Diagonalizing the Frobenius
نویسنده
چکیده
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 finitely generated projective modules) into eigenvectors for the Frobenius: the component with eigenvalue 1 describes the Dutta multiplicity of the element, and the remaining components describe the extent to which Serre’s vanishing conjecture fails to hold. As a consequence, we explicitly describe the Dutta multiplicity as a Q-linear combination of finitely many terms in a sequence of intersection multiplicities; and we show that, over a Cohen–Macaulay ring, a sufficient condition for the weak version of Serre’s vanishing conjecture (the one in which both modules are assumed to have finite projective dimension) to hold is that the Frobenius functor changes the length of modules of finite projective dimension by a factor p.
منابع مشابه
Frobenius kernel and Wedderburn's little theorem
We give a new proof of the well known Wedderburn's little theorem (1905) that a finite division ring is commutative. We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group theory to build a proof.
متن کاملPERRON-FROBENIUS THEORY ON THE NUMERICAL RANGE FOR SOME CLASSES OF REAL MATRICES
We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.
متن کاملNearly Rational Frobenius Groups
In this paper, we study the structure of nite Frobenius groups whose non-rational or non-real irreducible characters are linear.
متن کاملCompact weighted Frobenius-Perron operators and their spectra
In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.
متن کاملOn some Frobenius groups with the same prime graph as the almost simple group ${ {bf PGL(2,49)}}$
The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...
متن کامل