Amitsur's Conjecture: Algebra, Geometry & PI Theory

• Amitsur's Conjecture is a statement linking the birational geometry of Severi–Brauer varieties with cyclic Brauer classes of central simple algebras.
• Recent advances employ explicit geometric birational maps, Galois descent, and primary decomposition to resolve cases for cyclic algebras and indices that are not prime powers.
• In the context of PI theory, methods like cocharacter analysis and combinatorial bounds confirm the exponential growth of codimensions, establishing integer PI-exponents.

Amitsur's Conjecture is a fundamental statement at the intersection of the theory of central simple algebras, algebraic geometry, and the asymptotics of polynomial identities. It captures a rigidity phenomenon in both the birational geometry of Severi–Brauer varieties and the exponential growth rate of codimensions of polynomial identities for algebras possessing additional symmetries or structure.

1. Classical Formulation and Severi–Brauer Geometry

Let $F$ be a field and $A$ a central simple $F$-algebra of degree $n$. The Severi–Brauer variety $\SB(A)$ is an $F$-projective variety, with the property that $\SB(A) \otimes_F K \cong \mathbb{P}^{n-1}_K$ for any splitting field $K$ of $A$. The set of isomorphism classes of such algebras is classified by the Brauer group $\Br(F) \cong H^2(\operatorname{Gal}(F^{\operatorname{sep}}/F), (F^{\operatorname{sep}})^*)$.

Amitsur's conjecture predicts that for central simple $F$-algebras $A$ and $B$ of the same degree, the Severi–Brauer varieties $\SB(A)$ and $\SB(B)$ are $F$-birational if and only if the classes $[A]$ and $[B]$ generate the same cyclic subgroup of $\Br(F)$. Amitsur established the "birational implies same cyclic subgroup" direction, reducing the main challenge to constructing rational maps between Severi–Brauer varieties with equivalent cyclic Brauer classes (Ramachandran, 6 Dec 2025, Kollár, 30 May 2025).

2. Status, Explicit Results, and Recent Advances

Historically, the conjecture has been proved in many particular cases:

• Cyclic Algebras of Prime Degree: Roquette (Ramachandran, 6 Dec 2025) gave an algebraic proof using function field isomorphisms. Recent work has provided explicit geometric birational maps, utilizing Galois descent and explicit monomial coordinate transformations that respect the semilinear Galois cocycle data.
• Non-prime-power Index: The case when the index $\operatorname{ind}(A)$ is not a prime power was independently settled using a birational product formula, reducing the problem inductively to Severi–Brauer varieties attached to algebras of coprime degrees (Kollár, 30 May 2025). The geometric content here relies on primary decomposition in $\Br(F)$ and properties of Weil restrictions and Segre-type correspondences.

The table summarizes key results:

Algebra $A$	Main Case	Status	Key Reference
Cyclic, prime degree	Galois/cocycle model	Proved (explicit birational maps)	(Ramachandran, 6 Dec 2025)
Index not a prime power	Birational product	Proved (using coprime factorization)	(Kollár, 30 May 2025)
General (prime power index)	Unresolved	Open except for specific cases	(Ramachandran, 6 Dec 2025, Kollár, 30 May 2025)

3. Polynomial Identity Version: Growth of Codimension Sequences

In the context of PI theory, the conjecture takes the following form: for a finite-dimensional algebra $A$ (associative or Lie, possibly with additional structure such as group or Hopf action), the codimensions $c_n(A)$ of multilinear polynomial identities should exhibit exponential growth with a well-defined integer exponent, called the PI-exponent:

$$\exp(A) = \lim_{n\to\infty} \sqrt[n]{c_n(A)} \in \mathbb{Z}_{>0}.$$

Giambruno and Zaicev established the conjecture for ordinary associative PI-algebras. Major generalizations include:

• Generalized Hopf Actions: For associative or Lie algebras $A$ equipped with an $H$-action (where $H$ is a finite-dimensional semisimple Hopf algebra), the Hopf PI-exponent exists and equals a naturally defined structural invariant (Gordienko, 2012, Gordienko, 2012, Gordienko, 2012).
• Group Actions and Gradings: For $A$ with a finite (not necessarily Abelian) group action (by automorphisms and anti-automorphisms), or for $G$-graded $A$, the $G$-codimension exponent and graded PI-exponent coincide with the maximal dimension of the semisimple simple factors (or appropriate block) (Gordienko et al., 2012, Gordienko, 2011).

