The "Steenrod Tcl library" 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 of the author's PhD thesis

• Supports the primes $p=2,3,5,7,11$

• Written in C to maximize speed

• Comes with an interface to the Tcl programming language

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.

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.

Resolutions and chain maps are stored in an SQLite 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.

The Sage project was started by William Stein in 2005.

John Palmieri implemented support for Steenrod operations.

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)

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.

Yacop adds support for differential graded modules over the Steenrod algebra.

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 $$C_\ast \land M \quad \cong\quad C_\ast \otimes^L M$$

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

Yacop adds support for differential graded modules over the Steenrod algebra.

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 ${\mathbb R}P^\infty$:

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

Sage organizes its mathematics in "categories".

Yacop adds new categories and functors:

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]

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.

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

You can run Sage in multiple ways:

• download a binary package from the Sage download site

• 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.

Docker is the easiest way to get going:

~$ 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 is an interesting alternative on Linux. This is "rootless", hence less risky to allow for system administrators.

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

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!

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!

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

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.

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" to compute Ext of the localization $M\langle P_{3}^0\rangle$ = $\mathbb F_2\langle P_{3}^0\rangle$