Borger, James; Wieland, Ben Plethystic algebra. (English) Zbl 1098.13033 Adv. Math. 194, No. 2, 246-283 (2005). Let \(k\) be a commutative ring. A \(k\)-biring is a \(k\)-ring that represents a functor from the category of \(k\)-rings to itself. A \(k\)-plethory is a biring \(P\) with an associative map of birings \(\circ :P\odot _{k}P\rightarrow P\) and unit \(k\left\langle e\right\rangle \rightarrow P,\) where \(\odot \) is an analogue of tensor product on birings. A \(k\)-plethory is a non-linear generalization of a cocommutative bialgebra: along with the “product” and unitary maps above there are the notions of coaddition \(\Delta ^{+}\) and comultiplication \(\Delta ^{\times }\) with counits \(\varepsilon ^{+}\) and \( \varepsilon ^{\times }\) respectively. For any commutative \(k\)-bialgebra \(A\) the symmetric algebra \(S\left( A\right) \) is an example of a \(k\)-plethory.One of the motivations for this article is the following example. Let \( \Lambda \) be the ring of symmetric functions in infinitely many variables \( \left\{ x_{i}\right\} \) – this naturally has the structure of a \(\mathbb{Z}\) -plethory. Let \(\psi _{n}=x_{1}^{n}+x_{2}^{n}+\cdots .\) Then \(\psi _{n}=\sum_{d\mid n}w_{d}^{n/d}\) for some \(w_{1},w_{2},\dots \in \Lambda .\) These \(w_{i}\)’s are used in the Witt vector operations. In fact, given a plethory \(P\) if we define \(W_{P}\left( R\right) \) to be the set of \(k\)-ring maps \(P\rightarrow R\) for some \(k\)-ring \(R\) then \(W_{\Lambda _{p}}\left( R\right) \) is isomorphic (as rings) to the ring of \(p\)-typical Witt vectors, where \(\Lambda _{p}\) is a certain sub \(\mathbb{Z}\)-plethory of \(\Lambda .\) In this way one can obtain (from the paper): “a definition [of Witt vectors] given purely in terms of algebraic structure rather than somewhat mysterious formulas”.Given a morphism \(P\rightarrow Q\) of plethories, the author proves the following reconstruction theorem. Let \(\mathcal{C}\) be a category with all of its limits and colimits, and let \(U\) be a functor from \(\mathcal{C}\) to the category of rings. If \(U\) has a left and right adjoint as well as the property that a map \(f\) in \(\mathcal{C}\) is an isomorphism if \(U\left( f\right) \) is, then \(C\) is a category of \(P\)-rings for some plethory \(P\); furthermore \(U\) is the forgetful functor from \(P\)-rings to rings. Here a “\(P \)-ring” is a \(k\)-ring with an action by \(P\).Let \(\mathcal{O}\) be a Dedekind domain, \(\mathfrak{m\subset }\mathcal{O}\) an ideal, \(P\) an \(\mathcal{O}\)-plethory, \(Q\) an \(\mathcal{O}/\mathfrak{m}\) -plethory, and suppose we have a surjective map \(P\rightarrow Q\) of plethories. A \(P\)-ring \(R\) is a \(P\)-deformation of a \(Q\)-ring if it is \( \mathfrak{m}\) torsion free and the action of \(P\) on \(R/\mathfrak{m}R\) factors through \(P\rightarrow Q.\) Then there is an \(\mathcal{O}\)-plethory \( P^{\prime }\) which is universal among those which are \(P\)-deformations of \(Q\) -rings. This \(P^{\prime }\) is called the amplification of \(P\) along \(Q.\)Finally, the notion of a linearization of a plethory is described. From this we get \(A_{P}\), the set of elements of \(P\) acting additively on a \(P\)-ring, and \(C_{P},\) the cotangent space to the spectrum of \(P\) at zero. Then, under certain conditions, \(A_{P}\) is a cocommutative twisted \(k\)-bialgebra, there is a coaction of \(A_{P}\) on \(C_{P},\) and \(A_{P}\rightarrow C_{P}\) is \(A_{P}\) -coequivariant. Reviewer: Alan Koch (Decatur) Cited in 3 ReviewsCited in 24 Documents MSC: 13K05 Witt vectors and related rings (MSC2000) 13A99 General commutative ring theory 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 19A99 Grothendieck groups and \(K_0\) Keywords:ring scheme; Witt vector; Witt ring; plethory; biring × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Atiyah, M. F.; Tall, D. O., Group