6
$\begingroup$

Let $G_n$ be the elementary abelian 2-group with $2^n$ elements and $R=R_n:=KG$ the group algebra over the field with 2 elements.

Let $M_n$ be the direct sum of all non-projective modules of the form $uR$ for some element $u \in R$ up to isomorphism. Let $B_n:=End_{R_n}(M_n)$.

Question: For which $n$ does $B_n$ have finite global dimension?

Interestingly, $B_n$ has Cartan determinant equal to -1 for $n=3$ and has finite global dimension for $n=1$ and $n=2$. So in case $B_3$ has finite global dimension, this would give a counterexample to the Cartan determinant conjecture. Sadly $B_3$ has vector space dimension 813 and thus even with a computer it looks impossible to understand the algebra $B_3$ directly, but maybe there is a trick?

$\endgroup$
4
$\begingroup$

In my answer to does-this-algebra-have-finite-global-dimension-human-vs-computer I recalled how one can go about computing a projective resolution of the simple modules of an endomorphism ring of a module. There is a missing piece of code in QPA to do this in general, but now I have made an attempt to rectify this. The code is not made public yet, but hopefully soon it will. Using this new code it seems that for one simple $B_3$-module, the third and the fourth syzygy of this particular simple $B_3$-module has a common isomorphic direct summand. Hence the projective dimension of this particular simple $B_3$-module is infinite and consequently the global dimension of $B_3$ is also infinite. I am stressing that the calculations only indicating this, I don't have a proof of this.

Addition April 17, 2020: Here is a copy of the GAP-session computing the above (using the newly uploaded additions to QPA as of April 17th, 2020):

gap> G := ElementaryAbelianGroup( 8 );
<pc group of size 8 with 3 generators>
gap> A := GroupRing( GF( 2 ), G );
<algebra-with-one over GF(2), with 3 generators>
gap> B := AlgebraAsQuiverAlgebra( A )[ 1 ];
<GF(2)[<quiver with 1 vertices and 3 arrows>]/<two-sided ideal in <GF(2)[<quiver with 1 vertices and 3 arrows>]>, (6 generators)>>
gap> elements := Elements( RadicalOfAlgebra( B ) );;
gap> mods := List( elements, m -> RightAlgebraModule( B, \*, RightIdeal( B, [ m ] ) ) );;
gap> Mods := List( mods, m -> RightAlgebraModuleToPathAlgebraMatModule( m ) );
[ <[ 0 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 1 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]> ]
gap> n := Length( Mods );
128
gap> 
gap> non_isos := [ ];
[  ]
gap> isos := [ ];
[  ]
gap> multiplicities := [ ];
[  ]
gap> for i in [ 2..n - 1 ] do 
>     if not i in isos then
>         Add( non_isos, i );
>         testset := [ i + 1..n ];
>         SubtractSet( testset, isos ); 
>         num := 0;
>         for j in testset do 
>             if IsomorphicModules( Mods[ i ], Mods[ j ] ) then 
>                 num := num + 1;
>                 Add( isos, j );
>             fi;
>         od;
>         Add( multiplicities, num + 1 );
>     fi;
> od;
gap> 
gap> basiclist := List( non_isos, i -> Mods[ i ] );
[ <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 4 ]>, <[ 2 ]>, <[ 4 ]>, <[ 4 ]>, 
  <[ 2 ]>, <[ 2 ]>, <[ 4 ]>, <[ 2 ]>, <[ 2 ]>, <[ 2 ]>, <[ 1 ]> ]
gap> Length( basiclist );
22
gap> test := ProjectiveResolutionOfSimpleModuleOverEndo( basiclist, 1, 4 );
[ "projdim > 4", [ <[ 3 ]>, <[ 10 ]> ] ]
gap> U := test[ 2 ][ 1 ];
<[ 3 ]>
gap> V := test[ 2 ][2];
<[ 10 ]>
gap> decomp := DecomposeModule( V );
[ <[ 3 ]>, <[ 7 ]> ]
gap> IsomorphicModules( U, decomp[ 1 ] );
true
$\endgroup$

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