Global dimension of tiled orders over a discrete valuation ring

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Let R be a discrete valuation ring with maximal ideal m \mathfrak {m} and the quotient field K. Let Λ = ( m λ i j ) ⊆ M n ( K ) \Lambda = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K) be a tiled R-order, where λ i j ∈ Z {\lambda _{ij}} \in {\mathbf {Z}} and λ i i = 0 {\lambda _{ii}} = 0 for 1 ≤ i ≤ n 1 \leq i \leq n . The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in M n ( K ) {M_n}(K) of finite global dimension. Theorem 2. Tiled R-orders in M n ( K ) {M_n}(K) of finite global dimension satisfy DCC. Theorem 3. Let Λ ⊆ M n ( R ) \Lambda \subseteq {M_n}(R) and let Γ \Gamma be obtained from Λ \Lambda by replacing the entries above the main diagonal by arbitrary entries from R. If Γ \Gamma is a ring and if gl dim Λ > ∞ \dim \;\Lambda > \infty , then gl dim Γ > ∞ \dim \;\Gamma > \infty . Theorem 4. Let Λ \Lambda be a tiled R-order in M 4 ( K ) {M_4}(K) . Then gl dim Λ > ∞ \dim \;\Lambda > \infty if and only if Λ \Lambda is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order Γ = ( m λ i j ) ⊆ M 4 ( R ) \Gamma = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R) , where γ i i = γ 1 i = 0 {\gamma _{ii}} = {\gamma _{1i}} = 0 for all i, and γ i j = 1 {\gamma _{ij}} = 1 otherwise.