The Algebraic K-Theory of Finitely Generated Projective Supermodules P(R) Over a Supercommutative Super-Ring R

Ameer Jaber Department of Mathematics, the Hashemite University, Zarqa 13115, Jordan

Abstract: Problem statement: Algebraic K-theory of projective modules over commutative rings were introduced by Bass and central simple superalgebras, supercommutative super-rings were introduced by many researchers such as Knus, Racine and Zelmanov. In this research, we classified the projective supermodules over (torsion free) supercommutative super-rings and through out this study we forced our selves to generalize the algebraic K-theory of projective supermodules over (torsion free) supercommutative super-rings. Approach: We generalized the algebraic K-theory of projective modules to the super-case over (torsion free) supercommutative super-rings. Results: we extended two results proved by Saltman to the supercase. Conclusion: The extending two results, which were proved by Saltman, to the supercase and the algebraic K-theory of projective supermodules over (torsion free) supercommutative super-rings would help any researcher to classify further properties about projective supermodules.

Key words: Projective supermodules, superinvolutions, brauer groups, brauer-wall groups

INTRODUCTION with 1 ∈R0. An R-superalgebra A = A0 + A1 is called projective R-supermodule if it is projective as a module An associative super-ring R = R + R is nothing e o 0 1 over R. Define the superalgebra A = A ⊗R A, then A but a Z -graded associative ring. A Z -graded ideal e 2 2 is right A -supermodule. There is a natural map π from e I = I0+I1 of an associative super-ring is called a A to A given by deleting o,s and multiplying. superideal of R. An associative super-ring R is simple if In[2], Childs, Garfinkel and Orzech proved some it has no non-trivial superideals. Let R be an associative results about finitely generated projective supermodules [1] super-ring with 1 ∈ R0 then R is said to be a division over R, where R is a commutative ring. In , we super-ring if all nonzero homogeneous elements are generalized the same results about finitely generated −1 projective supermodules over R, where R is a invertible, i.e., every 0 ≠ rRαα∈ has an inverse rα , supercommutative super-ring. Here are the results: necessarily in Ra. If R = R0 + R1 is an associative super- ring, a (right) R-supermodule M is a right R-module Proposition 1: Let M be an R-supermodule and A an with a grading M = M0 + M1 as R0-modules such that R-superalgebra then there exist isomorphisms of R- mαrβ ∈ Mα+β for any mMαα∈ , rRββ∈ , αβ∈, Z2 . An R- superalgebras: supermodule M is simple if MR≠ {0} and M has no proper subsupermodule. Following[4] we have the A ⊗R EndR(M) ≅ EndR (M) ⊗R A following definition of R-supermodule homomorphism. Corollary 1: Let P and Q be a finitely generated Suppose M and N are R -supermodules. An R- projective supermodules over R, then: supermodule homomorphism from M into N is an R0- module homomorphism h:Mγ → N, γ∈Z2 , such that EndR(P) ⊗R EndR ( Q ) ≅ EndR (Q ) ⊗R EndR (P) ≅

Mhαγ⊆ N α+γ. Let K be a field of characteristic not 2. EndR(P ⊗R Q )

An associative superalgebra is a Z2 -graded associative Theorem 1: Let A be an R-superalgebra. The following K-algebra A = A0 +A1. A superalgebra A is central conditions are equivalent: simple over K, if Z(A) = K, where ( Z(A) )α = {αα ∈ αβ e Aα : ααbβ = (-1) bβαα∀ββ ∈Aβ} and the only • A is projective right A -supermodule superideals of A are (0) and A. Through out this study • 0 ker(π ) Ae π→→ A 0 splits as a sequence we let R be a supercommutative super-ring ( Z(A) = R) of right Ae -supermodules 171 J. Math. & Stat., 5 (3):171-177, 2009

