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**Introduction to non commutative algebraic geometry.**
*(English)*
Zbl 1377.14005

I cite the abstract from the paper: “Ordinary commutative algebraic geometry is based on polynomial algebras over an algebraically closed field \(k\). Here we make a natural generalization to matrix polynomial k-algebras which are non-commutative coordinate rings of non-commutative varieties”.

Note: The paper is 3 pages long.

In the first section of the paper the author defines the notion of an algebraic variety, the sheaf of regular functions on an algebraic variety and morphisms of algebraic varieties. In the second section of the paper titled “Local categories” he speaks of the Yoneda lemma, representable functors and pro-representability of functors. In the third section of the paper named “Algebraic Varieties Revisited (defined by local theory)” the author talks about fine moduli spaces and pro-representability. In the fourth section of the paper named “Non-commutative affine algebraic geometry” the author defines the notion of a “matrix polynomial algebra” and the “non-commutative deformation functor”. Then he gives some examples but gives no proofs. The reviewer wants to encourage the author to write a more detailed paper on the subject where he includes all proofs and all definitions – a “self-contained” paper. If the paper is supposed to be an introduction to non-experts it should contain elementary and explicit examples. Send it to a journal where the refereeing of the paper is more thorough.

Note: The paper is 3 pages long.

In the first section of the paper the author defines the notion of an algebraic variety, the sheaf of regular functions on an algebraic variety and morphisms of algebraic varieties. In the second section of the paper titled “Local categories” he speaks of the Yoneda lemma, representable functors and pro-representability of functors. In the third section of the paper named “Algebraic Varieties Revisited (defined by local theory)” the author talks about fine moduli spaces and pro-representability. In the fourth section of the paper named “Non-commutative affine algebraic geometry” the author defines the notion of a “matrix polynomial algebra” and the “non-commutative deformation functor”. Then he gives some examples but gives no proofs. The reviewer wants to encourage the author to write a more detailed paper on the subject where he includes all proofs and all definitions – a “self-contained” paper. If the paper is supposed to be an introduction to non-experts it should contain elementary and explicit examples. Send it to a journal where the refereeing of the paper is more thorough.

Reviewer: Helge Øystein Maakestad (Bergen)