On simplicial commutative algebras with Noetherian homotopy

✍ Scribed by James M. Turner

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 122 KB
• Volume: 174
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00044-0

✦ Synopsis

In this paper, we introduce a strategy for studying simplicial commutative algebras over general commutative rings R. Given such a simplicial algebra A, this strategy involves replacing A with a connected simplicial commutative k(˝)-algebra A(˝), for each ˝∈ Spec( 0A), which we call the connected component of A at ˝. These components retain most of the Andrà e-Quillen homology of A when the coe cients are k(˝)-modules (k(˝)=residue ÿeld of ˝in 0A). Thus, these components should carry quite a bit of the homotopy theoretic information for A. Our aim will be to apply this strategy to those simplicial algebras which possess Noetherian homotopy. This allows us to have sophisticated techniques from commutative algebra at our disposal. One consequence of our e orts will be to resolve a more general form of a conjecture of Quillen that was posed in Invent. Math.