e • (A )0 contains an element ε such that π()ε= 1 as morphisms. It is a category with product if we set ⊥ = ⊕ and ε⊗(1 aαα ) =ε (a ⊗ 1) for all aAαα∈ • The subcategory FP(R) of P(R) with finitely Definition 1: We say that A is R-separable if generated faithful projective supermodules as conditions (1-3) above hold. objects. Her we set ⊥ = ⊗R • The category Az(R) of Azumaya superalgebras Remarks: over R. Her we take ⊥ = ⊗R • Condition (3) states that A is R-separable if and only if it is R-separable of the sense of ungraded If C(R) denotes one of the categories mentioned algebras above and if R→R′ is a homomorphism of super-rings. • It is easy to see that ε defined above is idempotent. Then R′ ⊗R induces a functor C(R)→ C'(R ') A is a central separable R-superalgebra if it is preserving product. separable as an R-algebra, thus our Azumaya R- algebras A are those separable R-algebras which Definition 3: Let C be a category with product. The are superalgebras over R and whose supercenter is Grothendieck group of C is defined to be an abelian

R group K0 C, together with the map ( )C: obj(C) →K0C, which is universal for maps into abelian groups For any R-superalgebra A we have seen that A is satisfying: e naturally a right A -supermodule. This induces an R- superalgebra homomorphism µ from Ae to End (A) by R • if A ≅ B, then (A)C = (B)C associating to any element xy⊗ of Ae the element αβ • (A ⊥ B)C = (A)C + (B)C xyαβwhere for any aAγγ∈ : Definition 4: A composition on a category (C, ⊥ ) is a αγ composition of objects of C, which satisfies the a(xy)a.(xy)(1)xayγαµ⊗= β γαβ =− αγβ following condition: if AA' and BB' are defined

then so also is (A⊥ B) (A'⊥ B') and: Theorem 2: Let A be an R-superalgebra. The following conditions are equivalent: (A⊥ B) (A'⊥= B') (A A') ⊥ (B B') • A is an Azumaya R-superalgebra • A is finitely generated faithful projective R- Definition 5: If (C,⊥ , ) is a category with product and supermodule and µ is an isomorphism composition. Then the Grothendieck group of C is defined to be an abelian group K0 C, together with a MATERIALS AND METHODS map:

Suppose C is any category and obj(C) the class of ( )C : obj(C) →K0C all objects of C and let C(A,B) be the set of all morphisms A→B, where A,B ∈obj(C). A groupoid is a which is universal for maps into abelian groups category in which all morphisms are isomorphisms. satisfying the two conditions in Definition 3 and:

Definition 2: A category with product is a groupoid C, If AB is defined, then (A B)C = (A)C +(B)C together with a product functor ⊥ : C×C→C which is assumed to be associative and commutative. An easy computation gives us the following result. A functor F:(C,⊥→ ) (C', ⊥ ') of categories with product is a functor F:C→ C'which preserves the Proposition 2: Let (C,⊥ , ) be a category with product product. and composition. Then:

Examples: • Every element of K0 C has the form (A)C-(B)C for some A, B in obj(C) • Let R be any supercommutative super-ring and let • (A)C = (B)C if and only if ∃C ,D0,D1, E0, E1∈ P(R) denote the category of finitely generated obj(C), such that D0°D1 and E0°E1 are defined and

projective supermodules over Rwith isomorphisms A⊥⊥ C (D01 D ) ⊥ E 0 ⊥ E 1 ≅ B ⊥⊥ C D 0 ⊥ D 1 ⊥ (E 01 E ) 172 J. Math. & Stat., 5 (3):171-177, 2009

• If F: C→C′ is a functor of categories with product If Z ∈ spec(R) (i.e., ZR⊆ is a prime superideal) and composition, then F preserves the composition. and P∈ P(R), then PZ is a free RZ -supermodule and its Moreover, the map K0F: K0C→K0C′ given by (A)C rank is denoted by rkP (Z). The map: → (FA)C' is well-defined and makes K0F a functor into abelian groups rkP: spec(R) → Z

