Steenrod algebra cohomology with Yacop/Sage

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#Steenrod algebra cohomology with Yacop/Sage

eCHT seminar on machine computation in homotopy theory

Christian Nassau

Oct 28 2021
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# [Sage](https://www.sagemath.org)
### "Creating a viable free open source alternative to Magma, Maple, Mathematica, and Matlab"
- General purpose computer algebra system based on Python.
- Interfaces to many specialized computer algebra systems.
- Covers a lot of mathematics.

# [Yacop](https://github.com/cnassau/yacop-sage)
### ".underline[Y]et .underline[a]nother .underline[co]homology .underline[p]rogram"
- Specialized library for Steenrod algebra cohomology.
- Intended to be used through Sage.
- Meant to be powerful and easy to use.

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## Sage .padded[+] Yacop glue code .padded[+] .yellow[Steenrod Tcl library]
.left-column[
### .emph[Overview]
### Capabilities
### GFR only
### Storage
]
.right-column[
The "[Steenrod Tcl library](https://github.com/cnassau/steenrod)" is the computational work horse of the project.

Mostly developed around 2004 and very stable.

- Optimized code to compute minimal resolutions of subalgebras of the Steenrod algebra

- Uses the [algorithm](https://doi.org/10.4418/2021.76.1.1) of the author's [PhD thesis](https://dx.doi.org/10.30819/881)

- Supports the primes $p=2,3,5,7,11$

- Written in C to maximize speed

- Comes with an interface to the [Tcl](https://www.tcl.tk/) programming language
]
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## Sage .padded[+] Yacop glue code .padded[+] .yellow[Steenrod Tcl library]
.left-column[
### Overview
### .emph[Capabilities]
### GFR only
### Storage
]
.right-column[
The library is designed to do just a few things efficiently:

- compute the matrix of (parts of) a map $$d\_s:C\_s \rightarrow C\_{s-1}$$ in a free resolution.

- compute kernels, images and lifts of such matrices

These are the steps that are needed to compute a minimal resolution and
also to work with it (e.g. solve lifting problems to compute chain maps).

Currently everything is computed on the CPU.

Planned: delegate the computation of the $d_s$ to the graphics processing unit
using [OpenCL](https://www.khronos.org/opencl/).
]

---
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## Sage .padded[+] Yacop glue code .padded[+] .yellow[Steenrod Tcl library]
.left-column[
### Overview
### Capabilities
### .emph[GFR only]
### Storage
]
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Our philosophy is to compute everything from **one fixed minimal resolution**
$C\_\ast$ of the ground field:

- for other modules $M$ one has a functorial free resolution $C\_\ast \land M$

- for small $M$ and $N$ this resolution can be used to compute
$\operatorname{Ext}\_A(M,N)$ from the resolution
$$C\_\ast \land M \land N^\ast$$

We don't claim that this is the best approach for all $M$: for modules that are close to free it clearly isn't.

]

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## Sage .padded[+] Yacop glue code .padded[+] .yellow[Steenrod Tcl library]
.left-column[
### Overview
### Capabilities
### GFR only
### .emph[Storage]
]
.right-column[
Resolutions and chain maps are stored in an [SQLite](https://www.sqlite.org/index.html) database.

- SQLite is a very robust and widely used database engine

- A database is a single ordinary file

- Can be accessed through the SQL "structured query language"

- Support for SQLite exists in all popular programming languages.
]

---
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## Sage .padded[+] .yellow[Yacop glue code] .padded[+] Steenrod Tcl library
.left-column[
### .emph[Steenrod algebra]
### Modules
### Categories
### Resolutions
]
.right-column[
The Sage project was started by William Stein in 2005.

