2013/07/22 by Liran Shaul, Shaul, Liran · 1 citation
Mathematics · #13B35 #13C12 #13D03 #13J10 #14B15 #16E45 #18E30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AC #math.AG #math.KT #msc:13B35 #msc:13C12 #msc:13D03 #msc:13J10 #msc:14B15 #msc:16E45 #msc:18E30
paper · pdf · doi:10.48550/arxiv.1307.5658
31 pages. This revision: Added a result about a canonical isomorphism between topological Hochschild cohomology and discrete Hochschild cohomology
openalex publication_date 2013/07/22 · arxiv created 2013/08/27 · arxiv updated 2013/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \k be a commutative ring, and let (A,\mfraka) be an adic ring which is a \k-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of A over \k. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if \k is noetherian and (A,\mfraka) is essentially of finite type (in the adic sense) over \k, then there are formulas to compute them that stay inside the noetherian category. In the classical case, where \k is a field, we deduce that topological Hochschild cohomology and discrete Hochschild cohomology are isomorphic.