Now let (C,⊥ ) be a groupoid. For A ∈ obj(C), we given by Z→ rkP (Z) is continuous and is called the write G (A) = C(A,A), the group of automorphisms of rank of P. As R is a supercommutative super-ring, K0 A. If f: A → B is an isomorphism, then we have a P(R) and Q ⊗R K0P(R) = QK0P(R) are rings with homomorphism G(f): F(A) → G(B), given by multiplication induced by ⊗R . Since: G(f)(α) = fαf−1. We shall construct, out of C, a new category ΩC . rkPQ⊕ = rk P+ rk Q we take obj( ΩC ) to be the collection of all automorphisms of C. If α ∈ obj( ΩC ) is an and automorphism of A ∈C, we write (A,α) instead of α. A rkPQ⊕ = rk P rkQ morphism (A,α) → (B,β) in ΩC is a morphism R f:A→ Bin C such that the diagram in Fig. 1 is We have a rank homomorphism: commutative, that is G(f) (α) = β. The product in ΩC is defined by setting (A, α) ⊥ (β, β) = (A ⊥ B,α ⊥ β). rkP: K0P(R)→C The natural composition is defined in ΩC as follows: if α, β∈obj( ΩC ) are automorphisms of the where C is the ring of continuous functions same object in C, then α β = α β and: spec(R) → Z . The rank homomorphism rk is splitting by the ring (α ⊥ β) (α ′ ⊥ β′) = α α′ ⊥ β β′ homomorphism C→ K0 P(R), so that:

≅ Definition 6: If (C, ⊥ ) is a category with product, we K0 P(R) C ⊕KP(R)0 define: where, KP(R)0 = ker(rk) So: K C = K ΩC 1 0

If F: C→C′ is a functor, then ΩF : ΩC → ΩC ′, Q ⊗Z K0P(R) ≅ (C)Q ⊗Z ⊕ ( Q ⊗Z KP(R)0 ) preserving product and composition, so we obtain homomorphisms KiF: KiC → KiC′, i = 0,1. The next results generalize the results proved by H. If P(R) is the category of finitely generated Bass. projective R-supermodules, where R is a supercommutative super-ring and their isomorphisms Theorem 3: Suppose max(R), the space of maximal superideals of R, is noetherian space of dimension d, with ⊕ . Then the tensor product ⊗R is additive with then: respect to so that it induces on K P(R) a structure ⊕ 0 of commutative ring. • If x ∈ K P(R) and rk(x)≥ d , then x(p)= for The next following results are just the generalizing 0 P(R) of the results proved by H. Bass to the supercase. some P ∈ P(R) • If rk((P)P(R) )> d and if (PP(R)) )= (Q P(R ) , then PQ≈ d1+ • (K0 P(R))= 0

Proposition 3: The following conditions on R- supermodule P are equivalent:

• P is a finitely generated projective supermodule over R and has zero ahnihlator Fig. 1: Set of morphisms • P∈P(R) and has every where positive rank 173 J. Math. & Stat., 5 (3):171-177, 2009

• ∃ a supermodule Q and a positive integer n such product by a scalar. on Aσ is defined by λa = σ(λ)a for σ n all λ∈R. Then one easily check that A is a central that PQR⊗≈R separable R- superalgebra.

σσ RESULTS AND DISCUSSION Now let τ⊗→⊗:ARR A A A, be defined by αβ τ⊗(aα bββ ) =− ( 1) b ⊗ aα , then τ is a σ-linear Let P(R) be the category of finitely generated automorphism. In particular τ is S-linear. Define the projective supermodules over R, Az(R) the category of Corestriction: Azumaya superalgebras over R and Prog(R) the category of finitely generated faithful projective R- σ supermodules. Tr(A) = { xA∈⊗R A | τ=(x) x } A useful fact to be remember is that since R is supercommutative super-ring, P∈Prog(R) if and only if Obviously, Tr(A) is an S-superalgebra. But [3] P∈Prog(R) and P is faithful. If A,B∈Az(R) are by Tr(A) is an S-progenerator as an S-supermodule, if A is an R-progenerator as an R-supermodule. Moreover equivalent in BW(R) (the Brauer-Wall group of R), we [3] will write A ∼ B. If M is a supermodule over R, then if A is central separable over R then by Tr(A) is nM is the n-fold direct sum of M. If P∈P(R) let (P) be central separable over S. the image of P in K P(R) and {P} in Q ⊗ K P(R) = Q 0 Z 0 Lemma 1: Let R/S be a Galois extension of super- K P(R). The next results generalize the results proved 0 commutative super-rings with Galois group G{1,}= σ . by[6]. Let A, B be R-supermodules (superalgebras), P∈ Theorem 4: Let P,P',Q ∈ P(R). Then: Prog(R):

