Equivariant formal group laws and complex-oriented spectra over primary cyclic groups: elliptic curves, Barsotti–Tate groups, and other examples

Abstract

We explicitly construct and investigate a number of examples of ${\mathbb {Z}}/p^r$-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.

Appendix

In this section, we clarify the relationship between the isotropy separation diagram (27), which we used to construct equivariant elliptic and Barsotti–Tate cohomology, with the calculation of $(MU_{{\mathbb {Z}}/p^r})_*$ given in [1]. In [1], there is a “staircase-shaped diagram”

(64)

However, it should be pointed out that a general ${\mathbb {Z}}/p^r$-equivariant spectrum X is not the homotopy limit of the diagram (64). Rather, it is a homotopy limit of a larger “cube-like” diagram of which (64) is a part.

What happens for cobordism is that this homotopy limit has no higher derived functors, and therefore to calculate the coefficients of ${\mathbb {Z}}/p^r$-equivariant cobordism, we can just take the limit of the coefficients of the spectra in the cube-like diagram. This limit is then easily seen to be equal to the limit of the coefficients of the spectra (64) for $X=MU_{{\mathbb {Z}}/p^r}$, i.e.

(65)

In this case, the coeffients of the top term of (64) are

$$\begin{aligned} MU_*[u_\alpha ,u_{\alpha }^{-1},b^\alpha _j\mid j\in {\mathbb {N}}, 1\le \alpha <p^{i}] [[u_{p^i}]] /[p^{r-i}]u_{p^i}. \end{aligned}$$

(66)

while the coefficients of the bottom term are

$$\begin{aligned} {(MU_*[u_\alpha ,u_{\alpha }^{-1}, b_i^{\alpha }\mid 1\le \alpha \le p^{i-1}-1][[u_{p^{i-1}}]] [u_{p^{i-1}}^{-1}]/ [p^{r-i+1}]u_{p^{i-1}})^\wedge _{([p]u_{p^{i-1}})}.}\nonumber \\ \end{aligned}$$

(67)

(The completion symbol is actually missing in [1].)

For any ${\mathbb {Z}}/p^r$-equivariant spectrum X, there is an obvious comparison map from diagram (27) to diagram (64). In the case of $X=MU_{{\mathbb {Z}}/p^r}$, on coefficients, the map is injective. It is proved in [18] that for connectivity reasons, this map on coefficients is an isomorphism for $i\le 1$. It is also an isomorphism on (66) for $i=r$ (i.e., the top term (65)). Thus, for $r\le 2$, the comparison map from Diagram (27) to Diagram (64) is an equivalence on fixed points.

As it turns out, in all other cases, i.e. on (66) with $2\le i<r$ and on (67) with $2\le i\le r$, the comparison map fails to be an isomorphism. The reason is that in those cases, (66), (67) actually contain infinite series in $u_{p^i}$ with unbounded $u_{p^{i-1}}$-denominators, which can be dimensionally compensated by positive powers of $u_{p^j}$ with $j<i-1$. For $i\le 1$, there are no such compensating terms. A detailed proof of this is given in [18].

Therefore, in particular, while on coefficients, the diagram (27) has no higher derived limits, there are, in fact, non-zero higher derived limits of the diagram (65) for $r>2$, although its limit is still $(MU_{{\mathbb {Z}}/p^r})_*$. From this point of view, the diagram (27) is “minimal” on coefficients, and therefore it is a simpler model of isotropy separation for ${\mathbb {Z}}/p^r$ than the diagram of [1].