arXiv subjects

A. Tonolo

Publications and source records attributed to A. Tonolo.

3 recordsLinked to original sources

arXiv2026

The injective envelope of simple modules over Leavitt path algebras I: simple left ideals

Let $K$ be any field and $E$ any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra $L_K(E)$. Then, for each simple %(i.e., minimal) left ideal $I$ of %the Leavitt path algebra $L_K(E)$, we explicitly construct the injective envelope of $I$. This result generalizes to all graphs $E$ and all simple left ideals in $L_K(E)$ the construction presented previously by the three authors for the specific case of the Jacobson algebra $R=K\langle X,Y | XY=1\rangle$ and the simple left $R$-ideal $R(1-YX)$. Our method involves defining an $L_K(E)$-module structure on a $K$-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple $L_K(E)$-modules arising from two types of infinite emitters in $E$.

math.RA↗

arXiv2026

The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles

Let $K$ be any field, $E$ any directed graph, and $L_K(E)$ the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle $c$ in $E$ one can build the simple left $L_K(E)$-module $V_c^E$, and then more generally $V_{p(x),c}^E$ (where $p(x)$ is an irreducible polynomial in $K[x,x^{-1}]$). A cycle $c$ is called {\it exclusive} in case none of the vertices of $c$ is the base of any cycle other than $c$. In our main result we provide an explicit description of the injective envelope of $V_c^E$, and then more generally of $V_{p(x),c}^E$, for each exclusive cycle $c$. Our method involves defining an $L_K(E)$-module structure on an appropriately-built $K$-vector space of infinite series. Our main result significantly generalizes previous work of the authors, in that the result holds for all graphs (finite or not), and all exclusive cycles.

math.RA↗

arXiv2009

Derived Equivalence induced by $n$-tilting modules

Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.

math.RA↗