Cohomology of the Steenrod algebra

6. V. S. Drenski, "A minimal basis of identities of a matrix algebra of order two over a field of characteristic zero," Algebra Logika, 20, No. 3, 282-290 (1981). V. T. Filippov, "On the variety of Mal'tsev algebras," Algebra Logika, 20, No. 3, 300314 (1981). S. V. Okhitin, "The identities in two variables of the Lie algebra s~(2, K) over a field of characteristic zero are finitely based," Moscow (1985), Dept. at VINITI No. 2463-85 DEP. S. Yu. Vasilovskii, "A basis of identities of the Lie algebra s~ 2 over an infinite field," in: International Algebra Conference. Abstracts of Lectures in the Theory of Rings, Algebras and Modules [in Russian], Novosibirsk (1989), p. 29. Ao A. Premet and K. N. Semenov, "Varieties of residually finite Lie algebras," Mat. Sbornik, 137, No. i, 103-113 (1988). E. I. Zel'manov, On some problems in the theory of Lie groups and algebras," Mat. Sbornik, 180, No. 2, 159-167 (1989). E. I. Zelmanov, "On the restricted Burnside's problem," International Algebra Conference. Abstracts of Lectures in the Theory of Rings, Algebras and Modules, Novosibirsk (1989), p. 223. G. V. Sheina, "Varieties of metabelian A-algebras," Vestn. Mosk. Gos. Univ., Ser. i, Mat., Mekh., No. 4, 37-46 (1977). 7. 8. 9. i0. ii. 12. 13. COHOMOLOGY OF THE STEENROD ALGEBRA V. A. Smirnov Let A be a Steenrod algebra and BA: Z/2~-,4"-A~"-... * - A Z " ~ , . . . B is a construction over A. Then the cohomology A* of this complex is known as the cohomology of the Steenrod algebra A* = H*(BA) and it determines the E2-term of the Adams spectral sequence of homotopy groups of spheres [i]. In what follows we shall need a Steenrod algebra in which Sq ~ ~ i. To avoid the introduction of additional notation, we shall assume that A is an algebra satisfying this condition; in fact, we shall assume that its cohomology is isomorphic to that of the ordinary Steenrod algebra [2]. Our main problem is to describe this cohomology. of operads [3, 4]. To that end we shall use the language A family ~={~(7)} of chain complexes ~ (j), j e I, on which the symmetric groups lj are acting, is an operad if there are given maps ~: ~ ( k ) | 1 7 4 1 7 4 J=J,~...+J,, satisfying a certain condition of associativity and compatibility with the action of the symmetric groups [4]. Given a chain complex X and an operad ~, we can define a complex eJ ~X=~.~) i~U) | X X is known as an algebra over ~, if there is given a map of complexes ~: ~ X + X such that ~(~ • i) : ~(1 • ~). Vi I. Lenin Moscow State Pedagogical University. Translated from Matematicheskie Zametk$, Vol. 52, No. 2, pp. 120-126, August, 1992. Original article submitted March 25, 1992. 0001-4346/92/5212-0839512.50 9 1993 Plenum Publishing Corporation 839 The correspondence X + ~ X defines a monad in the category of chain complexes, and X is an algebra over ~, if it is an algebra over the corresponding monad [3]. Let us consider a few examples of operads and algebras over operads. I. ~={J(])}, where ~ ( j ) is the free Ej-module with one zero-dimensional generator a (j). The operator ~ is defined by j = j , + . . . +jk, ~(a(k)|174174 The algebras over ~ are just ordinary algebras. 2. S = {S(j)}, where S(j) is the trivial Ej-module with one zero-dimensional generator s(j). The operation ~ is defined as in the previous example. The algebras over S are the commutative algebras. 3. ~ = { ~ ( J ) } - the free and Ej-free operad, generated by elements ~i e ~ ( i dimension i. The differential is defined on the generators ~i by ( - - t)~? (~k | 1 | A l g e b r a s over ~ are known as ~ - a l g e b r a s | ~i-~-1 | + 2) of 9 9~ t): [5, 6 ] . The main result concerning ~ - a l g e b r a s is that the homology of a differential graded algebra can be endowed with the structure of an ~ - a l g e b r a , determined up to a certain equivalence relation [6]. In particular, this structure determines the Massey products, which are obtained from the operations ~i: x| § X by restriction to the elements zi+ 2 X| that satisfy the