The Radical-Graded Hochschild Cohomology of Truncated Cycle Algebras

Let \(Q_e\) be the oriented cycle on \(e≥2\) vertices and let \(A=kQ_e/J^N\), where \(J\) is the arrow ideal and \(N>e\). Suppose that \(chark N\), and put \( m=radHH^0(A)\). The positive-degree Hochschild cohomology groups are uniserial under the central cycle class; this determines both the annihilator filtration \(AnnHH^n(A)( m^r)\) and the radical-image filtration \( m^rHH^n(A)\), 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 \(gr_RHH^*(A)\). The BV operator associated with the standard Frobenius functional has zero positive-degree homology in characteristic zero; in characteristic \(p eN\), 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