Odd-primary operation coordinates and a faithful triangular test
Let \(p\) be an odd prime and \(k=\mathbb F_p\). This argument constructs the polynomial and exterior coordinates of the dual operation algebra, computes their coproduct, and proves a faithful triangular pairing with every admissible word. The completeness assertion is proved in the companion one-group calculation, Sections 1–9. Independence of familiar operations is not, by itself, proof that there are no others.
The reduced powers \(P^a\), including their normalization, naturality, additivity, Cartan formula, instability and suspension properties, are the ones constructed in Odd-prime reduced powers and the Wu classes, Sections G–I. This proof uses those established operations, not a second construction of them. The coefficient Bockstein is supplied below. All tensor products and products of odd classes use graded signs. Suspensions and coefficient connecting maps use the usual compatible graded cohomology convention; in particular, interchanging two degree-one connecting maps has a minus sign.
1. The Bockstein and the cyclic test space
For a mod-\(p\) singular cocycle \(a\), choose an integer-valued lift \(\widetilde a\). There is an integer cochain \(b\) with
\[ \delta\widetilde a=p b. \]The integer cochain groups have no \(p\)-torsion, so \(\delta b=0\). Define \(\beta[a]=[b\bmod p]\). Changing the lift adds an integral multiple \(p c\) and changes \(b\) by \(\delta c\). Changing the mod-\(p\) representative by a coboundary gives the same conclusion after lifting that coboundary. Hence this is a well-defined natural additive degree-one operation. It is the connecting operation of the coefficient sequence, or equivalently of its Eilenberg-Mac Lane spectra. The integral cocycle \(b\) itself lifts its reduction, so
\[ \beta^2=0. \tag{1} \]The cochain Leibniz rule gives its full product formula:
\[ \beta(ab)=\beta(a)b+(-1)^{|a|}a\beta(b). \tag{2} \]Indeed apply \(\delta\) to \(\widetilde a\widetilde b\), divide by \(p\), and reduce. Naturality and the coefficient exact sequence give the operation on pairs as well. These statements do not require an Adem relation.
For \(C_p=\langle g\rangle\), use the free integral resolution with one generator \(e_i\) in every degree and differentials alternating \(g-1\) in odd degrees and \(N=1+g+\cdots+g^{p-1}\) in positive even degrees. Its exactness over \(\mathbb Z[C_p]\) follows directly: \(\ker(g-1)=\mathbb ZN\), and \(Nf=\epsilon(f)N\) implies \(\ker N=\ker\epsilon=(g-1)\mathbb Z[C_p]\). The last equality follows by taking successive partial sums of the coefficients of an augmentation-zero element. This is the integral version of the resolution in Section G.2 of the reduced-power lesson. Its comparison with the homogeneous bar resolution uses the free-basis lifting construction of Section G.3; the bar resolution is the cellular chain model of the contractible universal \(C_p\)-space.
After applying trivial mod-\(p\) cochains all differentials are zero. The diagonal of that resolution, as given and checked in Section G.2, gives
\[ H^*(BC_p;k)=\Lambda(z)\otimes k[x], \quad |z|=1,\quad |x|=2,\quad \beta z=x. \tag{3} \]To see every part of this assertion, let \(z\) be the cochain dual to \(e_1\). Its integer lift has coboundary \(p\) times the cochain dual to \(e_2\), so the latter represents \(x=\beta z\). The even/even diagonal has coefficients one; thus \(x^j\) is dual to \(e_{2j}\). The even/odd diagonal likewise makes \(z x^j\) dual to \(e_{2j+1}\). The odd/odd coefficient for \(z^2\) is \(p(p-1)/2=0\) in \(k\). These classes give the unique basis vector in each degree, proving (3), including nonvanishing of every power of \(x\).
The class \(z\) is the homomorphism \(C_p\to k\) sending \(g\) to one. Under group addition \(BC_p\times BC_p\to BC_p\) it is primitive. Naturality and (2) make \(x\) primitive too. Also \(\beta x=0\) by (1).
