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Corollary 30 ([Idempotent stratifications]). For every field extension K/k and every integer L≥1 , there is an auxiliary finite-dimensional indecomposable symmetric K -algebra T_L admitting an idempotent stratification of length at least L …

Corollary 30 ([Idempotent stratifications]). For every field extension K/k and every integer L≥1 , there is an auxiliary finite-dimensional indecomposable symmetric K -algebra TL admitting an idempotent stratification of length at least L in the sense of Chen, Fang and Xi. In particular, TL has a nonzero proper stratifying ideal, and D(TL-Mod) admits a chain of at least L nontrivial standard recollements through successive corner algebras. Here Mod denotes all left modules and D is the unbounded derived category.

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证明
Corollary 29 ([Infinitely many tilting complements]). For every field extension…Proposition 28. Let B be a finite-dimensional symmetric algebra over a field…观点
The base-change argument above gives a symmetric K -algebra A_K and a nonprojective self-orthogonal module M_K . Choose a nonprojective indecomposable summand N in one block B of A_K . The block B is indecomposable and symmetric, and _B^i(N,N)=0 for every i>0 , since these groups are direct summands of _ A_K ^i(M_K,M_K) . Chen, Fang and Xi~ construct from such a pair, over any field, reduced mirror-reflective algebras T_n that are indecomposable and symmetric. Their Corollary~5.12(1), with Definition~4.7, gives an idempotent stratification of length at least n . Theorem~1.2(2) supplies the stated successive nontrivial standard recollements of unbounded derived categories. Taking n=L proves the claim.
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