• P∈Prog(R) if and only if there is a Q in P(R) such • If A and B have G-action, so does MA=⊗R B and GG G MA=⊗R B that PQ⊗R is free 1 • Tr( AB⊗R ) ≅ Tr(A) ⊗S Tr(B) • If x∈ Q K0P(R) and rk(x) > 0 then x{Q}=  for m • If E = EndR (P) , Tr(E) ≅ EndS ( Tr(P)) some Q∈ Prog(R), m0> an integer Theorem 5: Let A ∈Az(R) and P,Q∈ Prog(A) such • If {P}= {Q} , P∈Prog(R), then there is an integer that PQ≈ as R-supermodules. Then there is an integer n0> such that nP≈ nQ n > 0 such that nP≈ nQ as A-supermodules. • If Q∈ Prog(R) and ((P)−= (P'))(Q) 0 then there is

an integer n > 0 such that nP≈ nP' Proof: A⊗R EndA (P) ≅ EndR(P) ≅ EndR(Q) ≅ A⊗R • If P∈Prog(R) and rkP is a square then there is an integer n > 0 and Q∈ Prog(R) such that EndA(Q). Tensoring by A° yields that: 2 nP≈⊗ QR Q End (A) ⊗ End (P) ≅ End (A) ⊗ End (Q) R R A R R A Let R/S be Galois extension of supercommutative or super-rings with finite Galois Group G. M = M + M , 0 1 an R-supermodule, has a G-action if there is a group EndAR (A⊗ P) ≅ EndAR (A⊗ Q) injection ϕ→:G Aut(M) such that ϕσ() is σ-linear for all σ∈G. That is, ϕσ()(mr)αβ =ϕσ ()(m)(r) α σ β . Let where, A acts on AP⊗R (AQ⊗R ) by acting on P ( Q ). M{mM:()(m)mG =∈ϕσ= for all σ∈G} . The following Using[3], There is a rank one projective R-supermodule [1] G fact was proved in , if M∈ Prog(R), M Prog(S) then: I, such that AP⊗R ≅ AQ⊗R ⊗R I as A-supermodules.

≅ G Theorem 4(a) implies that mR⊗R P mR⊗R Q ⊗RI as RM⊗≅S M A-supermodules, for some m > 0 and m'R≅ m'R⊗R I Again let R/S be a Galois extension of super- as R-supermodules, for some different m′. Finally, commutative super-rings with Galois group G{1,}= σ . n = mm′ will satisfy the theorem. Let A be any central separable R-superalgebra, we On a superalgebra A, a map J:A→ A is called a define Aσ as follows, set Aσ = A as a super-ring, but the superinvolution if J2 is the identity and J is an 174 J. Math. & Stat., 5 (3):171-177, 2009

