# Gorenstein projective modules of a certain triangular matrix algebra

Let $$B$$ be a finite dimensional selfinjective algebra over a field $$k$$ with a finite dimensional non-projective $$B$$-module $$M$$ and $$A=\pmatrix{k&M\\0&B}.$$

A module $$N$$ over an algebra $$C$$ is called Gorenstein projective in case $$Ext_C^i(N,C)=0=Ext_C^i(D(C),\tau(N))$$ for all $$i >0$$.

Questions:

1. Is there a general description of Gorenstein projective $$A$$-modules depending on $$B$$ and $$M$$?

2. Can $$A$$ have only finitely many indecomposable Gorenstein projectives when $$M$$ satisfies $$Ext_B^j(M,M)=0$$ for all $$j >t$$ for some $$t$$?

Together with (the answer in) Question on Ext for finite dimensional algebras a positive answer to 2. would give a negative answer to the first question in chapter 8 of https://arxiv.org/pdf/1808.01809.pdf.

There is some literature on triangular matrix algebras and their Gorenstein projective modules, but it seems that the assumptions are always too strong to apply here.

A (right) $$A$$-module consists of a pair $$\pmatrix{V&X}$$, where $$V$$ is a vector space and $$X$$ a $$B$$-module, together with a $$B$$-module map $$V\otimes_kM\to X$$.
Another criterion for $$\pmatrix{V&X}$$ to be Gorenstein projective is that it has a complete projective resolution: i.e., an acyclic complex $$P^\bullet:=\dots\longrightarrow P^{-1}\longrightarrow P^0\stackrel{d^0}{\longrightarrow}P^1\longrightarrow\dots$$ of projective $$A$$ modules with $$\text{im}(d^0)\cong\pmatrix{V&X}$$, such that $$\text{Hom}_A(P^\bullet,A)$$ is also acyclic.
Up to removing projective summands of $$\pmatrix{V&X}$$ we can assume that $$P^\bullet$$ is minimal (i.e., contains no contractible summands), which means that it cannot involve the projective module $$\pmatrix{k&M}$$, and so $$V=0$$ and $$P^\bullet$$ is just a complete projective $$B$$-module resolution of some $$B$$-module $$X$$.
In order that $$\text{Hom}_A(P^\bullet,A)$$ is acyclic, it is necessary and sufficient that $$\widehat{\text{Ext}}^i_B(X,M)=0$$ for all $$i\in\mathbb{Z}$$.
So the non-projective indecomposable Gorenstein projective $$A$$-modules are those of the form $$\pmatrix{0&X}$$, where $$X$$ is an indecomposable non-projective $$B$$-module such that $$\widehat{\text{Ext}}^i_B(X,M)=0$$ for all $$i\in\mathbb{Z}$$.
• Thanks. One question: Is $\hat{Ext_B}^0(X,M))=\underline{Hom_B}(X,M)$? Ill try my luck a bit to look for question 2 with $B=K<x,y>/(x^2,y^2,xy-qyx)$ since there such modules $M$ exists when q is not a root of unity and maybe in some cases there can be only finitely many such X, at least it looks like a very strong condition that this ext vanishes for all $i \in \mathbb{Z}$. – Mare Nov 10 '18 at 12:17
• @Mare Yes, $\widehat{\text{Ext}}^0$ is stable $\text{Hom}$. – Jeremy Rickard Nov 10 '18 at 12:22