The Radical-Graded Hochschild Cohomology of Truncated Cycle Algebras

Let \(Q_e\) be the oriented cycle on \(e\geq2\) vertices and let \(A=kQ_e/J^N\), where \(J\) is the arrow ideal and \(N>e\). Suppose that \(\operatorname{char}k\nmid N\), and put \(\mathfrak m=\operatorname{rad}\operatorname{HH}^0(A)\). The positive-degree Hochschild cohomology groups are uniserial under the central cycle class; this determines both the annihilator filtration \(\operatorname{Ann}_{\operatorname{HH}^n(A)}(\mathfrak m^r)\) and the radical-image filtration \(\mathfrak m^r\operatorname{HH}^n(A)\), together with their rational bigraded Hilbert series. When \(N=me+1\) and \(m>1\), we compute the full filtered initial ideal of the Hochschild relation ideal and obtain an explicit presentation of \(\operatorname{gr}_R\operatorname{HH}^*(A)\). The BV operator associated with the standard Frobenius functional has zero positive-degree homology in characteristic zero; in characteristic \(p\nmid eN\), its homology is periodic and satisfies an exact mass formula. **Keywords** Hochschild cohomology; truncated cycle algebra; Nakayama algebra; Loewy filtration; associated graded algebra; Batalin–Vilkovisky operator. **Mathematics Subject Classification** The current scheme used in 2026 is **2020 Mathematics Subject Classification**: - 16E40 — Hochschild and cyclic homology and cohomology- 16G20 — Representations of quivers and partially ordered sets- 18G15 — Ext and Tor, generalizations, Künneth formula