# Finitistic dimension of an algebra

The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. (we look here only at finite dimensional modules).

It is an open problem wheter this dimension is always finite and perhaps the biggest open homological conjecture that says something about all finite dimensional algebras. See for example http://www.math.uni-bonn.de/people/schroer/fd-problems-files/FD-FinitisticDimConj.pdf .

I offer a 100 Euro prize (or Amazon gift card or something else worth that) together with 500 bounty points in case you show that the finitistic dimension is infinite for this algebra (or another algebra if you like), which is of course rather unrealistic.

I offer 300 bounty points if you can calculate the finitistic dimension in this example, which of course might be very easy and I just oversee something.

Let $r$ be a nonzero field element and $$A=A_r:=\langle x,y,z\rangle/(x^2,y^2,z^2,yx+rxy,zy+ryz,xz+rzx).$$ This is a 8-dimensional symmetric local algebra that can be viewed as a deformation of the exterior algebra of a 3-dimensional vector space. For nonzero field elements $c$ ,$M_c:=A/(x+c y)A$ is a 4-dimensional indecomposable module.

Let $c_1, ... , c_n$ be $n$ pairwise distinct field elements. Let $N:=A_r \oplus M_{c_1} \oplus ... \oplus M_{c_n}$ and $B:=\mathrm{End}_A(N)$.

Is the finitistic dimension of $B$ finite? If yes, can it be calculated in terms of the field parameters?

For the case $n=1$ the finitistic dimension should be equal to two. $n=2$ is the next interesting case.

One would show that the finitistic dimension is infinite in case one can find a module $W$ that has finite $\mathrm{add}(N)$-resolution dimension such that $\mathrm{Ext}^i(N,W)=0$ for all $i \geq 1$ if I made no mistake.

Background: If I made no mistake for field elements $r$ with $r^l \neq 1$ for all $l \geq 1$ this should be the first non-local algebras in the literature where the category of Gorenstein projectives does not coincide with the category of maximal Cohen-Macaulay modules and I think this is a good reason to test the homological conjectures on those algebras although this is the simplest class of examples I got from my construction.

Algebras with the category of Gorenstein projectives coinciding with the category of maximal Cohen-Macaulay modules or local algebras satisfy the strong Nakayama conjecture and thus also the generalized Nakayama conjecture.

My expectation is that the finitistic dimension is either trivial (for example always equal to two) to calculate or it is calculated by solving some complicated equatations in the field involving the parameters. The motivation for taking those modules $M=M_c$ is that they can satisfy that $Ext^i(M,M) \neq 0$ only finitely many often, but maybe it might be wise to add other modules to $N$ besides the $M_c$ in order to challenge the finitstic dimension conjecture...

• You probably mean $xz+rzx$ in the relations of the algebra? – Julian Kuelshammer Sep 1 '17 at 21:53
• It should perhaps be said, for those unfamiliar with the homological conjectures, that a conjecture from the 1960's states that the finitistic dimension is always finite for finite-dimensional algebras. Thus proving that the algebras in the post have infinite finitistic dimension would be rather spectacular. – Pierre-Guy Plamondon Sep 2 '17 at 11:50
• @Pierre-GuyPlamondon To be a little pedantic, I don't think that it's a conjecture from the 1960s. It's a question first published in 1960 (so probably actually from the late 1950s) that later morphed into a conjecture, apparently with no human intervention. – Jeremy Rickard Sep 2 '17 at 12:20
• @Mare, people do not get scared by providig context. – Mariano Suárez-Álvarez Sep 2 '17 at 17:06
• @MarianoSuárez-Álvarez Im just interested in finding the finitistic dimension of those algebra since they are somehow special as explained at the end of the text. – Mare Sep 2 '17 at 17:09