Let A/B be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications.

/pdf/gorenstein-flat-modules-of-hopf-galois-extensions-3ihs9svk.pdf

Gorenstein Flat Modules of Hopf-Galois Extensions

Let H be anite dimensional Hopf algebra over aeld k and A=B be a right H -Galois extension. In this note the relationship of cotorsion dimensions between A and B will be studied. We prove that r.cot.D( A) � r.cot.D( B) + l.D( H) . Moreover, we give some sufficient conditions for which r.cot.D( A) = r.cot.D( B) . As applications, we obtain some results about cotorsion dimension of the smash product.

https://journals.tubitak.gov.tr/math/issues/mat-16-40-1/mat-40-1-10-1408-59.pdf

Notes on cotorsion dimension of Hopf{Galois extensions

Let $H$ be a finite dimensional weak Hopf algebra and $A/B$ be a right faithfully flat weak $H$-Galois extension. Then in this note, we first show that if $H$ is semisimple, then the finitistic dimension of $A$ is less than or equal to that of $B$. Furthermore, using duality theorem, we obtain that if $H$ and its dual $H^*$ are both semisimple, then the finitistic dimension of $A$ is equal to that of $B$, which means the finitistic dimension conjecture holds for $A$ if and only if it holds for $B$. Finally, as applications, we obtain these relations for the weak crossed products and weak smash products.

/pdf/finitistic-dimension-of-weak-hopf-galois-extensions-9iyfkthzzc.pdf

Finitistic Dimension of Weak Hopf-Galois Extensions

Let H be a finite dimensional Hopf algebra over a field k and A/B be a right H-Galois extension. In this note the relationship of cotorsion dimensions between A and B will be studied. We prove that r.cot.D(A) <= r.cot.D(B) +1.D(H). Moreover, we give some sufficient conditions for which r.cot.D(A) = r.cot.D(B). As applications, we obtain some results about cotorsion dimension of the smash product.

Notes on Cotorsion Dimension of Hopf--Galois Extensions

Let H be a Hopf algebra over a field k and A/B a right H-Galois extension. Then in this note a spectral sequence for Ext will be constructed which yields the estimate for global dimension of A in terms of the corresponding data for H and B. As an application, we obtain the Maschke-type theorems for crossed products and twisted smash products. Finally, the relationship of finitistic dimensions between A and B will be given, if H is semisimple.

/pdf/global-and-finitistic-dimension-of-hopf-galois-extensions-49up7kvv7s.pdf

Global and finitistic dimension of Hopf-Galois extensions

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Global and finitistic dimension of Hopf-Galois extensions