The Frobenius exponent of Cartier subalgebras

Abstract

Let R be a standard graded finitely generated algebra over an F -finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that that Frobenius complexity of R is finite. Moreover, we extend this result to Cartier subalgebras of R.

Complexity of gauge bounded Cartier algebras

“…Alternatively, one can also avoid this by arguing as in [KZ14, Remark 2.3] to obtain the bound t(dim R + 1) via [EP19, Theorem 3.9]. (b) The strategy to apply Theorem 2.2 above is to use [KZ14, Lemma 3.2, Theorem 3.3] as outlined in [EP19]. Unfortunately, the proof of [KZ14, Lemma 3.2] contains a gap.…”

Symbolic analytic spread: upper bounds and applications

“…We remark that here we are using the alternative definition of Frobenius complexity as in [11,Section 4], which was later adopted in [26] and [9]. If R is F-finite and complete this definition coincides with the original one introduced by Enescu and Yao in [10].…”

“…In [9], the authors show that the Frobenius complexity is finite for standard graded rings over an F-finite field localized at the irrelevant maximal ideal. However, in general cx F (R) is not known to be finite, except when the anticanonical cover is Noetherian [10, 4.7], when dim R 2 (in this case cx F (R) 0) [10, 4.10], and in other particular cases [11,26].…”

Symbolic Analytic Spread: Upper Bounds and Applications

“…We remark that here we are using the alternative definition of Frobenius complexity as in [13,Section 4], which was later adopted in [11,28]. If R is F-finite and complete, this definition coincides with the original one introduced by Enescu and Yao in [12].…”

“…In [11], the authors show that the Frobenius complexity is finite for standard graded rings over an F-finite field localized at the irrelevant maximal ideal. However, in general, cx F (R) is not known to be finite, except when the anticanonical cover is Noetherian [12, 4.7], when dim R 2 (in this case, cx F (R) 0) [12, 4.10], and in other particular cases [13,28].…”