The Hilbert-kunz Function in Graded Dimension Two
نویسنده
چکیده
Suppose that (R,m) is a local Noetherian or a standard-graded ring of dimension d containing a field K of positive characteristic p. Let I denote an m-primary ideal and set I [q] = (f q : f ∈ I), q = p. The function e 7→ λ(R/I [p e]), where λ denotes the length, is called the Hilbert-Kunz function of the ideal I and was first considered by Kunz in [9]. Monsky showed in [12] that this function has the form (we write q for the argument, not e)
منابع مشابه
A Hilbert-kunz Criterion for Solid Closure in Dimension Two (characteristic Zero)
Let I denote a homogeneous R+-primary ideal in a twodimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I∗ if and only if eHK(I) = eHK((I, f)), where eHK denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the w...
متن کاملForcing Algebras, Syzygy Bundles, and Tight Closure
We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles. This work is based in understanding tight closure in terms of forcing algebras and the cohomological dimension of torsors of syzygy bundles. These geometric methods allowed to answer some fundamental questions of tight closure, in particular the equality between tight closure a...
متن کاملThe Rationality of the Hilbert-kunz Multiplicity in Graded Dimension Two
Abstract. We show that the Hilbert-Kunz multiplicity is a rational number for an R+−primary homogeneous ideal I = (f1, . . . , fn) in a twodimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the HilbertKunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of ...
متن کاملComputing Hilbert–kunz Functions of 1-dimensional Graded Rings
According to a theorem of Monsky, the Hilbert–Kunz function of a 1-dimensional standard graded algebra R over a finite field K has, for i 0, the shape HKR(i) = c(R) · p i + φ(i), where c(R) is the multiplicity of R and φ is a periodic function. Here we study explicit computer algebra algorithms for computing such Hilbert–Kunz functions: the period length and the values of φ, as well as a concre...
متن کاملRestriction of the Cotangent Bundle to Elliptic Curves and Hilbert-kunz Functions
We describe the possible restrictions of the cotangent bundle ΩPN to an elliptic curve C ⊂ P N . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R+primary ideal I ⊂ R in the graded section ring R = ⊕
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004