explicit applications of the CpMackeyFunctors package -- info about how this package can be applied to real-life examples

This package can be used to perform Tor and Ext computations for $C_p$-Mackey functors. It also implements Tor and Ext in the abelian subcategory of cohomological $C_p$-Mackey functors. Computations of Tor and Ext groups for (cohomological) Mackey functors are important for work in equivariant homotopy theory. As a demonstration, we present some computations which appear in the work of Mingcong Zeng [Zeng18]. As setup, let $B_1$ be the $C_p$-Mackey functor with fixed module $\mathbb{Z}/p$ and underlying module $0$. Let $Z$ denote the fixed point Mackey functor for $\ZZ$ as a trivial $C_p$-module (called $\square$ on the list here).

Proposition: For any prime number $p$ we have \[ \mathrm{Ext}^i_{Z}(B_1,Z) = \begin{cases} B_1 & i=3 \\ 0. & \textrm{else.} \end{cases} \] Note that by a theorem of Arnold (see [Arn] and also [BSW]) the category of cohomological Mackey functors has global dimension $3$, so it suffices to compute the first four Ext Mackey functors. For concreteness, we will do the computation at the prime 11.

i1 : B = cokernel(matrix({{11}}));

i2 : B1 = makeZeroOnUnderlyingMackeyFunctor(11,B)

o2 =                    Res : 0
     cokernel | 11 |  -----------> 0  -┐ Conj : 0
                     <-----------     <┘
                        Tr : 0

o2 : CpMackeyFunctor

i3 : Z = makeFixedPointMackeyFunctor(11,id_(ZZ^1))

o3 =        Res : | 1 |
       1  ---------------->   1  -┐ Conj : | 1 |
     ZZ  <----------------  ZZ   <┘
            Tr : | 11 |

o3 : CpMackeyFunctor

i4 : for i to 3 do (print (prune ExtCoh(i,B1,Z)))
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
                   Res : 0
cokernel | 11 |  -----------> 0  -┐ Conj : 0
                <-----------     <┘
                   Tr : 0

We can also recover some Tor computations which appear in Zeng's work. Let $DZ$ denote the Mackey functor which is $\mathbb{Z}$ at fixed and underlying, transfer and conjugation are the identity, and restriction is multiplication by $p$.

Proposition: For any prime number $p$ we have \[ \mathrm{Tor}^{Z}_i(B_1,B_1) = \begin{cases} B_1 & i=0,3 \\ 0 & \textrm{else.} \end{cases} \] and \[ \mathrm{Tor}^{Z}_i(DZ,DZ) = \begin{cases} DZ & i=0 \\ B_1 & i=1 \\ 0 & \textrm{else.} \end{cases} \] Again, we will perform these computations at the prime $p=11$.

i5 : B = cokernel(matrix({{11}}));

i6 : B1 = makeZeroOnUnderlyingMackeyFunctor(11,B)

o6 =                    Res : 0
     cokernel | 11 |  -----------> 0  -┐ Conj : 0
                     <-----------     <┘
                        Tr : 0

o6 : CpMackeyFunctor

i7 : DZ = makeOrbitMackeyFunctor(11,id_(ZZ^1))

o7 =        Res : | 11 |
       1  ---------------->   1  -┐ Conj : | 1 |
     ZZ  <----------------  ZZ   <┘
            Tr : | 1 |

o7 : CpMackeyFunctor

i8 : for i to 3 do (print (prune TorCoh(i,B1,B1)))
                       Res : 0
cokernel | 11 0 0 |  -----------> 0  -┐ Conj : 0
                    <-----------     <┘
                       Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
                                             Res : 0
cokernel | 11 0 0 0 0 0 0 0 0 0 0 0 0 0 |  -----------> 0  -┐ Conj : 0
                                          <-----------     <┘
                                             Tr : 0

i9 : for i to 3 do (print (prune TorCoh(i,DZ,DZ)))
       Res : | 11 |
  1  ---------------->   1  -┐ Conj : | 1 |
ZZ  <----------------  ZZ   <┘
       Tr : | 1 |
                                                                     Res : 0
cokernel | 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  -----------> 0  -┐ Conj : 0
                                                                  <-----------     <┘
                                                                     Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0
     Res : 0
0  -----------> 0  -┐ Conj : 0
  <-----------     <┘
     Tr : 0

[Arn] J.E. Arnold, The permutation projective dimension of odd p-groups. Comm. Algebra 13, 387-397 (1985).
[BSW] S. Bouc, R. Stancu, P. Webb, On the projective dimensions of Mackey functors Algebr. Represent. Theory 20 (2017), no. 6, 1467-1481.
[Zen] M. Zeng, Mackey functors, equviariant Eilenberg-Mac Lane Spectra and their slices. PhD. Thesis

explicit applications of the CpMackeyFunctors package -- info about how this package can be applied to real-life examples

See also