The Golod–Shafarevich inequality for finite-dimensional algebras

Let A be a finite-dimensional algebra over a field k and I a proper two-sided ideal of A, generated as a left ideal by x₁, …, x_d with d ≥ 1. Suppose that every relation ∑ aᵢ xᵢ = 0 is an A-linear combination of r vectors y₁, …, y_r ∈ A^d whose entries all lie in I. Then

d² < 4 r.

For a finite p-group G this applies to A = 𝔽_p[G], its augmentation ideal, a minimal generating tuple gᵢ of G with xᵢ = gᵢ - 1, and the relators of a minimal presentation of G, and gives the Golod–Shafarevich inequality d(G)² < 4 r(G).

The proof follows Roquette. Write P m for the ideal power I ^ m, so that P 0 = A and P (m + 1) = P m * I, and c m = dim_k (A ⧸ P m). Left multiplication of the xᵢ by a tuple of elements of P m has image P (m + 1), and a combination of the yⱼ with coefficients in P m has all its entries in P (m + 1). Counting dimensions for the induced linear maps

(A ⧸ P m)^r → (A ⧸ P (m + 1))^d → P 1 ⧸ P (m + 2),

with the kernel of the right-hand map contained in the range of the left-hand map, gives c (m + 2) + r c m ≥ c 1 + d c (m + 1) ≥ 1 + d c (m + 1). The sequence c is bounded by dim_k A, so TauCeti.sq_lt_four_mul_of_forall_add_mul_le forces d² < 4 r.

Main results

• TauCeti.card_sq_lt_four_mul_card_of_relations_le_span: the Golod–Shafarevich inequality d² < 4 r for a finite-dimensional algebra with a proper two-sided ideal generated by d ≥ 1 elements whose relations lie in the span of r vectors with entries in the ideal.

References

• E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964).
• P. Roquette, On class field towers, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter IX, §4.
• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.9.7).

theorem TauCeti.card_sq_lt_four_mul_card_of_relations_le_span {A : Type u_1} [Ring A] {ι : Type u_2} {κ : Type u_3} [Fintype ι] [Fintype κ] (k : Type u_4) [Field k] [Algebra k A] [FiniteDimensional k A] [Nonempty ι] (I : Ideal A) [I.IsTwoSided] (hI : I ≠ ⊤) (x : ι → A) (hx : Submodule.span A (Set.range x) = I) (y : κ → ι → A) (hy : ∀ (j : κ) (i : ι), y j i ∈ I) (hker : ∀ (a : ι → A), ∑ i : ι, a i * x i = 0 → a ∈ Submodule.span A (Set.range y)) : Fintype.card ι ^ 2 < 4 * Fintype.card κ

The Golod–Shafarevich inequality for finite-dimensional algebras. Let A be a finite-dimensional algebra over a field k and I a proper two-sided ideal of A, generated as a left ideal by a nonempty finite family x. If every relation ∑ aᵢ xᵢ = 0 among the generators is an A-linear combination of a finite family y of vectors in A^ι with all entries in I, then (#x)² < 4 · #y.