The character \(g\mapsto\exp(2\pi i/p)\) defines a complex line bundle over \(BC_p\). Its first Chern class reduces to \(x\). One can check the identity of these classes on the bar complex: represent a group element by an integer \(a\in\{0,\ldots,p-1\}\). The connecting cocycle of \(z\) is
\[ \frac{a+b-((a+b)\bmod p)}{p}. \]The real lifts \(a/p\) of the character's phases give exactly the same integer winding cocycle for its first Chern class. Hence the classifying map \(BC_p\to\mathbb{CP}^{\infty}\) pulls the universal degree-two class back to \(x\), with no unexplained nonzero scalar.
2. The generated operation algebra and its coaction
Let \(\mathcal S\) be the subalgebra of actual stable operations generated by the \(P^a\) and \(\beta\), with \(P^0=1\). It is connected and finite dimensional in each degree. For a fixed degree only finitely many positive-degree generator words can occur: \(|\beta|=1\), \(|P^a|=2a(p-1)\), and identity factors are discarded. Put \(H=\mathcal S^\#\).
Cartan and (2) give the coproduct on the generators of \(\mathcal S\):
\[ \Delta P^a=\sum_{b+c=a}P^b\otimes P^c, \qquad\Delta\beta=\beta\otimes1+1\otimes\beta. \tag{4} \]It extends multiplicatively with the graded tensor sign. This extension respects all actual relations: a zero operation acts as zero on every external product. Evaluation on classes separates each finite-dimensional operation space, by the definition of equality of natural operations. Products of such evaluations separate the tensor product of two such spaces, so a tensor operation which acts as zero on all external products is zero. This proves that (4) is well defined on \(\mathcal S\). The same three-factor argument gives coassociativity. Its counit is the scalar identity component, and connectedness gives the antipode by degree recursion. The coproduct is graded cocommutative on the generators and therefore on the algebra. Thus \(H\) is graded commutative.
The \(\mathcal S\)-action gives a right evaluation coaction \(\psi\) on cohomology. Its components in a specified operation degree are finite; its entire value can be a completed sum. Naturality and (4) make \(\psi\) multiplicative and compatible with maps of spaces.
The primitive polynomials in \(k[x]\), for \(x\) primitive, are precisely linear combinations of \(x^{p^j}\). Here is the coefficient argument. Writing \(m=\sum_jm_jp^j\) in base \(p\),
\[ (1+T)^m=\prod_j(1+T^{p^j})^{m_j}\quad\text{in }k[T]. \]If \(m\) is not a power of \(p\), choose a nonzero digit \(m_j\) with \(0<p^j<m\). The coefficient of \(T^{p^j}\) is \(m_j\ne0\), giving a nonzero cross term in \((x_1+x_2)^m\). Conversely Frobenius shows that a \(p\)-power has no cross term. Monomials of different total degree cannot cancel these tests. In \(\Lambda(z)\otimes k[x]\), the primitive elements are \(kz\) together with these same \(p\)-powers of \(x\): a term \(z x^m\) with \(m>0\) has a nonzero component \(z\otimes x^m\), and there is no other monomial that cancels that component.
All operations are additive and natural for addition on these classifying spaces. Consequently there are uniquely determined elements
\[ \xi_j\in H_{2(p^j-1)},\quad j\geq1, \qquad\tau_j\in H_{2p^j-1},\quad j\geq0, \]such that, with \(\xi_0=1\),
\[ \psi(x)=\sum_{j\geq0}x^{p^j}\otimes\xi_j, \qquad \psi(z)=z\otimes1+\sum_{j\geq0}x^{p^j}\otimes\tau_j. \tag{5} \]The \(\xi_j\) defined using \(\mathbb{CP}^{\infty}\) and \(BC_p\) agree by the last paragraph of Section 1, since their polynomial classes remain independent on \(BC_p\). The leading terms in (5) are fixed by the identity operation. These are degreewise identities of coactions; an infinite sum is never treated as an element of an uncompleted algebraic tensor product.
Coassociativity now computes the coproduct of these coordinates. Apply \(\psi\) twice to \(x\), use Frobenius on its even-degree terms, and compare the coefficient of \(x^{p^n}\). Apply the same procedure to \(z\). The result is
\[ \begin{aligned}[c] \Delta\xi_n&=\sum_{i=0}^n\xi_{n-i}^{p^i}\otimes\xi_i,\\ \Delta\tau_n&=\tau_n\otimes1+ \sum_{i=0}^n\xi_{n-i}^{p^i}\otimes\tau_i. \end{aligned} \tag{6} \]No coefficient is inferred from a guessed Hopf-algebra formula. These identities come from composition of the actual operations on the two typed test classes.
