We develop an induction theory for relative BiHom-Hopf modules, the special BiHom-Doi--Hopf case associated with the datum $(H,A,H)$ in which the final copy of $H$ carries its regular right $H$-module coalgebra structure. Let $H$ be a monoidal BiHom-Hopf algebra, let $A$ be a right $H$-BiHom-comodule algebra, and let $B=A^{\operatorname{co}H}$. We show that the balanced tensor product defines an induction functor $-\otimes_BA$ left adjoint to the coinvariant functor. If $H$ has a fixed Haar integral, untwisting yields a Haar identity for the BiHom setting and a canonical projection onto coinvariants. These constructions define a hereditary torsion theory with radical $\kappa$ and a torsion-free reflector $Q(M)=M/\kappa(M)$. The modified induction functor $Q(-\otimes_BA)$ then gives an equivalence between right $B$-BiHom-modules and torsion-free relative BiHom-Hopf modules generated by their coinvariants. Equal structure maps recover the corresponding Hom result, while identity structure maps recover the classical relative Hopf-module setting.
Let $A$ be a finite-dimensional basic hereditary algebra and let $T(A)=A\ltimes D(A)$ be its trivial extension. Building on the classification of indecomposable $\tau$-rigid $T(A)$-modules, we give explicit Hom-vanishing conditions characterizing arbitrary basic $\tau$-rigid $T(A)$-modules. For such a module $M$, we al...
Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\dim H=2\dim L$ is a normalized quadratic extension of $L$. For a fixed su...
We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of fi...
We show that for any algebra $R$ that is essentially of finite type over a field $k$, the derived functors of Smith-Van den Bergh admit a dg enhancement. In particular, for any $R$-module $M$ one can associate a dg algebra of differential operators $E_M$ whose cohomology is a nilpotent thickening of the (classical) rin...
J. Jeffries, Devlin Mallory, C. Miller et al.· 0 citations
Let $L/K$ be an arbitrary Galois extension with group $G$, equipped with the Krull topology. Let $C(G,K)$ denote the $K$-vector space of continuous functions $G\to K$, with $K$ discrete and $G$ acting by left translation. We prove that there exists a $G$-equivariant $K$-linear isomorphism $\varphi:C(G,K)\to L$ such tha...
Let $X$ be the good reduction of a projective hyperk\"ahler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof...
Hai-Tao Zou· 0 citations
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