John Palmieri implemented support for Steenrod operations.

```python
sage: A=SteenrodAlgebra(2,basis='adem')
sage: A
mod 2 Steenrod algebra, serre-cartan basis
sage: A(Sq(3,2,1))
Sq^11 Sq^4 Sq^1 + Sq^11 Sq^5 + Sq^13 Sq^3
sage: B=SteenrodAlgebra(2,basis='milnor')
sage: A.monomial((4,2,6,4))
Sq^4 Sq^2 Sq^6 Sq^4
sage: B(A.monomial((4,2,6,4)))
Sq(3,2,1) + Sq(7,3)
```
]

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## Sage .padded[+] .yellow[Yacop glue code] .padded[+] Steenrod Tcl library
.left-column[
### Steenrod algebra
### .emph[Modules]
### Categories
### Resolutions
]
.right-column[
Yacop adds support for differential graded modules over the Steenrod algebra.

Sage has a notion of "graded objects" but it's not quite suitable:

- we need **three** gradings $(s,t,e)$: homological, internal, Bockstein

- we want to **operate** on the gradings via suspensions & truncations

- we want **introspection** capabilities: for example, a module/chain complex
should know its bounding box.
]

Sage has a notion of "module over an algebra" but we prefer something more tailor-made:

Yacop modules can define their action through both
$$(a,m) \mapsto a \cdot m$$
and/or
$$(a,m) \mapsto \chi(a) \cdot m$$

An optimized computation of $\chi(a)\cdot m$ is desirable for the identification
$$C\_\ast \land M \quad \cong\quad C\_\ast \otimes^L M$$

$$ag \land m \mapsto \sum a'g \otimes \chi(a'')m$$
]

Sage has a notion of "module over an algebra" but we prefer something more tailor-made:

Yacop modules can define their action through both
$$(a,m) \mapsto a \cdot m$$
and/or
$$(a,m) \mapsto \chi(a) \cdot m$$

Example: projective space ${\mathbb R}P^\infty$:

$$\operatorname{Sq}(R) x^m = \binom{m}{r_1\,\cdots\,r_n} x^{m+\vert R\vert},
\quad
\chi\left(\operatorname{Sq}(R)\right) x^m = \binom{-1-m-\vert R\vert}{r_1\,\cdots\,r_n} x^{m+\vert R\vert}$$
]

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## Sage .padded[+] .yellow[Yacop glue code] .padded[+] Steenrod Tcl library
.left-column[
### Steenrod algebra
### Modules
### .emph[Categories]
### Resolutions
]
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Sage organizes its mathematics in "categories".

Yacop adds new categories and functors:

```python
sage: from yacop.modules.projective_spaces import
    ComplexProjectiveSpace
sage: ComplexProjectiveSpace().categories()
[Category of yacop left module algebras over              # Yacop
    mod 2 Steenrod algebra, milnor basis,
 Category of algebras with basis over Finite              # Sage
    Field of size 2,
 ...
 Category of magmas,                                      # Sage
 Category of yacop left modules over mod 2 Steenrod       # Yacop
    algebra, milnor basis,
 Category of yacop differential modules over mod 2        # Yacop
    Steenrod algebra, milnor basis,
 Category of vector spaces with basis over Finite         # Sage
    Field of size 2,
 ...
 Category of yacop graded objects,
 Category of yacop objects,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]
```
]

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## Sage .padded[+] .yellow[Yacop glue code] .padded[+] Steenrod Tcl library
.left-column[
### Steenrod algebra
### Modules
### Categories
### .emph[Resolutions]
]
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Yacop provides classes for ground field resolutions
$C\_\ast$ and smash product resolutions $C\_\ast\land M$.

It communicates with the Tcl Steenrod library through Python's "tkinter" package.

```python
sage: A=SteenrodAlgebra(5) ; A                                 # Python
mod 5 Steenrod algebra, milnor basis                           # Python
sage: C=A.resolution() ; C                                     # Python
minimal resolution of mod 5 Steenrod algebra, milnor basis     # Python
sage: C.compute(smax=10,nmax=100)                              # Python
> Working on C1 up to ideg=101, edeg=1                         # Tcl
> Last completed internal degree was 44                        # Tcl
...                                                            # Tcl
> Reading generators for C10                                   # Tcl
> 1 generators read                                            # Tcl
> Beginning with the computation                               # Tcl
  > Reading d9                                                 # Tcl
  > Reading d10                                                # Tcl
  > 10/88/8 found 1 new generator in 0.00 seconds              # Tcl
  > 10/89/9 found 1 new generator in 0.00 seconds              # Tcl
sage: C.differential(C.g(5,66))                      # Python -> Tcl -> SQLite ->
2*Q_0 P(1)*g(4,57) + Q_1*g(4,57) + P(3)*g(4,42)      #    -> Tcl -> Python
```
]

