When the Schur functor induces a triangle-equivalence between Gorenstein defect categories · arXivDesk
2003.06782Mar 15, 202018 pages. The necessary and sufficient conditions are given for the Schur functor to induce a triangle-equivalence of Gorenstein defect categories
When the Schur functor induces a triangle-equivalence between Gorenstein defect categories
Let R be an Artin algebra and e an idempotent of R. Assume that TorieRe(Re,G)=0
Nearby in the stack
for any
G∈GProjeRe
and
i
sufficiently large. Necessary and sufficient conditions are given for the Schur functor
Se
to induce a triangle-equivalence
Ddef(R)≃Ddef(eRe)
. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence
Dsg(R)≃Dsg(eRe)
to restrict to a triangle-equivalence
GProjR≃GProjeRe
. Applying these to the triangular matrix algebra
T=(AM0B)
, corresponding results between candidate categories of
T
and
A
(resp.
B
) are obtained. As a consequence, we infer Gorensteinness and CM-freeness of
T
from those of
A
(resp.
B
). Some concrete examples are given to indicate one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algabras.