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If $n$ does not divide $N$ and the characteristic of $k$ is fixed, then the value of $a$ determines whether $Q$ has finite or infinite projective dimension. If $Q$ has infinite projective dimension, then value of $r$, together with the parity of $a$, determines the periodic part of the infinite resolution. When $Q$ has infinite projective dimension we give an"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1012.1026","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-12-05T18:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"790a50cb9b36b0ed4c05dfa30cc6b7ca3a1044286bc47067df06b8912dc1ea01","abstract_canon_sha256":"feb6c0e265c83c432936f5ef9be8ef7b437f4e5f52b56df9c711a8f1aa0f3c23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:01.922818Z","signature_b64":"yRVCWRVsKad4ft10Gu6sVuBuhG08nKsQSGBRUmFkfuVR+u6tixDxznhjTQNFYtWO/sZeqt8Dfn5bWR3wff10DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73740032828563eac2eae9ded10bef2937b1bf483908134530eaf5d164b6d95f","last_reissued_at":"2026-05-18T04:34:01.922358Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:01.922358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The resolution of the bracket powers of the maximal ideal in a diagonal hypersurface ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Adela Vraciu, Andrew R. Kustin, Hamid Rahmati","submitted_at":"2010-12-05T18:59:56Z","abstract_excerpt":"Let $k$ be a field. For each pair of positive integers $(n,N)$, we resolve $Q=R/(x^N,y^N,z^N)$ as a module over the ring $R=k[x,y,z]/(x^n+y^n+z^n)$. Write $N$ in the form $N=a n+r$ for integers $a$ and $r$, with $r$ between $0$ and $n-1$. If $n$ does not divide $N$ and the characteristic of $k$ is fixed, then the value of $a$ determines whether $Q$ has finite or infinite projective dimension. If $Q$ has infinite projective dimension, then value of $r$, together with the parity of $a$, determines the periodic part of the infinite resolution. 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