| Russian Mathematical Surveys |
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Brief Communications Cohomological realization of the Buchstaber formal group law M. Bakuradzeab a A. Razmadze Mathematical Institute, Georgian Academy of Sciences b Iv. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences, Tbilisi, Georgia
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     In this paper we construct a commutative complex oriented cohomology theory whose coefficient ring is the coefficient ring of the Buchstaber formal group law with inverted 2. As shown in [11], after being localized away from 2, the forgetful map from the special unitary cobordism $\mathbf{MSU}_*[1/2]=\mathbb{Z}[1/2][x_2,x_3,\dots]$ to the complex cobordism $\mathbf{MU}_*[1/2]$ is an injection. We define the following ideal extensions in $\mathbf{MU}_*[1/2]$: the ideal $J_{{\rm SU}}^e$ generated by arbitrary polynomial generators $x_n$ of $\mathbf{MSU}_*[1/2]$, $n\geqslant 5$, viewed as elements of $\mathbf{MU}_*[1/2]$; the ideal $J_{T}^e$ generated by the ${\rm SU}$-flops [12] of dimension $\geqslant 10$, also viewed as elements of $\mathbf{MU}_*[1/2]$; and the extension $J_B^e$, where $J_B$ is the ideal in $\mathbf{MU}_*$ generated by the elements $\{A_{ij},i,j\geqslant 3\}$, defined in [3] and [2] as the coefficients of the formal power series $\sum A_{ij}x^iy^j=F(x,y)(x\omega(y)-y\omega(x))$, where $F(x,y)=\sum\alpha_{ij}x^iy^j$ is the universal formal group law and $\omega(x)= (\partial F(x,y)/\partial y)|_{y=0}$ is the invariant differential form of $F$. Proposition 1. (a) $J^e_B=J_{T}^e$; (b) when restricted to $\mathbf{MSU}_*[1/2]$, the classifying map of the Buchstaber formal group law localized away from 2 gives a genus with scalar ring $\mathbb{Z}[1/2][x_2,x_3,x_4]$, $|x_i|=2i$. One motivation for this result is the restricted Krichever–Höhn complex elliptic genus, studied in [7] and [12]. For another construction with the scalar ring $\mathbb{Z}_{(2)}[a,b]$, where $|a|=2$ and $|b|=6$, see [5]. The ideal $J_B$ is not prime as taking the quotient $f_B\colon\mathbf{MU}_*\to\mathbf{MU}_*/J_B=\Lambda_B$ by it does not yield an integral domain. But it contains only $2$-torsion elements of degree $2^k-2$, $k\geqslant 3$ [6]. Using the results of [6] on the structure of $\Lambda_B$, consider the quotient map $\Lambda_B/J=\Lambda_{\mathcal{B}}$, where $J$ is generated by the elements of order 2. The ideal $J$ is prime as $\Lambda_{\mathcal{B}}$ is an integral domain, and therefore so is $J_{\mathcal{B}}=f_B^{-1}(J)$ in $\mathbf{MU}_*$. Then $J_{\mathcal{B}}$ is the kernel of the composition $\mathbf{MU}_*\to \Lambda_B\to\Lambda_{\mathcal{B}}$. Proposition 2. There exists a multiplicative complex oriented cohomology $\mathbf{MU}^*_\Sigma$ with scalar ring the integral domain $\mathbf{MU}_*/\Sigma$, where $\Sigma=J_{\mathcal{B}}$. Theorem 3. There exists a commutative complex oriented cohomology which is a module over $\mathbf{MU}_*[1/2]$ and has the scalar ring $\mathbf{MU}_*[1/2]$ modulo $J^e_{\rm SU}$, the ideal generated by an arbitrary set of polynomial generators of $\mathbf{MSU}_*[1/2]$ of degree $\geqslant 10$ viewed as elements of $\mathbf{MU}_*[1/2]$ by the forgetful injection map. This quotient is identical to the scalar ring of the universal Buchstaber formal group law localized away from 2. We prove Proposition 2. (All the other proofs can be found in [4].) By Euclid’s algorithm, for any natural numbers $m_1,\dots, m_k$ one can find integers $\lambda_1,\dots, \lambda_k $ such that
$$
\begin{equation*}
\sum_{i=1}^{k}\lambda_im_i=\operatorname{gcd}(m_1,\dots,m_k).
\end{equation*}
\notag
$$
Set $d(m)=\operatorname{gcd}\bigl\{\binom{m+1}{1},\dots,\binom{m+1}{m-1}\bigr\}$, $m\geqslant 1$. By [8] one has $d(m)=p$ if $m+ 1= p^s$ for some prime $p$, and $d(m)=1$ otherwise. The elements $e_{m}=\sum_{i=1}^m \lambda_i\alpha_{im}$ are multiplicative generators in $\mathbf{MU}_*$: see [6]. By Theorem 9.9 in [6] or by [9], for
$$
\begin{equation*}
D(m)=\operatorname{gcd}\biggl\{\binom{m+1}{i}-\binom{m+1}{i-1}\biggr\},
\end{equation*}
\notag
$$
$2<i<m-1$, $m\geqslant 5$, we have that $D(m)/d(m)$ equals 2 if $m=2^k-2$; otherwise it equals $d(m-1)$. Let $m\geqslant 4$, and let $\lambda_2,\dots,\lambda_{m-2}$ be integers such that
$$
\begin{equation*}
d_2(m):=\sum_{i=2}^{m-2}\lambda_i\binom{m+1}{i}=\operatorname{gcd} \biggl\{\binom{m+1}{2},\dots,\binom{m+1}{m-2}\biggr\}.