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## .yellow[Sage] .padded[+] Yacop glue code .padded[+] Steenrod Tcl library
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### .emph[Running Sage]
### Docker
]
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You can run Sage in multiple ways:

- download a binary package from the [Sage download site](https://www.sagemath.org/download.html)

- download source and build yourself

- use your Linux package manager (e.g. on Ubuntu)

- use docker to run Sage without local installation

Yacop can be combined with all these methods.

]

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## .yellow[Sage] .padded[+] Yacop glue code .padded[+] Steenrod Tcl library
.left-column[
### Running Sage
### .emph[Docker]
]
.right-column[
[Docker](https://bit.ly/3asb6st)
is the easiest way to get going:

```bash
~$ docker run -e SAGE_BANNER=NO --rm -it cnassau/yacop-sage
Unable to find image 'cnassau/yacop-sage:latest' locally
latest: Pulling from cnassau/yacop-sage
297061f60c36: Pull complete
...
496abbf4c991: Pull complete
Digest: sha256:34f791d....e72ca886c5caf77
Status: Downloaded newer image for cnassau/yacop-sage:latest
sage: C=SteenrodAlgebra(2).resolution()
sage: C.differential(C.g(5,36))
Sq(1)*g(4,35) + Sq(14,1)*g(4,19) + Sq(32)*g(4,4)
sage:
```

Docker is supported on all popular OS.
For Windows you might need the "professional edition".

[Podman](https://podman.io/whatis.html) is an interesting alternative on Linux.
This is "rootless", hence less risky to allow for system administrators.

]

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# Intermission

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## Omissions, plans, hopes
.left-column[
### .emph[Missing Features]
### $\psi$-map
- Background
- Problem
- Ansatz
- Applications
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Missing features that should be easy to add

- chain maps $m\_g:C\_\ast \rightarrow C\_{\ast+s}$ corresponding to a
$g\in\operatorname{Ext}_A^s(k,k)$

- comparison maps $C\_\ast \rightarrow D\_\ast$ for algebra inclusions $A\subset B$

Possibly more involved

- support for motivic Ext computations over ${\mathbb C}$ and ${\mathbb R}$

- use the secondary Steenrod algebra of H. J. Baues to compute $d\_2$ differentials

Also highly desirable

- improve robustness of the code

- documentation & tutorials

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## Omissions, plans, hopes
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### Missing Features
### .emph[$\psi$-map]
- .emph[Background]
- Problem
- Ansatz
- Applications
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Q: Let $A(n)\subset A$ be the usual subalgebra.
Can you put a compatible $A$-module structure on $A(n)$, i.e. is there
$$A\_n\in A-Mod \quad \text{such that}\quad A\_n\cong\_{A(n)} A(n) ?$$

A: Yes!

| $n$ |    |
| --- | -- |
| $-1$, $0$ | problem is trivial |
| $1$ | $4$ easy solutions |
| $2$ | $1600$ solutions (M. Roth) |
| $\ge 3$ | at least one (highly symmetric) $A_n$ (Mitchell/Smith) |

]

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## Omissions, plans, hopes
.left-column[
### Missing Features
### .emph[$\psi$-map]
- Background
- .emph[Problem]
- Ansatz
- Applications
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Q: Let $D\_\ast$ be the minimal free $A(n)$-resolution.

Can you put a compatible $A$-module structure on $D\_\ast$, i.e. is there
$$E\_\ast\in \partial A-Mod \quad \text{such that}\quad E\_\ast\cong\_{A(n)} D\_\ast ?$$

A: open problem!

| $n$ |    |
| --- | -- |
| $-1$, $0$ | problem is trivial |
| $1$ | nice excercise |
| $\ge 2$ | seems difficult |

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## Omissions, plans, hopes
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### Missing Features
### .emph[$\psi$-map]
- Background
- Problem
- .emph[Ansatz]
- Applications
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The $A\_n$ can be written as $A\_n = A\cdot p\subset A\_\ast$ with $p\in A\_\ast$.