J2 antiautomorphism. More explicitly, (aα )= aα , perfect if every simple subsuperalgebra of A admits a JJ J J αβ JJ superinvolution of the second kind. (a+=+ b ) a b and (a b )=− ( 1) b a for all αβ αβ ααββ a,bαβ∈ A. Let C = Z(A) (the super-center of A) then J Theorem 8: Suppose R is a connected semilocal super- must preserve C. If J is the identity on C, J is a ring and A is a central separable R-superalgebra. superinvolution of the first kind. If not, J induces an Suppose R/S is a Galois extension with Galois group automorphism of C of order 2 and J is said to be of the {1, σ}. Then A has a superinvolution of the second second kind. Two superinvolutions J, J′ which agree on kind extending σ if and only if Tr(A) ∼ 1 and A is C are said to be of the same kind. perfect. The following theorem generalizes of[6]. Proof: Suppose A has a superinvolution, *, extending Theorem 6: If A∈ Az(R) and AA1⊗R ∼ , then there is a σ. Then it is easy to Check that A is perfect. Also * σ B∈ Az(R), such that BA∼ and BB≅ . induces an isomorphism AA≅ , so there is an isomorphism: Another way of viewing an isomorphism BB≅ is that B has an antiautomorphism, J, of the first kind. σ Now, we are ready to prove the following result. ϕ⊗→:AR A EndR (A)

ϕ αγ ∗ Theorem 7: Suppose A is a super-ring with given by x(aγα⊗=− b)β (1)axb αγβ . Set A'= A'01+ A' , antiautomorphism J such that J2 is inner, induced by a A'=∈ {a A :a∗ = a } JJ where α αααα. wA∈ such that w(w)== (w)w 1. Then M2(A) 00 00 0 0 Since ∗ is σ-linear R-supermodule automorphism has a superinvolution of the same kind. of A, the S-supermodule A' is an S progenerator as a Proof Let L be the inverse map to J. Since: module over S. ϕ induces an isomorphism Tr(A) ≅ End (A') , hence Tr(A) ∼ 1. 2 S waw−1J (a) 0 αα0 = Conversely, since R is a connected semilocal super-ring, one easily sees that S is a connected JJ JL semilocal super-ring also. Let Tr(A) ≅ End (P) . In other We have (a)(w)αα00= (w)(a) and S LJ σσ words let τ→⊗:A⊗ A AR A given by (aαα ) w00= w (a ) , so the map: R ταβ (aα ⊗=−⊗ bββ ) ( 1) b aα , be a σ-linear automorphism. JJL σ ab (d)(w)(b)  Then Tr(A) is the fixed super-ring of AA⊗R under τ . αα α0 α JL  cdαα w(c)0 α (a) α  Say Tr(A) ≅ EndS (P) , where Pis an S-progenerator as a supermodule over S. Then σ Is a superinvolution on M2A of the same kind of J. A⊗R A≅⊗ RRS EndS (P) ≅ EndR (R ⊗ P) and if Next we try to find the conditions on a central separable R-superalgebra A to have a superinvolution ϕ =σ⊗1:RPRP ⊗SS → ⊗ , then ϕ ϕ τ of the second kind, if R is a connected super-ring. In (xγα (a⊗= bββ )) x γα (a ⊗ b ) , for all xRPγ ∈⊗S and the next theorem we try to find the conditions on σ abAAαβ⊗∈⊗R . Since R is connected, A= EndR(P) to have a superinvolution of any kind, σ where P is an R-progenerator as a supermodule over R, rankRR (A)= rank (A ) , but: if R is a connected super-ring. The following theorem involves assuming that R, σ AAEnd(RP)⊗≅RS ⊗ the base super-ring, is semilocal. We will use the fact, R [5] from , that if A, B are central separable R-algebras, Therefore: AB∼ and the rank of A equals rank of B, then AB≅ , which is also true in the superalgebra case (i.e., if A, B σσ A⊗⊗RR (A⊗≅RSR A ) A End (A) ≅ End (R ⊗⊗ P) A are central separable R-superalgebras, AB∼ and the RR rank of A equals rank of B, then AB≅ ). Let M be the [5] σ Jacobson radical of R. Then AA/MA= is a finite So by , AA≅ , which implies that direct sum of simple superalgebras. We call A is EndRR (A)≅⊗ End (RS P) , but the R-rank of A equals the 175 J. Math. & Stat., 5 (3):171-177, 2009

