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Variation of moduli spaces and Donaldson invariants under change of polarization
 J. REINE ANGEW. MATH
, 1995
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Comparison of symbolic and ordinary powers of ideals
 Invent. Math
"... All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the lar ..."
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Cited by 17 (0 self)
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All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I, then I (hn) ⊆ In for all n ≥ 0. Here, if W is the complement of the union of the associated primes of
Homology over local homomorphisms
 Amer. J. Math
"... Abstract. The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ: R→S. Various techniques are developed to study the new invariants and to establish their basic propert ..."
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Cited by 10 (2 self)
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Abstract. The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ: R→S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite Rmodules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a ‘contracting ’ endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz’s characterization of regularity in prime characteristic.
DOUBLE AFFINE HECKE ALGEBRAS AND CALOGEROMOSER SPACES
"... Abstract. In this paper we prove that the spherical subalgebra eH1,τ e of the double affine Hecke algebra H1,τ is an integral CohenMacaulay algebra isomorphic to the center Z of H1,τ,andH1,τ e is a CohenMacaulay eH1,τ emodule with the property H1,τ =EndeH1,τ e(H1,τ e) whenτisnot a root of unity. ..."
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Cited by 9 (1 self)
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Abstract. In this paper we prove that the spherical subalgebra eH1,τ e of the double affine Hecke algebra H1,τ is an integral CohenMacaulay algebra isomorphic to the center Z of H1,τ,andH1,τ e is a CohenMacaulay eH1,τ emodule with the property H1,τ =EndeH1,τ e(H1,τ e) whenτisnot a root of unity. In the case of the root system An−1 the variety Spec(Z) issmoothand coincides with the completion of the configuration space of the RuijenaarsSchneider system. It implies that the module eH1,τ is projective and all irreducible finite dimensional representations of H1,τ are isomorphic to the regular representation of the finite Hecke algebra.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 9 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
The frobenius structure of local cohomology
"... Abstract. Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Fr ..."
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Cited by 8 (1 self)
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Abstract. Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that Fpure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition. 1.
The Lfunctions of Witt coverings
 Math. Z
"... Abstract. Results on Lfunctions of ArtinSchreier coverings by Dwork, Bombieri and AdolphsonSperber are generalized to Lfunctions of Witt coverings. Key words: Lfunctions, exponential sums, Newton polygon MSC2000: 11L07, 14F30 1 ..."
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Cited by 8 (7 self)
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Abstract. Results on Lfunctions of ArtinSchreier coverings by Dwork, Bombieri and AdolphsonSperber are generalized to Lfunctions of Witt coverings. Key words: Lfunctions, exponential sums, Newton polygon MSC2000: 11L07, 14F30 1
Numerical equivalence defined on Chow groups of Noetherian local rings
, 2004
"... In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensio ..."
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In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional Qvector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck’s standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a CohenMacaulay local ring of positive characteristic is a numerically Roberts ring if and only if the HilbertKunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.