{"title":"Floor, ceiling, slopes, and K-theory","authors":"Yuri Sulyma","abstract":"We calculate K_*(k[x]/x^e;Z_p) by evaluating the syntomic cohomology Z_p(i)(k[x]/x^e) introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case e=2 and p>2. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for e=2. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.","keywords":["algebraic K-theory","syntomic cohomology","topological Hochschild homology"],"year":2026,"slug":"floor-ceiling-slopes-k-theory","version":1,"url_hash":"6b5e5158ba7a","license":"cc-by-4.0","doi":"10.5281/zenodo.23260682","published_at":"2026-10-09T10:13:37.056207+00:00","subject":"Mathematics","storage_type":"archive","canonical_url":"https://scroll.press/2026/floor-ceiling-slopes-k-theory","version_url":"https://scroll.press/2026/floor-ceiling-slopes-k-theory/v1","paper_url":"https://scroll.press/2026/floor-ceiling-slopes-k-theory/paper","paper_version_url":"https://scroll.press/2026/floor-ceiling-slopes-k-theory/v1/paper","html_url":"https://scroll.press/2026/floor-ceiling-slopes-k-theory/paper","html_sha256":"6b5e5158ba7ab3ebc0f85ed70bb474839e0caf368771564c76a2713ccc6820a7","html_bytes":null,"cite_as":{"bibtex":"@misc{floor-ceiling-slopes-k-theory2026,\n  author = {Sulyma, Yuri},\n  title = {Floor, ceiling, slopes, and K-theory},\n  year = {2026},\n  publisher = {Scroll Press},\n  url = {https://scroll.press/2026/floor-ceiling-slopes-k-theory},\n  version = {1},\n  doi = {10.5281/zenodo.23260682}\n}","csl_json":{"type":"article","id":"6b5e5158ba7a","title":"Floor, ceiling, slopes, and K-theory","author":[{"family":"Sulyma","given":"Yuri"}],"publisher":"Scroll Press","URL":"https://scroll.press/2026/floor-ceiling-slopes-k-theory","version":"1","issued":{"date-parts":[[2026]]},"DOI":"10.5281/zenodo.23260682","license":"cc-by-4.0"}}}