Tootfinder

Neural category
Neha Gupta, Suhith K N
https://arxiv.org/abs/2403.17558 https://arxiv.org/pdf/2403.17558

Neural category
A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. Curto et al. \cite{curto2013neural} associated a ring $\mathcal{R}_{\mathcal{C}}$ (neural ring) to a neural code $\mathcal{C}$. A special class of ring homomorphisms between two neural rings, called neural ring homomorphism, was introduced by Curto and Youngs \cite{curto2020neural}. The main work in this paper comprises constructing two categories. First is the $\mathfrak{C}$ category, a subcategory of…

Relations between Poincar\'e series for quasi-complete intersection homomorphisms
Josh Pollitz, Liana M. Sega
https://arxiv.org/abs/2403.17079 https://…

Relations between Poincaré series for quasi-complete intersection homomorphisms
In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $φ\colon Q \to R$, where $Q$ is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series $\mathrm{P}^Q_M(t)$ of a finitely generated $R$-module $M$ to $\mathrm{P}^R_M(t)$ when the kernel of $φ$ is contained in $\mathfrak{m}\,\mathrm{ann}_Q(M)$. This generalizes a classical result of Shamash for complete intersection homomor…

Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring
Kyu-Hwan Lee, Se-jin Oh
https://arxiv.org/abs/2402.08140 https://

Quantum cluster algebra, braid moves and quantum virtual Grothendieck ring
In this paper, we study the quantum virtual Grothendieck ring, denoted by $\frakK_q(\g)$, which was introduced in [39], and further investigated in [26, 25]. Our approach involves examining this ring from two perspectives: first, by considering its connection to quantum cluster algebras of non-skew-symmetric types; and second, by exploring its relevance to categorification theory. We specifically focus on (i) the homomorphisms that arise from braid moves, particularly 4-moves and 6-moves, in th…

On Pro-zero homomorphisms and sequences in local (co-)homology
Peter Schenzel
https://arxiv.org/abs/2404.14018 https://arxiv.org/pdf/…

On Pro-zero homomorphisms and sequences in local (co-)homology
Let $\xx= x_1,\ldots,x_r$ denote a system of elements of a commutative ring $R$. For an $R$-module $M$ we investigate when $\xx$ is $M$-pro-regular resp. $M$-weakly pro-regular as generalizations of $M$-regular sequences. This is done in terms of Čech co-homology resp. homology, defined by $H^i(\check{C}_{\xx} \otimes_R \cdot)$ resp. by $H_i({\textrm{R}} \Hom_R(\check{C}_{\xx},\cdot)) \cong H_i(\Hom_R(\mathcal{L}_{\xx},\cdot))$, where $\check{C}_{\xx}$ denotes the Čech complex and $\mathcal{L…

On bi-amalgamated constructions
Federico Campanini, Carmelo Antonio Finocchiaro
https://arxiv.org/abs/2403.20224 https://arxiv.org/pd…

On bi-amalgamated constructions
Let $f:A\longrightarrow B, g:A\longrightarrow C$ be ring homomorphisms and let $\mathfrak{b}$ (resp., $\mathfrak{c}$) be an ideal of $B$ (resp., $C$) satisfying $f^{-1}(\mathfrak{b})=g^{-1}(\mathfrak{c})$. Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring $$A\bowtie^{f,g}(\mathfrak{b},\mathfrak{c}) :=\{(f(a)+b, g(a)+c)\mid a\in A, b\in\mathfrak{b}, c\in \mathfrak{c} \}$$ of $B\times C$, called the bi-amalgamation of $A$ with $(B,C)$ along $(\mathfrak{b}, \mathf…