A. – Let k be a field. We compute the set [ P,P ]N of naive homotopy classes of pointed k-scheme endomorphisms of the projective line P. Our result compares well with Morel’s computation in [11] of the group [ P,P ]A1 of A-homotopy classes of pointed endomorphisms of P: the set [ P,P ]N admits an a priori monoid structure such that the canonical map [ P,P ]N → [P1,P1]A1 is a group comple...

We show that among the endomorphisms of the free (non-abelian) group Fr of rank r, the set of monomorphisms (the injective endomorphisms) has density one. This contrasts with the known fact that the set of automorphisms has density zero. We show more generally that in the set of homomorphisms from one free group to another, the set of monomorphisms has density one whereas the set of epimorphism...

(c) μ(rs,m) = μ(r, μ(s,m)) (d) if 1 ∈ R, then μ(1,m) = m. We shall usually omit the notation of μ and simply write r ·m for μ(r,m). Thus axiom (c) would be written (rs) ·m = r · (s ·m), etc. Exercise 1. We denote by EndGrp(M) the set of group endomorphisms of M : An element φ ∈ EndGrp(M) is a group homomorphism φ : M → M . EndGrp(M) is naturally a ring, with addition and multiplication defined ...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p, then it also extends to the simple Cole extension with respect to p. W...

Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U...

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of met...

For each object in a tensor triangulated category, we construct a natural continuous map from the object’s support—a closed subset of the category’s triangular spectrum—to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tens...

An algebra is a vector space (over C) with a multiplication such that A is a ring with identity, i.e. there is a map A × A → A, (a, b) 7→ ab, which is bilinear and satisfies the associative and distributive laws. The following are examples of algebras: (1) The group algebra of a group G is the vector space CG with basis G and with multiplication forced by the multiplication in G (and the biline...

The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e. in the category of ...