Annotation of OpenXM/src/math-misc/twistedLogCohomology-en.tex, Revision 1.1
1.1 ! nisitani 1: \documentclass{article}
! 2:
! 3: \begin{document}
! 4: \begin{center}
! 5: {\bf twistedLogCohomology(List,List) -- twisted logarithmic cohomology groups in two variables}
! 6: \end{center}
! 7: \begin{flushleft}
! 8:
! 9: {\bf Synopsis}
! 10:
! 11: \begin{itemize}
! 12: \item Usage: twistedLogCohomology(F,A)
! 13: \item Function: twistedLogCohomology
! 14: \item inputs:
! 15: \begin{itemize}
! 16: \item F, a list of polynomials in two variables
! 17: \item A, a list of rational numbers
! 18: \end{itemize}
! 19: \item outputs:
! 20: \begin{itemize}
! 21: \item a hashtable, with entries $\{$ Bfunction, CohomologyGroups, LogBasis, OmegaRes, PreCycles, VResolution $\}$
! 22: \end{itemize}
! 23: \end{itemize}
! 24:
! 25: {\bf Description}
! 26:
! 27: Bases of twisted logarithmic cohomology groups are contained in a hashtable LogBasis.
! 28: Bases of $H^1$ and $H^2$ are outputted only numerators.
! 29: In following example, a basis of $H^1$ is
! 30: $\{ \frac{y^2dx-xydy}{x(x+y)}, \frac{2x^2dx+2xydx}{x(x+y)} \}$,
! 31: a basis of $H^2$ is $\{ \frac{ydxdy}{x(x+y)} \}$.
! 32:
! 33: {\footnotesize
! 34: \begin{verbatim}
! 35: i1 : loadPackage "Dmodules";
! 36:
! 37: i2 : load "twistedLogCohomology.m2";
! 38:
! 39: i3 : R = QQ[x,y];
! 40:
! 41: i4 : twistedLogCohomology({x,x+y},{-1,0})
! 42: Warning: not a generic weight vector. Could be difficult...
! 43:
! 44: o5 = HashTable{BFunction => (s - 1) }
! 45: 1
! 46: CohomologyGroups => HashTable{0 => QQ }
! 47: 2
! 48: 1 => QQ
! 49: 1
! 50: 2 => QQ
! 51: LogBasis => HashTable{0 => | x | }
! 52: 1 => | y2dx-xydy 2x2dx+2xydx |
! 53: 2 => | ydxdy |
! 54: 1 2 1
! 55: OmegaRes => (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- 0
! 56:
! 57: 0 1 2 3
! 58: PreCycles => HashTable{0 => | -x2-xy |}
! 59: | -x |
! 60: 1 => | 0 -2x |
! 61: | -y 0 |
! 62: | 0 0 |
! 63: 2 => | y |
! 64: 1 3 2
! 65: VResolution => (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- 0
! 66:
! 67: 0 1 2 3
! 68:
! 69: o5 : HashTable
! 70:
! 71: \end{verbatim}
! 72: }
! 73:
! 74: \end{flushleft}
! 75: \end{document}
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