equalities (~o| | +(-t)'1| (~,|174174174 | (~o| . . . @ 1 - . . . + ( - 1 ) ~ ' 1 @ ... | (z~+,). Conversely, i f t h e M a s s e y p r o d u c t s a r e d e f i n e d on a g r a d e d m o d u l e X a s p a r t i a l and manyvalued maps, then first, choosing representatives, we c a n make t h e s e maps s i n g l e - v a l u e d , a n d t h e n e x t e n d t h e m t o maps ~ i : x | " X that define an~=-structure on X. Of c o u r s e , this construction is unique only up to equivalence. We will need the operad ~ g in the category of graded modules, defined as the quotient of ~ by the image of the differential. Clearly, if a graded module X is an algebra over ~, it is also an algebra over ~ g . We now consider how to present a graded ~ - a l g e b r a . Let X be a graded J~-algebra and B(~=, X): X ~ X ~ 2 X ~- .... B a construction. Call an element x e X decomposable if it is homologous to zero in B(~=, X). In the language of Massey products, this definition means that an element x e X is decomposable if it is the image of some Massey product. Thus, the definition is analogous to that of a decomposable element in [7]. The quotient module of X by the module of decomposable elements will be denoted by X, and its elements will be called indecomposable elements. It was shown in [7], in particular, that the only indecomposable elements in the cohomology of the Steenrod algebra are the elements hi, i e 0. The other elements are decomposable, i.e., images of Massey products. The projection of the B-construction B ( ~ , X) onto its second term JW~X induces a homology map H . B ( ~ , X)-+~g~. The elements of the image of this map will be called generating relations in X. Translated into the language of Massey products, the definition states that the generating relations are determined by the elements not homologous to zero whose Massey products are equal to zero. To determine generating relations means specifying elements whose Massey products vanish. All other relations are consequences of these relations and the ~ - s t r u c ture. Having specified the indecomposable elements X and the generating relations, one can reconstruct the ~ = - a l g e b r a X. To that end, consider the ~=-algebra ~ g g and factorize it modulo the generating relations. This gives a certain ~ % Z ~ - a l g e b r a ~ g x . The submodule of this algebra generated by the Massey products will be the required module isomorphic to X. 840 We now continue our survey of examples of operads and algebras over operads. 4. E~ = {E=(j)} - the free, lj-free and acyclic operad with one generator U0 e E~(2) of dimension zero. For example, E~(2) is a E=-free and acyclic complex with generators U i of dimension i and differential dU i = Ui_ I + (-l)iUi_iT, T e E 2. We note that ~ + E~ [4]. 5. S= = {S~(j)} - the acyclic operad generated by elements U i e S~(2) = E~(2) and ~i e S~(i + 2) such that, as in ~ , ~0 = U0. All relations between these elements follow from the acyclicity condition of the operad S~. Clearly, one has a projection E~ + S~ and an injection 5g~ § S~. It was shown in [4] that a chain complex of a topological ever, a more general proposition can be proved. space is an E~-algebra. How- PROPOSITION i. The cochain complex of a simplicial object X,: X0 § X~ + ... + X n + ... in the category of commutative coalgebras is an E~-algebra. If the ground ring is a field, the cohomology of this complex is also an E~-algebra. In particular, one can give this cohomology the structure of an ~t~-aliebra, and if the ground ring is the field Z/2, one has the action of the Steenrod algebra Sqlx n = xnUn-ixn. These structures are compatible in the sense that the following equalities hold: 1) S q ~ (xo | 9| xi+~) = Z = 2a) a~(xo| . . . |174 =a~_~ (xo| ~i . . . |174 ... | (Sq nax o | ..-. +x~+,| U~x~+,+~_~ (xo | n~(xo| . . . |174174 ... | "... Sq"+lx~+~), | ...| ... | ... ... 2b) .. 9 9| +xoU,xi§174 ... +x,| | . . . |174 =xoU~a~-~ (x~ | . . . | +ni-~ (xoU~x,| 2~) ~ ~ (xi. | =Z + Z | xip | z~. | n._~ (x~. | a,-2 (x~. | ... | | 9 9 9+ x ~ | where the sums are taken over all reshuffles, | | z4) = x(k ' U~ z~t" | | x4~"U~xj~. | ... .. 