Combinatorial Algebra: Syntax and Semantics by Mark V. Sapir

Author:Mark V. Sapir
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

For every i = 1, 2, … choose a natural number n i . Let I be the ideal generated by . The question is how to make a choice of n i in order to obtain an infinite dimensional quotient algebra F∕I.

To solve this problem we first of all will make it a bit harder. Recall that every polynomial p in F is a sum of homogeneous components h 1 + … + h m where for every i all monomials in h i have the same length i (which is the degree of h i ).

Let | X |  = d ≥ 2. Then there exist exactly d n words over X of length n. Thus for every n the space F n of homogeneous polynomials of degree n from F has dimension d n (it is spanned by the words of length n in the alphabet X).

If an ideal I is generated by a set R of homogeneous polynomials, then it is spanned by homogeneous polynomials. Indeed by the definition I is spanned by X ∗ RX ∗. Each polynomial in X ∗ RX ∗ is homogeneous. Therefore in this case I = (I ∩ F 1) ⊕ (I ∩ F 2) ⊕ …. Thus I has a nice decomposition into a sum of finite dimensional subspaces (a subspace of a finite dimensional space is finite dimensional). This shows that ideals generated by homogeneous polynomials are easier to study than arbitrary ideals.

Unfortunately the ideal I generated by won’t be generated by homogeneous polynomials. Hence we shall make a sacrifice. Let us generate I not by but by all homogeneous components of . For example if p i  = x + y 2 and we choose n i  = 2 then p i 2 = x 2 + y 4 + xy 2 + yx 2, the homogeneous components will be x 2 (degree 2), xy 2 + y 2 x (degree 3), y 4 (degree 4). We will put all these components into I. It is clear that the ideal generated by these components is bigger than the ideal generated by , so the quotient algebra will be a nil-algebra also.

Let R be the subspace spanned by the homogeneous components of be the ideal generated by R, X′ be the subspace spanned by X.

Now let us introduce the key concept of the proof of Golod’s theorem.

With every subspace S of F spanned by homogeneous polynomials we associate the following Hilbert series:

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