arXiv subjects

Mahdi Majidi-Zolbanin

Publications and source records attributed to Mahdi Majidi-Zolbanin.

5 recordsLinked to original sources

arXiv2018

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

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.

math.AC↗

arXiv2014

On additivity of local entropy under flat extensions

Let $f\colon(R,\mathfrak{m})\rightarrow S$ be a local homomorphism of Noetherian local rings. Consider two endomorphisms \textit{of finite length} (i.e., with zero-dimensional closed fibers) $φ\colon R\rightarrow R$ and $ψ\colon S\rightarrow S$, satisfying $ψ\circ f=f\circφ$. Then $ψ$ induces a finite length endomorphism $\overlineψ\colon S/f(\mathfrak{m})S\rightarrow S/f(\mathfrak{m})S$. When $f$ is flat, under the assumption that $S$ is Cohen-Macaulay we prove an additivity formula: $h_{\mathrm{loc}}(ψ)=h_{\mathrm{loc}}(φ)+h_{\mathrm{loc}}(\overlineψ)$ for \textit{local entropy}.

math.AC↗

arXiv2011

Kunz Regularity Criterion via algebraic entropy

In arXiv:1109.6438v1 [math.AG] we introduced and studied a notion of algebraic entropy. In this paper we will give an application of algebraic entropy in proving Kunz Regularity Criterion for all contracting self-maps of finite length of Noetherian local rings in arbitrary characteristic. Some conditions of Kunz Criterion have already been extended to the general case by Avramov, Iyengar and Miller in arXiv:math/0312412v2 [math.AC], using different methods.

math.AC↗

arXiv2011

Dynamics and entropy in local algebra

We introduce and study a notion of algebraic entropy for self-maps of finite length of Noetherian local rings, and develop its properties. We show that it shares the standard properties of topological entropy. For finite self-maps we explore the connection between the degree of the map and its algebraic entropy, when the ring is a Cohen-Macaulay domain. As an application of algebraic entropy, we give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity.

math.AG↗

arXiv2011

Lifting finite self maps of equicharacteristic complete local rings

In the spirit of Fakhruddin (arXiv:math/0212208v1) and Szpiro-Bhatnagar (arXiv:1010.2715v1), we show that for an equicharacteristic complete local ring A, with a given embedding of Spec(A) in the prime spectrum Spec(R) of some complete regular local ring R, any finite self map of A can be lifted to a finite self map of R, keeping the given embedding.

math.AC↗