Diagonalizing the Frobenius

Diagonalizing the Frobenius

This paper explores the interplay between the Frobenius functor and Serre’s vanishing conjecture over a Noetherian local ring R of prime characteristic p. We show that the Frobenius functor induces a diagonalizable map on certain Q-vector spaces, which are tensor products of Q with quotients of Grothendieck groups. This allows us to decompose an element (representing a bounded complex of finite...

Diagonalizing Matrices

Diagonalizing Operators with Reflection Symmetry

Abstract. Let U be an operator in a Hilbert space H0, and let K ⊂ H0 be a closed and invariant subspace. Suppose there is a period-2 unitary operator J in H0 such that JUJ = U, and PJP ≥ 0, where P denotes the projection of H0 onto K. We show that there is then a Hilbert space H (K), a contractive operator W : K → H (K), and a selfadjoint operator S = S (U) in H (K) such that W W = PJP , W has ...

Diagonalizing by Fixed-Points

A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema. Here, we fit more theorems in the universal schema of diagonalization, such a...

Diagonalizing the genome II: toward possible applications

In a previous paper, we showed that the orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. In this draft of a sequel, we construct a related (stacky) resolution of a space of real quadratic forms, and suggest, perhaps without much ...