(F(a)) = f(φ(a)).
Remark 5.11. As in Remark 5.4, we use capitalization to differentiate formal objects from others. That is, f is any particular linear function, while F is the formal linear function, of the algebra C X, F .
Theorem 5.12. The multilinear part of the ideal of identities of C[ε] X (g) , sTr with linear function sTr is generated by: Note that the Σ-superidentity F{a, b} = 0 of Definition 5.2 is not in the list, as it is not an (ordinary) identity.
To prove the theorem we require a few lemmas. We begin by proving a lemma analogous to Lemma 3.8: Lemma 5.13. The following identities with linear function hold in C[ε] X (g) , sTr : Proof. The identities (7a) and (7b) follow immediately from the definition of the Σ-supertrace (Definition 5.2).
We will now show that the identities (7c) and (7d) are indeed satisfied by any Σ-supertrace, using the fact that the Σ-supertraces Σ-supercommute with everything and a product of two elements inside a Σ-supertrace behaves as if it Σsupercommutes. Thus, for the purpose of checking (7c) and (7d), one can assume that everything Σ-supercommutes. But the Σ-supercommutative Σ-superalgebra G satisfies the Grassmann identity, which thus implies these two identities.
Proof of Theorem 5.12. We will use the following equalities: for all words s = x 1 · · · x n and t, we have [s, t] = [x 1 , x 2 · · · x n t] + [x 2 , x 3 · · · x n tx 1 ] + · · · + [x n , tx 1 x 2 · · · x n−1 ], (9a) 0 = [x 1 , x 2 · · · x n ] + [x 2 , x 3 · · · x n x 1 ] + · · · + [x n , x 1 x 2 · · · x n−1 ]. (9b) The strategy of our proof greatly resembles that of Lemma 3.10. We will use the above identities to bring an arbitrary polynomial f ∈ C X, F to a specified standard form, and then use substitutions to show that the coefficients are 0. This will be done via substitutions from the matrix algebras M n (G) over G, with the Σ-supertraces str associated with the usual traces in M n (C).
We begin by specifying the standard form we will use. Note that we are working with multilinear polynomials. The form is a sum of terms of the form: where w, w 1 , . . . , w m , v 1 , . . . , v n , u 1 , . . . , u k and t 1 , . . . , t are all words in the x i , and the s 1 , . . . , s are letters. However, many of these forms are trivially equal, so we require that: the words u 1 , . . . , u k are alphabetically ordered; the words v 1 , . . . , v n are alphabetically ordered; the pairs (s i , t i ) are also alphabetically ordered; for every i, the letter s i is smaller than some letter of t i ; and the words v i and u i are cyclically minimal, where a word is cyclically minimal if it is the first among its cyclic rotations.
Lemma 5.13, Lemma 5.14 and (9a), (9b) imply that every multilinear polynomial can be brought to this form. Now, we will show that the coefficients of the terms containing no F's are zero. Indeed, substitute matrix units x i → e σ(i),σ(i+1) into all x i , where σ is some permutation. Then only the monomial in which the x i are ordered according to σ contributes, and thus its coefficient is 0.
Next, rather than substitute a path as we just did, we choose some subset of the variables and substitute a cycle into them and a path into the rest. Since the standard trace is zero off-diagonally, the only terms contributing are those that have no more than one appearance of F, corresponding to the cycle. We thus have three options for the terms that contribute: w · F(v 1 ), w · [w 1 , F(u 1 )] and w · F[s 1 , t 1 ].
Note that the last two do not contribute at all if the coefficients of the matrix units are central. Thus the coefficient of the first is 0. Now, substitute coefficients from G to two edges of the loop, such that exactly one edge has e 1 as the coefficient, and another has e 2 as the coefficient. Then only the term w · F[s 1 , t 1 ] contributes -and hence has coefficient equal to 0. Finally, substitute e 1 to just one of the variables of the loop, and e 2 to an edge of the path. Then the term w · [w 1 , F(u 1 )] gives a non-zero contribution, unless it too has coefficient zero.
