On a $q$-Skew Amitsur's Theorem

Technical Summary: On a q-Skew Amitsur's Theorem

Problem Statement
The paper addresses the structure of the Jacobson radical, denoted $J(R[x; \sigma, \delta])$, in Ore extensions $R[x; \sigma, \delta]$. Specifically, it investigates whether the intersection of this radical with the base ring $R$, denoted $J(R[x; \sigma, \delta]) \cap R$, is a nil ideal (an ideal where every element is nilpotent). This inquiry is framed as a generalization of Amitsur's 1956 theorem for standard polynomial rings, where $J(R[x]) = (J(R[x]) \cap R)[x]$ and the intersection is a nil ideal.

The paper focuses on the case where the Ore extension is defined by an automorphism $\sigma$ and a $\sigma$-derivation $\delta$. The specific conditions under study are:

1. $R$ is an algebra over an uncountable field $k$.
2. $\sigma$ is a locally torsion automorphism (for every $a \in R$, $\sigma^n(a) = a$ for some $n$).
3. $\delta$ is a locally nilpotent $\sigma$-derivation (for every $a \in R$, $\delta^n(a) = 0$ for some $n$).
4. The derivation is $q$-skew, meaning $q\sigma\delta = \delta\sigma$ for some nonzero scalar $q \in k$.

This work partially answers an open question posed by Greenfeld, Smoktunowicz, and Ziembowski (2019) regarding whether $J(R[x; \sigma, \delta]) \cap R$ is nil under the assumptions of locally nilpotent $\delta$ and locally torsion $\sigma$.

Methodology
The proof strategy follows the approach established by Smoktunowicz and Ziembowski (2014) for the case of derivations ($\sigma=1$), adapted to the $q$-skew setting. The methodology relies on several key technical components:

• Quasi-invertibility in Power Series: The author utilizes the fact that if an element is in the Jacobson radical of the polynomial ring, it is quasi-invertible. By extending the multiplication to the Ore power series ring $R[[x; \sigma, \delta]]$ (which is well-defined due to the local nilpotence of $\delta$), the quasi-inverse can be expressed as a formal series $f = \sum (-ax^n)^i$.
• Degree Analysis and Coefficient Extraction: The proof analyzes the coefficients of powers of elements of the form $ax^n$. Using a generalized $q$-binomial expansion (Proposition 4) and iterative formulas for $(ax^n)^m$ (Lemma 5), the author derives bounds on the nilpotency degree of elements under the action of $\delta$ and $\sigma$ (Lemma 6).
• Linear Independence over Uncountable Fields: A critical step involves considering the quasi-inverses of $\lambda ax^n$ for varying scalars $\lambda \in k$. Since $k$ is uncountable, the author employs a Vandermonde matrix argument. By showing that the degree of the quasi-inverse is bounded for infinitely many $\lambda$, it is deduced that specific high-degree coefficients in the expansion must vanish.
• Local Torsion Property: The vanishing of these coefficients leads to a relation of the form $a\sigma^n(a)\sigma^{2n}(a)\cdots\sigma^{(d-1)n}(a) = 0$. The locally torsion nature of $\sigma$ is then used to select an $n$ that is a multiple of the order of $a$, forcing the product to become a power of $a$ (specifically $a^d$), thereby proving nilpotence.

Key Results
The paper establishes the following main theorem and corollary:

• Theorem 1: Let $R$ be a $k$-algebra over an uncountable field $k$, $\sigma$ a locally torsion automorphism, and $\delta$ a locally nilpotent, $q$-skew $\sigma$-derivation. Then the set $I = \{a \in R : ax^t \in J(R[x; \sigma, \delta])\}$ is a nil ideal of $R$ for any $t \in \mathbb{N}$.

  • Significance: This confirms that for any element $a$ in the intersection $J(R[x; \sigma, \delta]) \cap R$, $a$ is nilpotent. This provides a partial affirmative answer to Question 6.17 in [GSZ19].

• Corollary 10 (q-Skew Amitsur's Theorem): If the field $k$ is additionally assumed to be of characteristic zero, then the full Amitsur property holds:
  $$J(R[x; \sigma, \delta]) = (J(R[x; \sigma, \delta]) \cap R)[x; \sigma, \delta]$$
  and $J(R[x; \sigma, \delta]) \cap R$ is a nil ideal of $R$.

  • Derivation: This result combines Theorem 1 with a 2024 result by Shin, which characterizes the Jacobson radical in characteristic zero for $q$-skew extensions.

Significance and Scope
The paper claims to provide a partial resolution to the question of whether the Jacobson radical of an Ore extension intersects the base ring in a nil ideal, specifically under the conditions of an uncountable field, locally torsion automorphism, and $q$-skew locally nilpotent derivation.

The author explicitly notes that the result is "partial" because it relies on the field being uncountable and, for the full Amitsur property (Corollary 10), on the field having characteristic zero. The paper does not claim to solve the general case for arbitrary fields or for cases where the $q$-skew condition ($q\sigma\delta = \delta\sigma$) does not hold.

Open Questions and Future Directions
The paper concludes by highlighting remaining open problems, maintaining a modest stance on the current scope of the results:

1. Whether $J(R[x; \sigma, \delta]) \cap R$ is always the nilradical (the largest nil ideal) in the context of Corollary 10.
2. Whether Theorem 1 holds if the $q$-skew condition is removed ($q\sigma\delta \neq \delta\sigma$).
3. Whether Corollary 10 holds if $\sigma$ is only locally finite rather than locally torsion.
4. Whether $J(R[x; \delta]) \cap R$ is nil for algebras over fields of characteristic zero when $\sigma=1$ (the standard derivation case).

The author conjectures that these questions may be answered affirmatively but notes that they remain open.