Idempotent characters and equivariantly multiplicative splittings of K-theory

We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude that the idempotent summands of $G$-equivariant topological $K$-theory and the corresponding summands of the $G$-equivariant sphere spectrum admit exactly the same flavors of equivariant commutative ring structures, made precise in terms of Hill-Hopkins-Ravenel norm maps. This paper is a sequel to the author's earlier work on multiplicative induction for the Burnside ring and the sphere spectrum, see arXiv:1802.01938.Comment: 19 pages. Comments welcome! v2: Updated references. Removed a lemma that is no longer relevant. v3: Changes in response to a referee repor

Solving Polynomial Equations with Equation Constraints: the Zero-dimensional Case

A zero-dimensional polynomial ideal may have a lot of complex zeros. But sometimes, only some of them are needed. In this paper, for a zero-dimensional ideal $I$, we study its complex zeros that locate in another variety $\textbf{V}(J)$ where $J$ is an arbitrary ideal. The main problem is that for a point in $\textbf{V}(I) \cap \textbf{V}(J)=\textbf{V}(I+J)$, its multiplicities w.r.t. $I$ and $I+J$ may be different. Therefore, we cannot get the multiplicity of this point w.r.t. $I$ by studying $I + J$. A straightforward way is that first compute the points of $\textbf{V}(I + J)$, then study their multiplicities w.r.t. $I$. But the former step is difficult to realize exactly. In this paper, we propose a natural geometric explanation of the localization of a polynomial ring corresponding to a semigroup order. Then, based on this view, using the standard basis method and the border basis method, we introduce a way to compute the complex zeros of $I$ in $\textbf{V}(J)$ with their multiplicities w.r.t. $I$. As an application, we compute the sum of Milnor numbers of the singular points on a polynomial hypersurface and work out all the singular points on the hypersurface with their Milnor numbers

Linear equations over noncommutative graded rings

We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function $D(d)$. For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensively study so--called universally coherent algebras, that is, such that the function $D(d)$ is bounded by 2d for $d >> 0$. For example, finitely presented monomial algebras belong to this class, as well as many algebras with finite Groebner basis of relations.Comment: 22 pages; corrections in Propositions 2.4 and 4.3, typos, et