arXiv subjects

Shantanu Sardar

Publications and source records attributed to Shantanu Sardar.

8 recordsLinked to original sources

arXiv2026

Hammock coordinates for silting subcategories of derived-discrete algebras

We introduce hammock coordinates to study the combinatorial structure of partial silting configurations in the bounded derived category of a derived-discrete algebra $Λ(r,n,m)$. By extracting exact integer inequalities from the Bobiński--Geiß--Skowroński Hom-hammock formula, we establish a closed-form criterion for pairwise partial silting both within any $\mathcal Z$-component and across distinct components. Using this coordinate framework, we establish three primary results. First, we prove the universal upper bound $|S| \le n+m$ on the cardinality of any partial silting configuration $S$ of $\mathcal Z$-objects ---both globally via Bongartz completion in the silting-discrete setting, and combinatorially within a single component via a $Σ^r$-invariant staircase argument. Second, we prove that this bound is universally sharp by constructing an explicit L-shaped partial silting configuration of size $n+m$ for every $Λ(r,n,m)$, alongside specialized diagonal configurations. Third, we show that the configuration lattice carries a faithful action of the cyclic group $\mathbb Z/g$ where $g = \gcd(r+m, n-r)$. We conclude by showing a structural obstruction to local determination: pairwise partial silting of consecutive objects in the lattice ordering does not imply global partial silting, due to the strict additivity of a coordinate defect along chains.

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arXiv2026

Syzygy-Quiver Formulas and Finitistic Dimensions of Special Biserial Algebras

Let $Λ$ be a finite dimensional algebra over a field $\mathbf{k}$. We introduce the reachable syzygy quiver of $Λ$, a directed multigraph whose vertices are the isomorphism classes of indecomposable nonprojective direct summands of iterated syzygies of simple modules, and whose arrows record nonprojective summands of first syzygies. The projective dimension of a vertex is one plus its directed path height, allowing the global and finitistic dimensions of $Λ$ to be read off from this height function. This formula holds for every finite dimensional algebra, but its states are arbitrary modules. We show that when the first syzygy of an indecomposable module admits a closed combinatorial description from the bound quiver, this state space reduces to a smaller combinatorial quiver, making the global and finitistic dimensions exact computable invariants. We carry this out in three stages. Over a special biserial algebra, the Liu--Morin closure theorem identifies the reachable syzygy quiver with a string-syzygy quiver whose vertices are string modules. Over a string algebra, this compresses to the relation-chain quiver of Xin and Zhang, recovering their global dimension formula along with a parallel formula for the finitistic dimension. Over an arbitrary biserial algebra, the universal formula holds, but uniform compression to paths or strings is impossible, with the Skowroński--Waschbüsch algebra providing a witness. Finally, we show two string algebras may share the same Gabriel quiver but have distinct global dimensions, proving the ideal of relations is indispensable for these invariants, and we prove that a pointwise homological reduction to a string algebra on the same quiver preserving the monomials of the relations is impossible.

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arXiv2026

Super-decomposable pure-injective modules over some Jacobian algebras

Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations over an algebra. Such an existence occurs when an algebra is non-domestic; a conjecture due to M. Prest. G. Puniski confirms the conjecture for non-domestic string algebras. Geiß, Labardini-Fragoso and Schröer show that every Jacobian algebra associated with a triangulation of a closed surface with marked points is finite-dimensional and tame. We show that, excluding only the case of a sphere with four (or fewer) punctures, there exists a special family of pointed modules, called an independent pair of dense chains of pointed modules. In the process, we show the existence of such an independent pair in a non-domestic skew-gentle algebra and (skew) Brauer graph algebras by showing that the Galois semi-covering functor and trivial extension preserve such pairs. Then it follows from a result of M. Ziegler that there exists a superdecomposable pure-injective module if the algebraically closed field is countable.

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arXiv2026

Krull-Gabriel dimension of Skew group algebras

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional algebra A and AG is the associated skew group algebra. The author with S. Trepode and A. G. Chaio introduced the notion of a Galois semi-covering functor to study the irreducible morphisms over skew group algebras. In this paper, we establish a Galois semi-covering functor between the morphism categories as well as the functor categories over the algebras A and AG and prove that their Krull-Gabriel dimension are equal. This computation confirms Prests conjecture on the finiteness of Krull-Gabriel dimension and Schroers conjecture on its connection with the stable rank (the least stabilized radical power) over skew gentle algebras. Moreover, we determine all posible stable ranks for (skew) Brauer graph algebras.

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arXiv2025

On the Irreducible Morphisms for Skew group algebras

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of covering that we call a Galois semi-covering functor, which becomes a Galois covering when the group action is free. We study the module category of its skew group algebra under this functor. As an application, we obtain a complete description of the irreducible morphisms and almost split sequences of skew group algebras and show that the (stable) rank is preserved under skewness. In particular, we determine the stable rank of skew-gentle algebras.

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arXiv2022

On the computation of order types of hammocks for domestic string algebras

For the representation-theoretic study of domestic string algebras, Schröer introduced a version of hammocks that are bounded discrete linear orders. He introduced a finite combinatorial gadget called the bridge quiver, which we modified in the prequel of this paper to get a variation called the arch bridge quiver. Here we use it as a tool to provide an algorithm to compute the order type of an arbitrary closed interval in such hammocks. Moreover, we characterize the class of order types of these hammocks as the bounded discrete ones amongst the class of finitely presented linear orders--the smallest class of linear orders containing finite linear orders as well as $ω$, and that is closed under isomorphisms, order reversal, finite order sums and lexicographic products.

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arXiv2022

Variations of the bridge quiver for domestic string algebras

In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schröer--the \emph{bridge quiver} whose vertices are (representatives of cyclic permutations of) bands and whose arrows are certain band-free strings--has been used extensively. There is a natural but ill-behaved partial binary operation, $\circ$, on the larger set of \emph{weak bridges} such that bridges are precisely the $\circ$-irreducibles. With the goal of computing hammocks up to isomorphism in a later work we equip an even larger set of \emph{weak arch bridges} with another partial binary operation, $\circ_H$, to obtain a finite category. Each weak arch bridge admits a unique $\circ_H$-factorization into \emph{arch bridges}, i.e., the $\circ_H$-irreducibles.

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arXiv2020

On the stable radical of some non-domestic string algebras

We introduce the concept of a prime band in a string algebra $Λ$ and use it to associate to $Λ$ its finite bridge quiver. Then we introduce a new technique of `recursive systems' for showing that a graph map between finite dimensional string modules lies in its stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have `meta-torsion-free' string algebras that are characterized by a dichotomy statement for ranks of graph maps between string modules--such maps either have finite rank or are in the stable radical. Their stable ranks can only take values from $\{ω,ω+1,ω+2\}$.

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