4. Methodology: Representation Theoretic and Combinatorial Tools

The verification of Amitsur's conjecture in the PI context employs a blend of:

• Structural Decomposition: Use of invariant Wedderburn–Mal'cev or Levi decomposition allowing reduction to simple or semisimple blocks, possibly equipped with additional symmetry (e.g., Hopf action, group grading).
• Cocharacter Theory: Analysis of the $S_n$-module structure of the multilinear component, decomposing the space into irreducible $S_n$-modules indexed by partitions $\lambda \vdash n$ whose Young diagrams are controlled by the dimensions of simple blocks or invariants.
• Upper and Lower Bounds via Alternating Polynomials: Construction of explicit non-identities alternating in large blocks, securing a lower bound, and hook-length/combinatorial bounds for the upper bound. The multiplicity of large rectangular diagrams provides both the exponential growth rate and the sharpness of the exponent (Gordienko, 2012, Gordienko, 2011, Gordienko, 2013, Gordienko, 2017).
• Product Structure and Induction: For birational Severi–Brauer questions, coprime decomposition and Segre-type embedding allow reduction to products of varieties associated to smaller algebras (Kollár, 30 May 2025).

5. Extensions: Hopf, Graded, and Differential Identities

The analog of Amitsur's conjecture holds in much broader settings, formalized as follows:

• Hopf Module Algebras: For an $H$-module algebra (associative or Lie, finite-dimensional, char $0$), with $H$ finite-dimensional semisimple, the $H$-codimension sequence satisfies

$$C_1\,n^{r_1}\,d^n \leq c_n^H(A) \leq C_2\,n^{r_2}\,d^n,$$

with integer exponent $d$ equal to the maximal dimension of an $H$-simple component or analogous invariant (Gordienko, 2012, Gordienko, 2012, Gordienko, 2017).

• Graded Algebras and Group Actions: For $A$ graded by an arbitrary group or with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms, the graded and $G$-codimensions, as well as the exponents, are integer and coincide with the ordinary PI-exponent (Gordienko et al., 2012).
• Representations: For representations $\rho: L \to \mathfrak{gl}(V)$ of Lie algebras, the sequence of codimensions $c_n(\rho)$ also exhibits exponential growth with integer exponent given by suitable invariants involving the image algebra and chains of irreducible modules (Gordienko, 2011).

6. Special Cases and Examples

Taft Algebra Actions on Lie Algebras

Let $H_{m^2}(\zeta)$ denote the $m$-th Taft algebra. For finite-dimensional $H_{m^2}(\zeta)$-module Lie algebras $L$ which are $H$-simple, the $H$-codimensions of polynomial $H$-identities satisfy

$$\lim_{n \to \infty} \sqrt[n]{c_n^{H_{m^2}(\zeta)}(L)} = \dim L,$$

even if $L$ is not semisimple in the ordinary sense. This demonstrates the robustness of the exponential rate even when radical structures are allowed, provided the Hopf–action simplicity constraints are imposed (Gordienko, 2017).

Sweedler's Hopf Algebra and Hopf PI-Exponents

For algebras (associative or Lie) simple with respect to an action of Sweedler's 4-dimensional Hopf algebra $H_4$, the analog of Amitsur’s conjecture holds with $PIexp^{H_4}(A) = \dim A$ (Gordienko, 2013, Gordienko, 2017).

7. Open Problems and Ongoing Directions

The remaining principal open case for the birational Amitsur conjecture is the situation where the index is a prime power $p^e > 1$ and the algebra is not cyclic. While recent geometric and algebraic constructions have extended the scope substantially (Kollár, 30 May 2025, Ramachandran, 6 Dec 2025), explicit birational maps for composite index with complex splitting field structure and nonclassical types remain a central challenge.

For Hopf–module algebra and PI-exponent settings, the conjecture is settled for broad classes (semisimple Hopf, group actions, derivations), but infinite-dimensional or nonsemisimple Hopf algebra actions may present further complexity.

References to Major Results

• (Ramachandran, 6 Dec 2025) "A geometric perspective on Amitsur's conjecture"
• (Kollár, 30 May 2025) "Birational equivalence of Severi-Brauer varieties"
• (Gordienko, 2012) "Amitsur's conjecture for associative algebras with a generalized Hopf action"
• (Gordienko, 2012) "Amitsur's conjecture for polynomial H-identities of H-module Lie algebras"
• (Gordienko, 2012) "Asymptotics of H-identities for associative algebras with an H-invariant radical"
• (Gordienko et al., 2012) "Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities"
• (Gordienko, 2011) "Codimensions of polynomial identities of representations of Lie algebras"
• (Gordienko, 2011) "Graded polynomial identities, group actions, and exponential growth of Lie algebras"
• (Gordienko, 2013) "Algebras simple with respect to a Sweedler's algebra action"
• (Gordienko, 2017) "Lie algebras simple with respect to a Taft algebra action"