9| | x~,~,U ~ , xi~) + | 9 9| z~q) + .~ xU1y = 0 if x ~ y, p, q ~ 2 [8]. The operations Sq i and ~i with relations (I), (2) form an S~-algebra of course, does not exhaust the entire E~-algebra structure. Since the Steenrod algebra follows that the B-construction coalgebras and, by what we have algebra. Fortunately, it turns describe A*. structure, which, A is a Hopf algebra with commutative comultiplication, it BA is a simplicial object in the category of commutative stated, its cohomology A* is an E~-algebra and hence an S~out that the S~-algebra structure alone is sufficient to A graded S~-algebra is presented in a manner analogous to that of an 5r To be precise: Given a differential graded module X over the Steenrod algebra, we let S~X denote the quotient complex of ~r modulo the relation (2), on which the action of the Steenrod algebra is defined by the relation (1). Clearly, the correspondence X + S~X is a monad in the category of differential graded A-modules. Given a graded A-module X, we let S~gX denote the quotient module of S~X by the image of th e differential. An element x of the graded S~-algebra X is said to be decomposable if it islhomologous to zero in B(S~, X). Denote the quotient module of X modulo the decomposable elements by X. Its elements will be called indecomposable elements in X. The projection of the B-construction B(S~, X) onto its second term S~X induces a homology map H,B(S~, X) + s~gx. The elements of the image of this map will be called generating relations. $ust as in the case of 3g~-algebras, the indecomposable tions.uniquely determine the S~-algebra X. elements and generating rela- Our main result is that A*, as an S~-algebra, is generated by a unique one-dimensional elemeot h 0 and the relations ~i(h0 | ... | hi+ I) = 0, where hi+ I = Sq~ i ~ 0. 841 To prove this assertion, consider the diagram BA: A CBA: CA A| o:-- ~- <- CA @2 ........ A@ll (- CA | <- 4-- 9 ....... i C.BA: C"A , 9 9 <- "- C' A r~''' <-- ~ , (y~ A | <-- (- , ,~, , where C denotes the comonad in the category of graded modules that associates to any module X the free commutative coalgebra CX generated by X. Note that, since the Steenrod algebra A is the free commutative coalgebra generated by elements $i of dimension 2 i - i, it follows that the cohomology of the first column of the diagram vanishes in all dimensions except zero, in which dimension it is generated by the elements $i" Similarly, A | A is the free commutative algebra generated by the elements $i | ~0, $0 ~ $i, and, consequently, the cohomology of the second column of the diagram vanishes in all dimensions except zero, in which it is generated by the elements $i ~ $0, $0 | $i, etc. Going over now to the row cohomologies, we conclude that the cohomology of the total complex of the diagram vanishes in all dimensions except the first, in which it is generated by the elements $i" We now consider computation of the cohomology of the total complex in another order: first rows and then columns. The cohomology of the first row is, by definition, A*. rows we will use the following proposition, from [9]. To compute that of the next PROPOSITION 2. For any simplicial complex X, there is an isomorphism of cohomologies H*(CX) ~ SAH*(X), where AH*(X) is the free unstable A-module generated by H*(X) and SAH*(X) a free commutative algebra over AH*(X). Thus, the row cohomologies of our diagram are A*, ( S A ) A * . . . . . (SA)"A* .... It follows from results of Berikashvili [i0] and myself [ii] that in this case we can define in the B-construction B(SA, A*) higher differentials that convert this B-construction into a complex B(SA, A*) chain equivalent to the original total complex. Moreover, these differentials in B(SA, A*) are determined by the E~-algebra structure on A*. To describe the homology of the B-construction B(SA, A*), we will need the following general proposition about B-constructions [12]. PROPOSITION 3. Let F, G be two monads in the category of chain complexes. Assume given the permutation map T: FG + GF, and let the chain complex X be an algebra over F and G. Then X is an algebra over the monad GF and one has the chain equivalence B(GF, X)~B(F. B(G, X)). To prove this proposition, one uses the fact that the complex B(G, GX) is the simplicial resolution of X and the complex B(F, X) is a retract of the complex B(GF, GX). Applying the functor B(GF, -) to the simplicial resolution B(G, GX), we get the simplicial resolution B(GF, GX) § B(GF, GiX) § ... of the complex B(GF, X). Now, using the above retractions, we see that the complex B(F, B(G, X)): B(F, X)*-B(F, GX)~ is a retract of the simplicial resolution of B(GF, X). Consequently, B ( G F , X ) ~ - B ( F , B ( G , X ) ). 842 ... This proposition implies a chain equivalence B(SA, A*) = B(A, B(S, A*)). Higher differentials in B(SA, A*) determine higher differentials in B(A, B(S, A*)), which convert it into a complex B(A, B(S, A*)) chain equivalent to the complex B(SA, A*). Denoting X, = H,B(S, A*), we will use the chain equivalence B(SA, A*) = B(A, X,). The cohomology of the B-construction B(A, X,): X, + AX, + ... + Anx, + ... is thus isomorphic to that of the total complex. By dimensional arguments, these cohomologies are trivial in all but the zero dimension, in which they are generated by the elements $i and therefore X i = HiB(S , A*) is the free unstable A-module generated by the element ~i+z. We now consider the B-construction B ( S , A*): A**-SA*+- . . . + - S " A * ~ . . . , whose differential is determined by the S~-algebra structure on A*. Note that, since the projection S~X + SX is a chain equivalence, it induces a chain equivalence of B-constructions B(S~, A*) -~ B(S, A*). Therefore, the zero-dimension homology of the B-construction B(S, A*) is the module of indecomposable elements in A*. And that, as we have seen, is the free unstable A-module with one generator Sz = h0. The i-dimensional homology of the B-construction B(S, A*) is a free unstable A-module with one generator $i+l e SiA*,determined by the tuple (h0, .... hi+l). The fact that these generators are cycles means that the Massey products ~i(h0 | ... | hi+z) vanish and therefore the generating relations are relations of the form ~i(h0 | ... | hi+l) = 0. This completes the proof. We now consider a few computations in the cohomology of the Steenrod algebra A*. As we have shown, the indecomposable elements are the elements hi, i _> 0, hi+ 1 = Sq~ . Applying the operation Sq ~ to the relation h0.h I = 0, we obtain relations hi.hi+ I = 0. Again applying the operations Sq I, Sq 2, etc., to these relations, we obtain hi2hi+2 + hi+13 = 0, hi4hi+3 = 0 . . . . . hi 2 hi+k+m = 0, k _> 2, and these are in fact all the relations that are derivable from the relation h0.h I = 0 by means of the Steenrod operations. To obtain other relations, we use the operations ~z and the relations a (~| n~ (xo| 1+ I | = ~ , (~| i | I - l | 1 7 4 i + I | I | |174174174174 = = (x~xi)U,z2+xo(x,U,x2+(xoUlx2)x,, It follows from the second relation t h a t ~ l ( h l | h0 | h i ) = h 0 " h 2 . The f i r s t , t o t h e e l e m e n t h i | h0 | h~ | h 2, i m p l i e s a new r e l a t i o n h 0 . h 2 2 = O. A p p l y i n g t h e operations, we o b t a i n r e l a t i o n s hil~2hi+k+2 2 = O. Using these relations, one can get new elements of A*. applied Steenrod These elements are Massey prod- ucts 2t hi+~ +~), k ~ 1. The least of them is the ll-dimensional element vz(h z | h 0 | h22). To obtain further relations and elements, one uses the operations ~2, ~3, ..., and the relations among them. LITERATURE CITED i. 2. G. F. Adams, "On the structure and applications of the Steenrod algebra," Comm. Math. Helv., 32, 180-214 (1958). J. P. May, "A general algebraic approach to Steenrod operations," Lect. Notes Math., 168, 153-231 (1970). 3. 4. 3. P. May, The Geometry of Iterated Loop Spaces, Lect. Notes Math., 271, Springer, Berlin (1972). V. A. Smirnov, "On the chain complex of a topological space," Mat. Sbornik, 115, No. i, 146-158 (1981). 