\end{equation*}
\notag
$$
Then by Lemma 9.7 in [6], for $m\geqslant 3$ we have $d_2(m)=d(m)d(m-1)$. For the elements $A_{ij}$, $i,j\geqslant 3$, $i+j-2=m$, and the integers $\lambda_2,\dots,\lambda_{m-2}$ corresponding to $D(m)$ consider the linear combinations
$$
\begin{equation*}
T_{m}=\lambda_2 A_{3,m-1}+\lambda_3 A_{4,m-2}+\dots+\lambda_{m-2}A_{m-1,3}.
\end{equation*}
\notag
$$
By [6], Corollary 2.10, the $f_B(e_i)$ form a minimal set of multiplicative generators of $\Lambda_B$. Set $A_l=\mathbb{Z}[e_1,e_2,\dots,e_l]$ and $A^{l+1}=\mathbb{Z}[e_{l+1},e_{l+2},\dots]$, that is, $\mathbf{MU}_*=A_l\otimes A^{l+1}$. Let $J_{\mathcal{B}}(l)$ be the ideal in $\mathbf{MU}_*=\mathbb{Z}[e_1,e_2,\dots]$ generated by only those generators in $J_{\mathcal{B}}$ with degree $\geqslant-2l$. The preimage of $J_{\mathcal{B}}(l)$ under the obvious inclusion defines an ideal of $A_l$, denoted by the same symbol, so that $\mathbf{MU}_*/J_{\mathcal{B}}(l)=A_l/J_{\mathcal{B}}(l)\otimes A^{l+1}$. The proof of Proposition 6.5 in [6] implies that $A_l/J_{\mathcal{B}}(l)$ is an integral domain, and therefore so is $\mathbf{MU}_*/J_{\mathcal{B}}(l)$ and $J_{\mathcal{B}}(l)$ is prime. To prove this we need only to modify the generating monomials of $\Lambda_{\mathcal{B}}\otimes \mathbb{F}_p$ by replacing the extra factors for $A_l/J(l)$, $p^r\leqslant l<p^{r+1}$, namely, $\beta_{p^{r+1}}^{k_1}\beta_{p^{r+2}}^{k_2}\cdots \beta_{p^{r+s}}^{k_s}\cdots$ , $k_1,k_2,\dotsc \leqslant p-1$, by the factors $\beta_{p^r}^{pk_1+p^2k_2+\cdots+p^sk_s+\cdots}$. The rank of $\Lambda^{-2m}\otimes \mathbb{F}_p$ remains the same since there is no relation (6.2) from [6] in our ring. Then by Theorem 6.1 and Proposition 6.5 in [6] we have $J_{\mathcal{B}}=J_B+(e_{2^k-2})$, $k\geqslant 3$. Let $\Sigma=(\mathcal{T}_i)=(T_n, e_m)$, where $n\ne m=2^k-2$, $n\geqslant 5$ and $k\geqslant 3$. Then $\Sigma(l)=J_{\mathcal{B}}(l)$ for all integers $l\geqslant 5$. It is clear that $\Sigma(l)\subset J_{\mathcal{B}}(l)$. We prove that $ J_{\mathcal{B}}(l)\subset \Sigma(l)$ using induction on $l$. This is obvious for $l=5$ because $T_5=A_{34}$. To prove that $J_{\mathcal{B}}(l)=(J_{\mathcal{B}}(l-1),\mathcal{T}_l)$ note that $s_{i+j-2}(A_{ij})=\binom{i+j-1}{j-1}-\binom{i+j-1}{j}$. Indeed, modulo the decomposable elements, $A_{ij}=\alpha_{i-1j}-\alpha_{ij-1}$ and $s_{i+j-1}(\alpha_{ij})=-\binom{i+j}{i}$. Now apply Euclid’s algorithm to $m_i=s_l(A_{i,l+2-i})$, fix some integers $\lambda_i$, and consider the elements $T_l$. The above identities imply that $s_l(T_l)=D(l)$ is the greatest common divisor of the integers $s_l( A_{ij})$ for $A_{ij}\in J_B$, $i+j-2=l$. It follows that $A_{ij}=(s_n(A_{ij})/D(l))T_l+P(e_1,\dots,e_{l-1})$ for some polynomial $P$. Therefore, $P\in \ker f_{\mathcal{B}}$, that is, $P\in J_{\mathcal{B}}(l-1)=\Sigma(l-1)$. It also follows that the sequence $\Sigma=(\mathcal{T}_5,\dots)$ is regular. The construction in [1] of a cobordism with singularities gives a cohomology theory $\mathbf{MU}^*_{\Sigma}(-)$, which admits an associative multiplication by [10] (Theorems 4.3 and 4.5). Moreover, all obstructions to commutativity are in $\Lambda_\mathcal{B}\otimes \mathbb{F}_2$.
Citation in format AMSBIB
\Bibitem{Bak22} |
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