We prove the injectivity of Frobenius endomorphism on $\mathsf {B}_{\operatorname {cris}}$, {B}_{\max }$ and {st}}$.

We observe some new characterizations of $n$-presented modules. Using the concepts of $(n,0)$-injectivity and $(n,0)$-flatness of modules, we also present some characterizations of right $n$-coherent rings, right $n$-hereditary rings, and right $n$-regular rings.

In the current paper, we present search plans which can guide pattern matching for advanced graph patterns with edge identities, containment constraints, type variables, negative application conditions, attribute conditions, and injectivity constraints. Based on a generic search graph representation, all search plan operations (e.g. checking the existence of an edge, or extending a matching can...

We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injec...

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf...

The injectivity of the restriction homomorphism on divisor class groups to hypersurfaces has been studied by Grothendieck, Danilov, Lipman, and Griffith & Weston, among others. In particular, when A is a Noetherian normal domain of equicharacteristic zero and A/fA satisfies R1, Spiroff established a map Cl(A) → Cl((A/fA)), where (A/fA) represents the integral closure of A/fA, and gave some cond...

Birkhoff (quasi-)variety categorical axiomatizability results have fascinated many scientists by their elegance, simplicity and generality. The key factor leading to their generality is that equations, conditional or not, can be regarded as special morphisms or arrows in a special category, where their satisfaction becomes injectivity, a simple and abstract categorical concept. A natural and ch...

The generation of piecewise linear maps has received a lot of attention from the scientific community. In particular, locally injective maps are very important in many fields, from geometry processing and shape modelling to computer games and movies. We propose a mesh-based method for computing positionally constrained piece-wise linear maps which are locally injective and have low distortion. ...