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It is naturally an object in the category ${\\operatorname{Mod}}{(G)}$ of all smooth $k$-representations of $G$.\n  We study the graded Ext-algebra $E^*={\\operatorname{Ext}}_{{\\operatorn"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.09503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-08-28T19:15:58Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"b5e1d30b2b015bfd0c60c4d02130b879e9fdf11b3488281c0a92d25e7bbd1c13","abstract_canon_sha256":"b8c65a427ced04e1e578442d07071be511aef37e385ba6a345623b2ffa7da6d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:55.370543Z","signature_b64":"Pn5Rnly0ZJ2OV7c32yUSUkZqxyxfK90uWN3Z5xAlDwNG0YyuHILdvuYqt04vpzmsRIUFXW+WuT8YZVUpasLnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d09eb387264a2c2df58508741ed4714098ea53fa9287ed4595d0716a3815962c","last_reissued_at":"2026-05-18T00:06:55.369953Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:55.369953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The modular pro-$p$ Iwahori-Hecke ${\\operatorname{Ext}}$-algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Peter Schneider, Rachel Ollivier","submitted_at":"2018-08-28T19:15:58Z","abstract_excerpt":"Let $\\mathfrak F$ be a locally compact nonarchimedean field of positive residue characteristic $p$ and $k$ a field of characteristic $p$. 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