Idempotent plethories

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• 作者：Jesse Elliott ^{jesse.elliott@csuci.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13A99 ; 16W30 ; 16W99 ; 13G05 ; 13F20 ; 13F05 ; 18C15 ; 18C20
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：463
• 期：Complete
• 页码：33-79
• 全文大小：792 K

文摘

Let k be a commutative ring with identity. A k-plethory is a commutative k-algebra P   together with a comonad structure W_P${\mathbb{W}}_P$, called the P-Witt ring functor, on the covariant functor that it represents. We say that a k-plethory P is idempotent   if the comonad W_P${\mathbb{W}}_P$ is idempotent, or equivalently if the map from the trivial k  -plethory k[e]$k[e]$ to P is a k-plethory epimorphism. We prove several results on idempotent plethories. We also study the k  -plethories contained in K[e]$K[e]$, where K is the total quotient ring of k  , which are necessarily idempotent and contained in Int(k)={f∈K[e]:f(k)⊆k}$\mathrm{Int}(k)=\{f\in K[e]:f(k)\subseteq k\}$. For example, for any ring l between k and K we find necessary and sufficient conditions—all of which hold if k   is a integral domain of Krull type—so that the ring Int_l(k)=Int(k)∩l[e]${\mathrm{Int}}_l(k)=\mathrm{Int}(k)\cap l[e]$ has the structure, necessarily unique and idempotent, of a k  -plethory with unit given by the inclusion k[e]⟶Int_l(k)$k[e]⟶{\mathrm{Int}}_l(k)$. Our results, when applied to the binomial plethory Int(Z)$\mathrm{Int}(\mathbb{Z})$, specialize to known results on binomial rings.