… The aim of this chapter is to present the relation between the finite dimensional complex
simple … Lie group) homomorphism if it is both holomorphic and a homomorphism of the abstract …

… An n-FC ring is a coherent ring with self-FPinjective dimension n, ie, FP-idR(R) ⩽ n… ring R,
M and N be R-modules. Since we need localization, we consider the following homomorphism …

… Inducting over the length of the zig-zag, we conclude that η is natural, as desired. … x) : HX,
then for any i : Fin(n) we have the morphism in HX on the left, whose filler is the reflexivity path: …

Among topological modular forms with level structure, $${{\,\mathrm{TMF}\,}}_0(7)$$ TMF 0
( 7 ) at the prime 3 is the first example that had not been understood yet. We provide a …

… Secondly, for a ring homomorphism R → S, we prove that if S satisfies (SAC) (resp. (ARC)), …
Let R be a ring. An R-module M is said to have finite flat dimension if there exists an integer n …

… We study the Tannakian category of finite-dimensional (flat) vector bundles on \(X\) and
its … can be written as reflexive coequalizers of finitely generated polynomial rings over \(K\). …

… with the structure of modules via the ring homomorphism … To be rigorous, we have to impose
some condition on the size of … Then we see that a topos may be defined as a reflexive left-…

… In our setting, the reflectivity is ensured by the definability of the members of the sequence. …
The maximal immediate extension is not uniquely determined as a ring homomorphism, but it …

… ring homomorphism O S → F ⁎ O S . By our assumption, the graded ring R is F-finite, ie, the
Frobenius on Z = Spec R is a finite … e R of the graded ring R is identified with the ring R 1 / p …

… colimits and reflexive coequalizers to Pro(Vectω k ). This can be identified with finite-product-…
, where we define the dimension of a profinite vector space to be the dimension of its dual. …