3. Admissible words and the degree-preserving indexing
Use words
\[ W(\epsilon,s)=\beta^{\epsilon_0}P^{s_1}\beta^{\epsilon_1} \cdots P^{s_L}\beta^{\epsilon_L}, \quad \epsilon_i\in\{0,1\}, \quad s_i\geq p s_{i+1}+\epsilon_i, \tag{7} \]where \(s_{L+1}=0\). Trailing pairs \((s_i,\epsilon_i)=(0,0)\) are discarded. The word of length zero is \(\beta^{\epsilon_0}\). Define
\[ r_i=s_i-p s_{i+1}-\epsilon_i\geq0. \tag{8} \]Conversely, any finitely supported nonnegative sequence \((r_i)_{i\geq1}\) and finitely supported zero-one sequence \((\epsilon_i)_{i\geq0}\) determine the word uniquely by
\[ s_i=\sum_{j\geq i}p^{j-i}(r_j+\epsilon_j). \tag{9} \]There are no hidden words with \(P^0\) in the interior: if \(s_i=0\), nonnegativity in (9) makes all later \(r_j,\epsilon_j\) zero too. Its degree is exactly
\[ \begin{aligned}[c] d(W)&=2(p-1)\sum_i s_i+\sum_i\epsilon_i\\ &=\sum_{i\geq1}2(p^i-1)r_i+ \sum_{i\geq0}(2p^i-1)\epsilon_i. \end{aligned} \tag{10} \]The second equality sums the finite geometric progression in (9). Thus these words are in degree-preserving bijection with the formal monomials
\[ \xi^R\tau_{j_1}\cdots\tau_{j_e}, \qquad j_1>\cdots>j_e,\quad\epsilon_{j_a}=1. \tag{11} \]Descending exterior order is chosen for the sign calculation below. It is as valid a basis convention as increasing order.
4. A complete triangular independence proof
For a label \(J=(R,\epsilon)\), take one independent degree-two variable for each of its \(r_j\) copies at level \(j\), and one independent cyclic-space pair \((z_j,x_j)\) for each \(\epsilon_j=1\), including level zero. Form the class \(v_J\) consisting of all the chosen \(z_j\) in decreasing order of \(j\), followed by all the degree-two variables to their first powers. This is a class on a finite product of copies of \(BC_p\) and \(\mathbb{CP}^{\infty}\).
The distinguished output monomial \(w_J\) has no \(z\)'s: each variable at level \(j\), whether originally a \(z\) or an \(x\), has become \(x^{p^j}\). By (5), the coefficient of \(w_J\) in \(\psi(v_J)\) is exactly (11). Every output factor is even, so collecting these coefficients introduces no tensor-interchange sign. In particular, pairing (11) with an operation \(W_I\) is precisely the coefficient of \(w_J\) in \(W_I(v_J)\).
The only action formulas needed to compute this coefficient are
\[ \begin{gathered} P^a z=0\ (a>0),\quad P^0z=z,\quad \beta z=x,\quad\beta x=0,\\ P^a(x^b)=\binom ba x^{b+a(p-1)}. \end{gathered} \tag{12} \]The first line follows from instability, the identity operation and Section 1. The second follows by applying Cartan to the product of \(b\) copies of \(x\), since only \(P^0x=x\) and \(P^1x=x^p\) are nonzero. In particular, the base-\(p\) coefficient calculation of Section 2 gives
\[ P^a(x^{p^h})=0\quad\text{unless }a=0\text{ or }p^h. \tag{13} \]In the two nonzero cases the variable stays as it is or its exponent is multiplied by \(p\), with coefficient one. Each Bockstein converts one remaining \(z\) into its \(x\); it never creates another \(z\).
