+++ title = “The Module Builder” section = “features” +++

The Module Builder

The Module Builder lets you define new modules over the Steenrod algebra and compute their Adams spectral sequences.

Defining a module

A module definition specifies:

• Prime — the prime $p$ (currently 2, 3, 5, or 7)
• Type — the type of module (e.g., “finite dimensional module”)
• Generators — a list of generators with their degrees
• Actions — the Steenrod algebra actions on the generators ($\operatorname{Sq}^i$ for $p = 2$, or $P^i$ and $\beta$ for odd primes)

The JSON format

Module definitions are JSON objects. See Module Definition Format for the complete specification. A minimal example for $p = 2$:

{
  "type": "finite dimensional module",
  "p": 2,
  "gens": { "x0": 0, "x1": 1 },
  "actions": ["Sq1 x0 = x1"]
}

Cell complexes and cofibers

You can build modules as cell complexes — attaching cells to an existing module. See Cell Complexes and Cofibers for the full workflow.

The “Take Cofiber” workflow:

1. In the viewer, select a class at filtration 1 on the $E_\infty$ page.
2. Click “Take Cofiber” to open the builder with a pre-filled module definition.
3. The builder auto-checks the Steenrod operations suggested by the attaching map.
4. Submit to compute the cofiber’s spectral sequence.

The resulting module tracks its provenance — the parent module and attaching data — which enables the deduction engine to compute naturality maps automatically.

Computation range

When you submit a module definition, htpy resolves it to compute Ext groups. The default computation range is $n \leq 30$, $s \leq 7$, but you can adjust this when recomputing from the viewer.

Limitations

• The module builder is currently available for the Adams spectral sequence only. Motivic Adams, CESS, and algebraic Novikov modules must be provided as JSON definitions directly.
• At most $p = 7$ is supported (limited by the bit-packing in the linear algebra layer).