Suppose we are given finite dimensional semisimple $k$-algebras $A_1,..., A_r$. now we consider the matrix algebra$A=\begin{pmatrix} A_1 & M_{1,2} & \dots & M_{1,r} \\ 0 & A_2 & \dots & M_{2,r} \\ \vdots & 0 & \ddots & \vdots \\ 0 & \dots & 0 & A_r \end{pmatrix}$ with finitely generated bimodules $M_{i,j}$. Suppose $A=\mathrm{End}(\mathcal{T})$, for some $G$-equivariant tilting bundle on some smooth projective variety $X$ over a field $k$ of characteristic zero on which $G$ acts. One has $\mathrm{gldim}(A)$ is finite. Furthermoe suppose we are given a linearly reductive group $G$ acting on $X$. Do we have $\mathrm{gldim}(\mathrm{End}_G(\mathcal{T}))<\infty$?
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1$\begingroup$ What is the difference between this and your previous question? $\endgroup$– Dag Oskar MadsenJan 24, 2014 at 9:54
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1$\begingroup$ $G$ is an algebraic group and linearly reductive and not supposed to be finite. $\endgroup$– user45766Jan 24, 2014 at 11:16
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