2010/03/31 by Karl Schwede, Kevin Tucker · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algorithm #Annotation #Artificial intelligence #Coding theory and cryptography #Computer science #Discrete mathematics #Mathematics #Morphism #Polynomial and algebraic computation #math.AC #math.AG #msc:13A35 #msc:14B05 #msc:14F18
paper · pdf · doi:10.1090/s1056-3911-2013-00610-4
published as J. Algebraic Geom. 23 (2014), 399-443 · 33 pages. The appendix has been removed (it will appear in a different work). Minor changes and typos corrected throughout. To appear in the Journal of Algebraic Geometry
arxiv created 2012/01/05 · openalex publication_date 2013/09/09 · arxiv updated 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper Y right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo> : </mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">π \colon Y → X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of normal varieties in prime characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Specifically, given a ℚ -divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript upper X"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Δ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal OX</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -linear map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T colon upper K left-parenthesis upper Y right-parenthesis right-arrow upper K left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mo> : </mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak T \colon K(Y) → K(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we associate a ℚ -divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript upper Y"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Y</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Δ Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T left-parenthesis pi Subscript asterisk Baseline tau left-parenthesis upper Y semicolon normal upper Delta Subscript upper Y Baseline right-parenthesis right-parenthesis equals tau left-parenthesis upper X semicolon normal upper Delta Subscript upper X Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msub> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Y</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi>