The Radical-Graded Hochschild Cohomology of Truncated Cycle Algebras

Let \ (Qₑ\) be the oriented cycle on \ (e2\) vertices and let \ (A=kQₑ/JN\), where \ (J\) is the arrow ideal and \ (N>e\). Suppose that \ (chark N\), and put \ (m=radHH⁰ (A) \). The positive-degree Hochschild cohomology groups are uniserial under the central cycle class; this determines both the annihilator filtration \ (Ann₇₇䂞 (₀) (mʳ) \) and the radical-image filtration \ (mʳHHⁿ (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 \ (grRHH^* (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