Canonical Homomorphism Research Articles

Elementary properties of free lattices

Abstract We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive ∃ ∀ {\exists\forall} -sentence true in 𝐅 3 {\mathbf{F}_{3}} and false in 𝐅 4 {\mathbf{F}_{4}} . Secondly, we show that every model of Th ⁢ ( 𝐅 n ) {\mathrm{Th}(\mathbf{F}_{n})} admits a canonical homomorphism into the profinite-bounded completion 𝐇 n {\mathbf{H}_{n}} of 𝐅 n {\mathbf{F}_{n}} . Thirdly, we show that 𝐇 n {\mathbf{H}_{n}} is isomorphic to the Dedekind–MacNeille completion of 𝐅 n {\mathbf{F}_{n}} , and that 𝐇 n {\mathbf{H}_{n}} is not positively elementarily equivalent to 𝐅 n {\mathbf{F}_{n}} , as there is a positive ∀ ∃ {\forall\exists} -sentence true in 𝐇 n {\mathbf{H}_{n}} and false in 𝐅 n {\mathbf{F}_{n}} . Finally, we show that DM ⁢ ( 𝐅 n ) {\mathrm{DM}(\mathbf{F}_{n})} is a retract of Id ⁢ ( 𝐅 n ) {\mathrm{Id}(\mathbf{F}_{n})} and that for any lattice 𝐊 {\mathbf{K}} which satisfies Whitman’s condition ( W ) {\mathrm{(W)}} and which is generated by join prime elements, the three lattices 𝐊 {\mathbf{K}} , DM ⁢ ( 𝐊 ) {\mathrm{DM}(\mathbf{K})} , and Id ⁢ ( 𝐊 ) {\mathrm{Id}(\mathbf{K})} all share the same positive universal first-order theory.

Algebras with a bilinear form, and idempotent endomorphisms

The category k-BFAlg of all k-algebras with a bilinear form, whose objects are all pairs (R,b) where R is a k-algebra and b:R×R→k is a bilinear mapping, is equivalent to the category of unital k-algebras A for which the canonical homomorphism (k,1)→(A,1A) of unital k-algebras is a splitting monomorphism in k-Mod. Call the left inverses of this splitting monomorphism “weak augmentations” of the algebra. There is a category isomorphism between the category of k-algebras with a weak augmentation and the category of unital k-algebras (A,bA) with a bilinear form bA compatible with the multiplication of A, i.e., such that bA(x,y)=bA(z,w) for all x,y,z,w∈A for which xy=zw.

On the image of the trivial source ring in the ring of virtual characters of a finite group

We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Co\c{s}kun's theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc's theory of rational $p$-biset functors.

Classical and variational Poisson cohomology

We prove that, for a Poisson vertex algebra \({\cal V}\), the canonical injective homomorphism of the variational cohomology of \({\cal V}\) to its classical cohomology is an isomorphism, provided that \({\cal V}\), viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

Gorenstein flat representations of left rooted quivers

We study Gorenstein flat objects in the category Rep(Q,R) of representations of a left rooted quiver Q with values in Mod(R), the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep(Q,R) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φiX:⊕a:j→iX(j)→X(i) is injective, and the left R-modules X(i) and CokerφiX are Gorenstein flat. As an application, we obtain a Gorenstein flat model structure on Rep(Q,R) in which we give explicit descriptions of the subcategories of trivial, cofibrant and fibrant objects.

Integral trace form of extensions of degree pq

In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text]. The integral trace form of [Formula: see text] allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of [Formula: see text].

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

Topological groups without infinite precompact continuous homomorphic images

Let P be a property of topological groups. We say that a topological group G is MinAP moduloP if its image f[G] under each continuous homomorphism f:G→K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n(G) of every topological group G with respect to its von Neumann kernel n(G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n(G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

Interpolation in the Automorphism Group of a Polynomial Ring

Let R be a commutative ring with unity and SAn(R) be the group of volume-preserving automorphisms of the polynomial R-algebra R[n]. Given a proper ideal 𝔞 of R, we address in this paper the question of whether the canonical group homomorphism SAn(R) → SAn(R/𝔞) is surjective. As an application, we retrieve and generalize, in a unified way, several known results on residual coordinates in polynomial rings.

A remark on connective K-theory

Let X be a smooth algebraic variety over an arbitrary field. Let φ be the canonical surjective homomorphism of the Chow ring of X onto the ring associated with the Chow filtration on the Grothendieck ring K(X). We remark that φ is injective if and only if the connective K-theory CK(X) coincides with the terms of the Chow filtration on K(X). As a consequence, CK(X) turns out to be computed for numerous flag varieties (under semisimple algebraic groups) for which the injectivity of φ had already been established. This especially applies to the so-called generic flag varieties X of many different types, identifying for them CK(X) with the terms of the explicit Chern filtration on K(X). Besides, for arbitrary X, we compare CK(X) with the fibered product of the Chow ring of X and the graded ring formed by the terms of the Chow filtration on K(X).

Relative group homology theories with coefficients and the comparison homomorphism

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu’s theory to Adamson’s theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.

Canonical Submersion and Lie Homomorphisms Normal Subgroup

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

Constructions of Dense Lattices over Number Fields

In this work, we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2,3,4,5,6,8 and 12, which are rotated versions of the lattices Lambda_n, for n =2,3,4,5,6,8 and K_12. These algebraic lattices are constructed through canonical homomorphism via Z-modules of the ring of algebraic integers of a number field.

Realizing spaces as path-component spaces

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group is trivial.

Homomorphisms between rings with infinitesimals and infinitesimal comparisons

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of non-nonstandard analysis. In particular, we will show that there exists a nonzero element in Henle's ring that is too small to be registered as nonzero in Paolo Giordano's ring, while it is seen as a nonzero infinitesimal in ${^\ast\mathbb{R}}$. This result suggests that some hyperreal infinitesimals are smaller than the nilpotent infinitesimals. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ${^{\bullet}\mathbb{R}}$.

A topological characterization of the strong disk property on open Riemann surfaces

In this paper, we give a topological characterization of a subdomain $G$ of an open Riemann surface $R$ which has the strong disk property. Namely, we show that the domain $G$ satisfies the strong disk property in $R$ if and only if the canonical homomorphism $\pi_1(G) \to \pi_1(R)$ is injective.

Motivic spectral sequence for relative homotopy K-theory

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$.

Chow Filtration on Representation Rings of Algebraic Groups

Abstract We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

On Automorphisms of Enveloping Algebras

Abstract Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathcal{L}_{\infty }(\mathfrak{g})$ defined over $\mathbb{C}_{\infty }$, the reduction of $\mathbb{C}$ modulo the infinitely large prime, and show that for a class of Lie algebras, $\mathcal{L}_{\infty }(\mathfrak{g})$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a non-trivial semi-simple automorphism. As a corollary, we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ is isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$

Zeroth canonical homomorphism from singular to Milnor–Thurston homology is injective

Milnor–Thurston homology also known as “measure homology” is a generalisation of singular homology where chains are defined to be measures (possibly supported on infinitely many singular simplices). In this paper we prove that the canonical homomorphism from singular to Milnor–Thurston homology is injective in dimension zero for metrizable spaces.