[5] J1− JL− 1 L R-rank of RP⊗S . So again by , AR≅⊗S P. In other where zuα = α and hence zzαα== (u) α, so that ϕϕ'' α −1L L α L1−− 1L words, A has a σ -linear antiautomorphism J such that (xγ ) = (-1) xγα (u ) u α= x γ since (1)(u)−=αα (u). It αλ J for all a,x,bαγβin A, setting x(aγα⊗=− b) β (1)axb αγβ ϕ suffices to find uα of A such that (uαα )+ u is a unit, σ yields the isomorphism AAEnd(A)⊗≅RR and the ϕ for if uα is a preimage of uα , then (uαα )+ u will be a 2 map ϕσ⊗:1:A(RP)A ≅ ⊗S → satisfies ϕ =1 and ϕ fixed unit of Aa. Since A is perfect, it suffices to ϕϕ τ (xγ . (aαβ⊗= b )) x γαβ .(a ⊗ b ) . Therefore: prove.

αλ J ϕϕαβ β(α+γ)J ϕ Lemma 3: Let A be a finite dimensional central (-1) (a x b ) = (-1) x .(b⊗ a ) = (-1) b x a αγβ γ βαα βγ simple superalgebra over a field F with a superinvolution J of the second kind and any associated ϕ (ϕ respects the grading). For w=1∈ A0 we have ϕ to J then there is an element aα in Aα such that JJ −1J2 ϕ α α ww = w w = 1 and waαα w = a and: (a )+ a is a unit.

ϕ 2 J ϕϕαλ β(α+γ)J Proof: The element w=1 is central since J is a ϕ=1, (aαγβ x b ) = (-1) (-1) b βγα x a (1) superinvolution. If w1≠ − , then a10 = will do. If ϕ JJ w = −1, then (a)αα= (a)w=-(a) α . Since J is of order Lemma 2: Let A be a central separable R-superalgebra, J with J and ϕ satisfying (1). Then A has a 2 on F, there is f in F such that f-f≠ 0, so again take J superinvolution agreeing with J on R if ϕ fixes a unit of a=f-f0 .

Aa. e Lemma 4: Suppose Q is a right A=A⊗R A - ϕ Proof: If ua is a unit in Aa such that u=u then supermodule, then: α α u(1.u)uw==ϕ J , so (uJ1 )− u = w , but ααα αα Q=M⊕ I J-1α− 1J α−1J (uαα )=− ( 1) (u ) , therefore w(1)(u)u=− αα, implying

' that xuxuJ1J= − is a superinvolution since J' is an where, M is the R-subsupermodule of Q generated by γαγα all elements of the form (a⊗⊗ 1-1 a )q , where antiautomorphism on A and: ααβ aAα ∈ α and qQβ ∈ β . If Q is R-projective as a JJ'' -11J−α−− J 1 J J2 1J (xγααγααααγαα ) = u (u x u ) u = (-1) u (u x (u ) )u supermodule over R then: −−1J 1 =uαα u (wxw γ )w rankRR (A).rank (I) = rank R (Q) . −−1J 1 =xγαα ,sinceu u =w Proof: Consider the well-known split exact sequence of Continuing proof of the theorem: Let M be the Ae -supermodules: jacobson radical of R. Then AA/MA= is a finite e µ direct sum of simple superalgebras. On A , ϕ and J 0JA→→ →→ A0 induce maps ϕ and J satisfying (1). Every preimage of where, µ(aα ⊗ bβαβ ) = a b and J is a right super-ideal of a unit uα of A is a unit ua of Aa. Thus we can change J e A generated by all elements of the form a1-1aα ⊗ ⊗ α by conjugation with a unit u , to make J any desired a where aA∈ . Suppose Q is a right Ae -supermodule. A ' α α antiautomorphism of of the same kind. In fact, if J e ' Tensoring by Q over A yields a split exact sequence is defined by x=uxuJ1J− , we can find a corresponding γαγα of R-supermodules: ϕ' so that J,''ϕ satisfy (1). Specifically if L is the