representations, \( \lambda \)-rings and the \(J\)-homomorphism, Topology, 8, 253-297 (1969) · Zbl 0159.53301 [2] Bergman, G. M.; Hausknecht, A. O., Co-groups and Co-rings in Categories of Associative Rings, (Mathematical Surveys and Monographs, vol. 45 (1996), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0857.16001 [3] Brzeziński, T.; Militaru, G., Bialgebroids, \( \times_A\)-bialgebras and duality, J. Algebra, 251, 1, 279-294 (2002) · Zbl 1003.16033 [4] Cartier, P., Groupes formels associés aux anneaux de Witt généralisés, C. R. Acad. Sci. Paris Sér. A-B, 265, A49-A52 (1967) · Zbl 0168.27501 [5] Dăscălescu, S.; Năstăsescu, C.; Raianu, Ş., Hopf algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 235 (2001), Marcel Dekker: Marcel Dekker New York, (An introduction) · Zbl 0962.16026 [6] Deligne, P., La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math., 322, 4, 321-326 (1996) · Zbl 0910.22008 [7] Fontaine, J.-M., Le corps des périodes \(p\)-adiques, Astérisque, 223, 59-111 (1994), (With an appendix by Pierre Colmez, Périodes \(p - \operatorname{adiques}, \operatorname{Bures} - \operatorname{sur} - \operatorname{Yvette}, 1988)\) · Zbl 0940.14012 [8] Hazewinkel, M., Formal Groups and Applications, Pure and Applied Mathematics, vol. 78 (1978), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: Academic Press Inc. [Harcourt Brace Jovanovich Publishers] New York · Zbl 0454.14020 [9] Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup., 12, 4, 501-661 (1979) · Zbl 0436.14007 [10] Joyal, A., \( \delta \)-anneaux et vecteurs de Witt, C. R. Math. Rep. Acad. Sci. Canada, 7, 3, 177-182 (1985) · Zbl 0594.13023 [11] Knutson, D., \( \lambda \)-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, vol. 308 (1973), Springer: Springer Berlin · Zbl 0272.20008 [12] Lang, S., Algebra (1965), Addison-Wesley Publishing Co., Inc.: Addison-Wesley Publishing Co., Inc. Reading, MA · Zbl 0193.34701 [13] Lazard, M., Commutative Formal Groups, Lecture Notes in Mathematics, vol. 443 (1975), Springer: Springer Berlin · Zbl 0304.14027 [14] Mac Lane, S., Categories for the Working Mathematician (1998), Springer: Springer New York · Zbl 0906.18001 [15] I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995 (With contributions by A. Zelevinsky, Oxford Science Publications).; I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995 (With contributions by A. Zelevinsky, Oxford Science Publications). · Zbl 0824.05059 [16] Schauenburg, P., Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures, 6, 2, 193-222 (1988) · Zbl 0908.16033 [17] M.E. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ. Math. (44) (1974) 79-189.; M.E. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ. Math. (44) (1974) 79-189. · Zbl 0314.16008 [18] Takeuchi, M., Groups of algebras over \(A \otimes \overline{A} \), J. Math. Soc. Japan, 29, 3, 459-492 (1977) · Zbl 0349.16012 [19] Tall, D. O.; Wraith, G. C., Representable functors and operations on rings, Proc. London Math. Soc., 20, 3, 619-643 (1970) · Zbl 0226.13007 [20] Wilkerson, C., Lambda-rings binomial domains and vector bundles over \(C P(\infty)\), Comm. Algebra, 10, 3, 311-328 (1982) · Zbl 0492.55004 [21] Witt, E., Zyklische Körper und Algebren der Charakteristik \(p\) vom Grad \(p^n\). Struktur diskret bewerter perfekter Körper mit vollkommenem Restklassen-körper der charakteristik \(p\), J. Reine Angew. Math., 176 (1937) · JFM 62.1112.03 [22] Witt, E., Collected Papers, Gesammelte Abhandlungen (1998), Springer: Springer Berlin, (With an essay by Günter Harder on Witt vectors, Edited and with a preface in English and German by Ina Kersten) · Zbl 0917.01054 [23] Wraith, G. C., Algebras over theories, Colloq. Math., 23, 181-190 (1971) · Zbl 0226.18003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.