1. What is Maschke's theorem and its relation to group algebra kG?
Maschke's theorem states that the group algebra kG of a finite group G over a field k is k-separable if and only if the order of G is invertible in k. This theorem is significant in the study of group algebras and their separability. In the modular case, where char(k) divides |G|, the separability of kG is questioned. The Donald-Flanigan (DF) problem explores whether kG can still be deformed into a separable algebra in this scenario. The DF Conjecture posits that the group algebra kG admits a k((t))-separable deformation. The conjecture has been proven for solvable groups obtained through sequences of extensions, as demonstrated in the provided papers. Theorem A and Theorem B are key results in the study of separable deformations of group algebras, particularly in the context of the DF problem.
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2. What is a k((t))-algebra A t?
A k((t))-algebra A t is a deformation of A, defined by a k[[t]]-algebra [A] t. It satisfies two conditions: A t ~ = k((t)) k[[t]] [A] t as k((t))-algebras, and A ~ = [A] t /t[A] t as k-algebras. This concept allows for the study of algebraic structures and their extensions in a more flexible manner, providing insights into the behavior of algebras under deformations.
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3. What is a separable deformation of A?
A separable deformation of A is a separable algebra over k((t)). In the given context, A t is identified with k[[t]] k[[t]] [A] t A t. This means that A t is a deformation of A, where A t is a separable algebra over k((t)). This concept is crucial in understanding the algebraic structure and behavior of A t in relation to A. The separable deformation allows for the study of the deformation of polynomials and automorphisms within the algebraic framework, providing insights into the properties and transformations of A t.
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4. How does Theorem A become immediate when char(k) = p?
When char(k) = p, Theorem A becomes immediate because B * C p is already k((t))-separable, as shown in [1, Theorem 3.3]. In this case, q t (y) = 0 serves the purpose. From this assumption, char(k) = p is taken. The polynomial q t (y) is constructed in two steps. In SS3.1, each summand in (3.1) is handled separately, and then in SS3.2, a global solution is provided. For each component B e [y; e t ]/ y p - u e, a polynomial q e (y) B e [y; e t ]/ y p - q e (y) - u e is defined, making it L e -separable. Since L e is a separable field extension of k((t)), this algebra is k((t))-separable as well. By setting q t (y) := eE eq e (y), it is established that B[y; e t ]/ y p + q t (y) - u e is separable over k((t)). Tensoring it over L e with the L e -separable algebra B e results in an L e -separable, and hence also k((t))-separable algebra B e Le L e [z]/ z p - t m z - v e ~ = B e [z]/ z p - t m z - v e ~ = B e [y; e t ]/ y p - t m q 1-p y - u e. In this scenario, q e (y) is taken as -t m q 1-p y for any positive integer m.
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