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Diagonalizing the Frobenius

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Abstract

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 dim R. 1.

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... The Grothendieck spaces GD f (X), GP f (X) and GI f (X) are topological Q-vector spaces modelled on these categories. The first two of these spaces were introduced in [11] but were there modelled on ordinary non-derived categories of complexes. The construction of Grothendieck spaces is similar to that of Grothendieck groups but targeted at the study of intersection multiplicities. ...
... The main result of [11] is a diagonalization theorem in prime characteristic p for the automorphism on GP f (X) induced by the Frobenius functor. A consequence of this theorem is that every element α ∈ GP f (X) can be decomposed as ...
... Even in arbitrary characteristic, R satisfies vanishing if and only if all elements α ∈ GP f (X) are self-dual in the sense that α = (−1) codim X α * ; and R satisfies weak vanishing if all elements α ∈ GP f (X) are numerically self-dual, meaning that α−(−1) codim X α * is in the kernel of the homomorphism GP f (X) → GD f (X) induced by the inclusion of the underlying categories (see Theorem 7.4). Rings for which all elements of the Grothendieck spaces GP f (X) are numerically self-dual include Gorenstein rings of dimension less than or equal to five (see Proposition 7.11) and complete intersections (see Proposition 7.7 together with [11,Example 33]). ...
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Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.
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The new intersection theorem states that, over a Noetherian local ring R , for any non-exact complex concentrated in degrees n ,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R . One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if P d (length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d ,…0, the inclusion P d (length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of P d (length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.
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Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for which this equality fails to hold. This then provides an example of a nonzero Todd class , and of a bounded free complex whose local Chern character does not vanish on this class.
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