ИСТИНА

The proof of local finite basis property and local representability of relatively free algebras was done, according the Kemer programm. In order to make the proof more accepteble, following question arises. What does it provide for the community? Why should people read it? Joint work of speaker with Louis Rowen and Uzi Vishne discovered relations with non-commutative algebraic geometry and provide new insights for representaion theory. Consider representation $\rho$ of $k$-algebra $A$ to matrix algebra over algebraic close field $K$, wich is an affine space. Then Zarissky closer of $\rho(A)$ satysfy the same identities and its natural to investigate representations up to Zarissky clousure wich usually not a linea span if $k$ is finite. $\rho(A)$ decomposes into sum of prime components and Pierce components of radical. First canonization theorem says that it can be reduced to upper block-triangular case. Blocks are either {\it glued} (may be up to {\it Frobenious twist}) or independant. Second canonization theorem provides information for quiver or pseudoquiver. Third canonisation theorem says about quiver transformation under factorizing by representable $T$-ideal. And Fourth (projective) canonization theorem provide existence of non-identities of some canonical structure inside non-zero $T$-ideal and Phoenix property (``hiking'') saying that any element of $T$-ideal generated by hiked polynomial can be restituted to the same form. Because one can provide via substitutions on hiked polynomial structure of Notherean module, Specht properties and local representability follows.