Example: $A\_2 = A\cdot p$ with $p = \xi\_{1}^{7}\xi\_{2}^{3}\xi\_{3} $ + $ \xi\_{1}^{2}\xi\_{2}^{7} $ + $ \xi\_{1}^{4}\xi\_{2}^{4}\xi\_{3} $ + $ \xi\_{1}^{5}\xi\_{2}^{6} $ + $ \xi\_{1}^{8}\xi\_{2}^{5} $ + $ \xi\_{1}^{10}\xi\_{2}^{2}\xi\_{3} $ + $ \xi\_{1}^{11}\xi\_{2}^{4} $ + $ \xi\_{1}^{13}\xi\_{2}\xi\_{3} $ + $ \xi\_{1}^{14}\xi\_{2}^{3} $ + $ \xi\_{1}^{17}\xi\_{2}^{2} $ + $ \xi\_{1}^{20}\xi\_{2}
$ + $ \xi\_{1}^{23}$

Imagine an algorithm that

1. takes an approximate $p = \xi\_{1}^{7}\xi\_{2}^{3}\xi\_{3} + \cdots$

2. decomposes all $Sq(8r,4s) p = \sum a\_{k,l}^{r,s} \left(\xi\_1^{8k}\xi\_2^{4l}\cdot p\right)$
with $a\_{k,l}^{r,s}\in A(2)$.

3. magically turns the error terms $a_{k,l}^{r,s}$ for $(k,l)\not=(0,0)$ into correction terms

4. adds correction terms to $p$ and goes back to 2. until errors are zero

]

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## Omissions, plans, hopes
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### Missing Features
### .emph[$\psi$-map]
- Background
- Problem
- .emph[Ansatz]
- Applications
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This algorithm could work for a resolution as well:

- $p$ gets replaced by a chain map
$$\psi: A\otimes\_{A(2)} D\_\ast \rightarrow A\_\ast\otimes\_{A(2)} D\_\ast$$
with $\psi(g) = \xi\_{1}^{7}\xi\_{2}^{3}\xi\_{3}\otimes g + \cdots$
for $A(2)$-generators $g$ of $D\_\ast$

- the image of the corrected $\psi$ would be $E\_\ast$

Remarks:

- can replace $A\_\ast\otimes\_{A(2)} D\_\ast$ by $A^\infty \otimes\_{A(2)} D\_\ast$
with $A^\infty = {\mathbb F}_2[\xi\_1,\xi\_2]\otimes E(\xi\_3)$

- for $n=1$ every $\psi: A\otimes\_{A(1)} D\_\ast \rightarrow A^\infty\otimes\_{A(1)} D\_\ast$ seems to work.

]

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## Omissions, plans, hopes
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### Missing Features
### .emph[$\psi$-map]
- Background
- Problem
- Ansatz
- .emph[Applications]
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### Application 1

Compute $A\_2$-filtration of $\operatorname{Ext}\_A({\mathbb F}\_2)$ using that $E\_\ast$ is small(ish).

This is a complement to the tmf-filtration.

### Application 2

Hope that the $A$ action on $E\_\ast$ is asymptotically compatible with $v\_2$ periodicity.
Let $K\_n$ be the kernel of $d:E\_n \rightarrow E\_{n-1}$
and compute the limit

$$M = \operatorname{lim}\limits\_{k \rightarrow\infty} \Sigma^{-48k} K\_{8k}$$

This $M$ would be $A(1)$-free and have the same $P\_3^0$-localization as ${\mathbb F}_2$

Finally, use Palmieri's "[Margolis Adams spectral sequence](https://doi.org/10.1090/conm/158/01459)"
to compute Ext of
the localization $M\langle P\_{3}^0\rangle$ = $\mathbb F\_2\langle P\_{3}^0\rangle$

]

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# The End

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