A-identities for the {2 \times 2} matrix algebra

Let K be an algebraically closed field of characteristic zero. We prove that the minimum degree for A-identities of \({M_{2}(K)}\) is 6. We also construct an explicit A-polynomial of degree 6 and prove it is an A-identity.     

Logarithmic Picard groups,chip firing,and the combinatorial rank

Mathematische Zeitschrift - Illusie has suggested that one should think of the classifying group of $$M_X^{gp}$$ -torsors on a logarithmically smooth curve X over a standard logarithmic point as a...