Entropy in the category of perfect complexes with cohomology of finite length

Abstract

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen-Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.

On the categorical entropy of the Frobenius pushforward functor

“…In this section, we give in terms of local entropies a lower bound on the categorical entropy h t (φ) of the pushforward φ * : D b (R) → D b (R) along a finite local endomorphism φ of a local ring R. The following lemma plays a key role in showing the main theorem in this section, ideas of whose proof are taken from [9,Lemma 2.1]. Denote by K(−) the Koszul complex.…”

“…One is called the Frobenius pushforward F * on the bounded derived category D b (R) of finitely generated R-modules and the other is called the Frobenius pullback LF * on the derived category D perf (R) of perfect R-complexes. As to the latter, Majidi-Zolbanin and Miasnikov [9] considered the full subcategory D perf fl (R) of D perf (R) consisting of perfect complexes with finite length cohomologies, and computed the categorical entropy h D perf fl (R) t (LF * ). The aim of this paper is to study the Frobenius pushforward F * on D b (R) and compute its categorical entropy.…”

“…Then one has the equalities ℓ R (R/(F m ) e * (m)R) = Θ(m de ) and h loc (F m ) = d log m. In this paper, we are mainly concerned with the Frobenius functor. Let (R, m, k) be an F -finite local ring with characteristic p. Then the Frobenius endomorphism F : R → R induces the Frobenius pushforwardF * = RF * : D b (R) − −−−−−− → D b (R) ⊆ ⊆ D b fl (R) − −−−−−− → D b fl (R), and the Frobenius pullback (see[9, Proposition 1.10])LF * : D perf (R) − −−−−−− → D perf (R) ⊆ ⊆ D perf fl (R) − −−−−−− → D perf fl (R).Here, D b fl (R), D perf fl (R) stand for the subcategories of D b (R), D perf (R) consisting of complexes with finite length homologies, respectively. The categorical entropy of the Frobenius pullback on D perf fl (R) is computed by Majidi-Zolbanin and Miasnikov: Theorem 2.19 ([9, Corollary 2.6]).…”

On the categorical entropy of the Frobenius pushforward functor

“…One is called the Frobenius pushforward 𝐹 * on the bounded derived category 𝖣 𝖻 (𝑅) of finitely generated 𝑅-modules and the other is called the Frobenius pullback 𝕃𝐹 * on the derived category 𝖣 𝗉𝖾𝗋𝖿 (𝑅) of perfect 𝑅-complexes. As to the latter, Majidi-Zolbanin and Miasnikov [11] considered the full subcategory 𝖣…”

“…One is called the Frobenius pushforward $$F_*$$ on the bounded derived category $${\mathsf {D^b}}(R)$$ of finitely generated $$R$$‐modules and the other is called the Frobenius pullback $$\mathbb {L}F^*$$ on the derived category $${\mathsf {D^{perf}}}(R)$$ of perfect $$R$$‐complexes. As to the latter, Majidi‐Zolbanin and Miasnikov [11] considered the full subcategory $${\mathsf {D^{perf}_{fl}}}(R)$$ of $${\mathsf {D^{perf}}}(R)$$ consisting of perfect complexes with finite length cohomologies, and computed the categorical entropy $$\mathbf {h}_t^{{\mathsf {D^{perf}_{fl}}}(R)}(\mathbb {L}F^*)$$.…”