We use induction on N = n + k + to show that all coefficients are 0. We substitute matrix elements such that there is one path, and N = n + k + loops. We are now left with the liberty to choose their coefficients from G. Now, we must be able to tell how they are partitioned into v i 's, u i 's and (s i , t i )'s. So, at first we substitute only central coefficients. This gives us the case of: k = = 0, so its coefficient is zero. Now, we will use induction on k + . We choose n = N − (k + ) loops, and substitute central coefficients. This forces them to be v 1 , . . . , v n , and by induction, no coefficient with any other v i 's contributes. Now, we substitute coefficients e i into all elements of the path, and we substitute one coefficient into the generators in each remaining loop (out of the k + loops left). This gives us the case where = 0. We use induction on . Choose k loops, and substitute one coefficient into each one of them, in addition to the substitution into elements of the path. This forces these loops to be the u 1 , . . . , u k . We are left with two things to find out: how is the path split into the w, w 1 , . . . , w m , and how are the remaining loops partitioned between the s i and the t i .
Choosing the partition of each remaining loop into s i and t i is easy, and will be done via induction on the position of the letter s i relative to the largest letter of t i . Indeed, the base of the induction is this: substitute a coefficient e i 1 to the largest letter and e i 2 to the letter before it. Then the only contribution to the coefficient of the product ε i 1 ε i 2 comes from the cases in which the largest letter itself is s i , or the one before it is s i (otherwise e i 1 and e i 2 appear in their correct order). But because the largest letter is never s i , we see that s i is also never the letter before that. Proceeding by induction, we are done. Therefore, we have almost isolated all coefficients of the form; we must now isolate one specific way to break down the path to w, w 1 , . . . , w m , for an arbitrary (but known) choice of s i . This is done as follows. We use induction on m. Now, we already know that the associated loop, u i , has one coefficient, say e 1 , and we know which loop it is. Also recall that we substituted coefficients into the elements of the path. So, after the substitution, look for the largest number of ε i 's appearing. This information determines which elements of the path belong to w (their ε i 's never appear). Now look for the smallest number of ε i 's from the path appearing. This is the case where each w i contributes one ε i . So, sort these ε i 's, and put the element of the path corresponding to the j-th ε i into w j . This gives us all elements of w j , and only the case where m is the smallest value we have not considered, contributes.
This isolates everything -only one term contributes, and thus has a coefficient of zero, which completes all of the above inductive steps.
Note that incidently, just like in Lemma 3.10, we also obtain the co-dimension sequence. Then there is some Σ-superalgebra A with Σ-supertrace str, such that A and A have the same multilinear identities with linear function f and str respectively.

Concluding remarks.
We have seen how the structure of the generalized Grassmann algebra can be used to generalize the notions of superalgebras and supertraces to arbitrary characteristics and rings. In a similar manner, one can define a Lie Σ-superalgebra: Definition 5.16. Let L be a C[ε]-module with a Σ-superalgebra grading. Suppose that {·, ·} is a bi-linear form that respects the grading (if a ∈ L g , b ∈ L h then {a, b} ∈ L gh ). Then L will be called a Lie Σ-superalgebra if for every homogenous x, y, z ∈ L: (1) {x, y} = − exp(ε x ε y ){y, x}, Note that (3) is superfluous when 3 is invertible in C. This new object is obviously equivalent to an ordinary Lie superalgebra whenever 2 is invertible. However, the interesting property of this definition is that it yields non-trivial behavior in characteristic 2, where (unlike ordinary Lie superalgebras) it does not degenerate to an ordinary Lie algebra.
In this paper we only considered Σ-supertheory from the point of view of PItheory. In a similar manner, one can consider all of Σ-supertheory in characteristic 2. The cost we pay for this is that since the grading is over an infinite group, we must consider infinite-dimensional objects; therefore, in order to replicate the study of finite dimensional objects, one should consider Σ-superobjects that are locally finite-dimensional, in the sense that their graded components are each finite dimensional and isomorphic to one another in a sufficiently strong sense (so infinite-dimensional behavior is not "hidden" across multiple graded components).
One hopes that this construction can be used to yield characteristic-free results over arbitrary rings, such as Theorem 5.12.