' e1⊗µ inverse map to J, we can set x=uxuϕ−ϕ1L, to show that 0Q→⊗AAee JQ →⊗ A →⊗ Q A e A0 → γαγα we have: e e of course, QAQ⊗A ≅ under the map azαβ⊗ az αβ. ϕϕ'' −11 − ϕ L ϕ L (xγααγαα ) = u (u x u ) u Under this isomorphism QJ⊗Ae is mapped onto M

α−1LJ L = (-1) uααγαα (u x z )u defined above. Thus QMI≅ ⊕ , where IQ≅⊗Ae A. But: 176 J. Math. & Stat., 5 (3):171-177, 2009

IAQ⊗≅⊗⊗RARARe (AA)Q ≅⊗e (AA) ⊗ CONCLUSION

The extended two results proved by Saltman[6] to e as an R-supermodules, therefore, IAQ⊗≅⊗RAe A ≅ Q. the supercase and the algebraic K-theory of projective Suppose R is a local supercommutative super-ring, supermodules over (torsion free) supercommutative σ an automorphism of R of order 2, P is an R- super-rings would help any researcher to classify progenerator as a supermodule over R and I a rank one further properties about projective supermodules. R-projective supermodule. A morphism e:P⊗R P→ I is σ REFERENCES called a bilinear I form on P, a morphism e:P⊗R P→ I is called a σ bilinear I form on P. The image e(p⊗ q ) α β 1. Jaber, A., 2003. Superinvolutions of Associative is often written as e(pαβ ,q ) and in either case, e can be Superalgebras, P.h. D. thesis, University of Ottawa, thought of as a map e:P×→ P I. Such a form induces a Ottawa, Canada, 2003. ∗ σ 2. Childs, L.N., G. Garfinkel and M. Orzech, 1973. map e:P→ Hom(P,I) (P→ Hom (P,I)) given by R R The brauer group of graded azumaya algebras, ∗ e (pαβ )(q ) = e(p αβ ,q ) . In a similar manner, we define Trans. Am. Math. Soc., 175: 299-326. DOI: σ e:P∗ → Hom(P,I)R (P→ HomR (P ,I)) given by 10.1007/BFb0077340 3. DeMeyer, F. and E. Ingraham, 1971. Separable e (p )(q ) = e(p ,q ) . If e∗ and e are isomorphisms ∗α β αβ ∗ Algebras Over Commutative Rings, L.N.M. 181, then we say e is nondegenerate. The next final result Springer-Verlag, New York, USA., ISBN: shows that the existence of superinvolutions on 9783540053712, pp: 172. EndR (p) , where EndR (p) is an R-progenerator as a 4. Racine, M.L., 1998. Primitive superalgebras with supermodule over R, is equivalent to the existence of superinvolution, J. Algebra, 206: 588-614. DOI: forms on P and this result was proved in[1]. 10.1006/JABR.1997.7412 5. Roy, A. and R. Sridharan, 1967. Derivations in Theorem 9: Let R be a connected super-ring and azumaya algebras. Math. Kyot. Univ., 72: 161-167. 6. Saltman, D.J., 1978. Azumaya algebras with A = EndR (p) be a central separable R-superalgebra such that A is an R-progenerator as a supermodule over involution. J. Algebra, 52: 526-539. DOI: R, then: 10.1016/0021-8693(78)90253-3

• A has a superinvolution of the first kind if and only if there is a rank one R-projective I, a

nondegenerate bilinear I form e on P and a δ∈ R0 2 αβ such that δ=1 and e(xαβ , y ) = (-1)δ e(y βα ,x ) for

all x,yαβ in P • Let σ be an automorphism of R of order 2. Then A has a superinvolution of the second kind extending σ if and only if there is a rank one R-projective I with a σ-linear automorphism of order 2 (also called σ) a σ-bilinear I form e on P and an element

δ in R0 such that σδδ=() 1 and αβ σδ(e(xαβ , y )) = (-1) e(y βα ,x ) for all x,yαβ in P

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