5. G. D. Stasheff, "Homotopy associativity of H-spaces," Trans. Am. Math. Soc., i08, No. 2, 275-312 (1963). 843 6. 7. 8. 9. i0. ii. 12. T. V. Kadeishvili, "On the homology theory of fibered spaces," Usp. Mat. Nauk, 35, No. 2, 183-186 (1980). V. K. A. M. Gugenheim and J. P. May, "On the theory and applications of differential torsion products," in: Mem. Am. Math. Soc., 142, American Mathematical Society, Providence, R.I. (1974). J. P. May, "Matric Massey products," J. Algebra, 12, 533-568 (1969). V. A. Smirnov, "Homology of symmetric products," Mat. Zametki, 49, No. i, 104-113 (1991). N. A. Berikashvili, Differentials of the Spectral Sequence [in Russian], Tr. Tbilisi. Mat. Inst. im. A. M. Razmadze, 51 (1976). V. A. Smirnov, "Functional homological operations and weak homotopy type," Mat. Zametki, 45, No. 5, 76-86 (1989). V. A. Smirnov, "Homotopy theory of coalgebras," Izv. Akad. Nauk SSSR, Ser. Mat., 40, No. 6, 1103-1121 (1985). ON REPRESENTATION OF FINITE PSEUDO-BOOLEAN ALGEBRAS AND ONE OF ITS APPLICATIONS V. I. Khomich The present paper is devoted primarily to the study of properties of the method proposed in [1-4] for representation ol finite pseudo-Boolean algebras [5] in terms of partially ordered sets or implicatures of a special form [6, 2]. They can be used to obtain certain results concerning the superintuitionistic propositional logic 8 I constructed in [8, 4].* We will prove a result announced in [9] to the effect that the logic 81 coincides with the logic LM consisting of propositional formulas that are finitely general-valued in the sense of [10-12]. In order to do this, we will establish a relationship between methods of constructing pseudo-Boolean algebras yielding these logics. I. Let v be a set of propositional logical symbols containing ~. Below we will use the notion of v-algebra introduced in [13]. It is also defined in [i, 3, 4]. We denote a distinguished element in a v-algebra by I. Let 0 and @ be v-algebras. We will say that 0 is imbeddable in ~ if there exists an isomorphism between % and some v-subalgebra of the valgebra ~. We define a partial ordering on 0 by setting $ ~ ~ ($, ~ e 0) when $ ~ N = I. If @ contains a unique immediate predecessor (we denote it by ~) of its distinguished element, we say that 0 is a Godel v-algebra. We say that element $ and ~ of @ are incomparable elements of 0 if ~ m ~ ~ 1 and ~ ~ $ ~ i. An element $ of % is said to be an indecomposable element of % if ~, ~ O , ~ = I , ~=I and ~ ~ (~ ~ ~) = 1 implies that ~ = ~ or $ = cp. As in [i], we use I(0) to denote the set of all indecomposable elements of 0. We use the notation {$i . . . . , $n} m ~, where n, $i, $2, .... ~ n a r e e l e m e n t s of 0, to denote the expression gn ~ ( " " ~ ($i ~ ~)...). If 0 contains a-smallest element, we denote it by 0. If 0 is finite, then, for each $ of 0, we can construct a unique nonempty set r($) of incomparable and indecomposable elements of 0 such that r($) ~ $ = 1 and ~ ~ n = 1 (~ e r(~)).% Suppose is a subset of the set 0. An element g of ~ is said to be a minimal element of ~ if, for any ~ (~ e ~) it is not true that q m $ = 1 and $ ~ ~ ~ i. We denote the set of all minimal elements of ~ by ~(~). Let f be a mapping of a set ~ into a set A, and let A be a subset of ~. We use f(A) to denote the set of images of A under the mapping f, and we use IEI to denote the number of elements in a finite set E. *By a superintuitionist propositional logic we mean a set of propositional formulas that contain all of the axioms of the intuitionist propositional calculus [7] and are c l o s e d under a substitution rule and the law of modus ponens. %This was proved in [4] (see the footnote on p. 386 in Russian edition). Computing Center, Russian Academy of Sciences. Zametki, Vol. 52, No. 2, pp. 127-137, August, 1992. 1991. 844 0001-4346/92/5212-0844512.50 Translated from Matematicheskie Original article submitted March 26, 9 1993 Plenum Publishing Corporation