Fix a total operation degree and pad every word with trailing zero pairs to a common length \(L\). Order the words lexicographically by the finite tuple
\[ (\epsilon_L,s_L,\epsilon_{L-1},s_{L-1},\ldots, \epsilon_1,s_1,\epsilon_0). \tag{14} \]We prove that the matrix of coefficients of \(w_J\) in \(W_I(v_J)\) is triangular, with diagonal entries one. Operations act from right to left. At level \(L\), an output variable of exponent \(p^L\) must have its exponent multiplied by \(p\) at every one of the \(L\) reduced-power stages. If it starts as \(z_L\), it must therefore be converted by the rightmost Bockstein. A nonzero coefficient is impossible if \(\epsilon_L^I<\epsilon_L^J\). When these two entries agree, the reduced power must allocate at least one unit to each of the \(r_L^J+\epsilon_L^J=s_L^J\) level-\(L\) variables. It is impossible if \(s_L^I<s_L^J\).
If both entries agree, the allocation is forced: exactly those highest-level variables are raised, no lower-level variable is changed, and the Bockstein, when present, converts exactly the required \(z_L\). Its sign is positive because that is the first remaining odd factor in the chosen order.
Continue downwards. Suppose all entries at levels above \(i\) agree. The operations already performed are then forced as just described. A variable originally at level \(j>i\) currently has exponent \(p^{j-i}\); to reach \(p^j\) it must be raised at every remaining reduced-power stage. The new level-\(i\) variables must likewise be raised at all remaining \(i\) stages. If there is a \(z_i\), conversion cannot be postponed to a later Bockstein, because it would then have at most \(i-1\) opportunities for exponent multiplication. Thus a nonzero coefficient requires first \(\epsilon_i^I\geq\epsilon_i^J\). If these are equal, it requires
\[ s_i^I\geq\sum_{j\geq i}p^{j-i}(r_j^J+\epsilon_j^J)=s_i^J. \tag{15} \]Equality exhausts the entire Cartan budget on the required variables, so the next step is again uniquely determined and all lower-level variables remain untouched. A strict inequality at the first differing entry is on the permitted side of (14); a strict inequality on the opposite side makes the coefficient zero. At level zero the remaining Bockstein, if required, converts the sole remaining \(z_0\). This proves triangularity.
On the diagonal every required Cartan coefficient is one by (13); every required Bockstein has no odd factor before it, by the decreasing order of the \(z_j\)'s. There is exactly one sequence of nonzero choices producing the distinguished output. The diagonal coefficient is therefore one, not a factorial and not merely an unspecified nonzero scalar.
It follows in each total degree that the admissible operations (7) are linearly independent as actual operations, and that the coordinate monomials (11) are linearly independent in \(\mathcal S^\#\). This proof includes repeated polynomial factors, arbitrarily many variables, the degree-one Bockstein and all odd primes. Finite calculations are useful regression checks of the indexing and signs, but are not its general proof.
5. What the faithful test does and does not establish
Graded commutativity of \(H\) and odd characteristic imply \(\tau_j^2=0\). Consequently the coordinates define an algebra map
\[ k[\xi_1,\xi_2,\ldots]\otimes\Lambda(\tau_0,\tau_1,\ldots) \longrightarrow\mathcal S^\#. \tag{16} \]Section 4 proves it is injective, and Section 2 computes its coproduct. It does not, on its own, prove surjectivity or identify \(\mathcal S\) with every stable operation. In particular a successful finite pairing matrix is not authority to declare the full operation algebra computed.
Here is the precise sufficient additional input:
SB. In each degree, every stable mod-\(p\) cohomology operation is a linear combination of the admissible words (7).
Given SB, the actual full operation algebra equals \(\mathcal S\). Independence from Section 4 makes its dimension in each degree exactly the number of labels (8)-(10), which is also the dimension of the source of (16). The injective map (16) is then an isomorphism in every degree. Equations (5)-(6) give its root action and coproduct. This proves the entire operation input O in The odd-primary Thom module from SB, not from an unsupported count alone.
6. Completion and sources
The one-group calculation, Sections 8–9, proves SB for all spectrum operations. Thus (16) identifies the full dual operation algebra, with the coproduct and root action computed above. The argument does not infer spanning merely from independence.
René Thom, Travaux de Milnor sur le cobordisme, pp. 172–173, describes the dual-coordinate calculation. J. Peter May, A general algebraic approach to Steenrod operations, Section 10, states the one-group theorem.