o1 = $\\left\\{1,\\:9,\\:25,\\:49,\\:81,\\:121,\\:169,\\:225,\\:289\\right\\}$
\n", "o1 : List
\n" ], "text/plain": [ "o1 = $\\left\\\\{1,\\:9,\\:25,\\:49,\\:81,\\:121,\\:169,\\:225,\\:289\\right\\\\}$\n", "\n", "o1 : List\n", "\n" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "for i from 1 to random(20) list if i%2==1 then i*i else continue" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can write text and mathematics directly along your code.\n", "Jupyter supports Markdown fully and $\\TeX$ to a degree.\n", "For example:\n", "\n", "The cell below defines the twisted cubic $C$ over the rationals in two ways —\n", "as $\\text{Proj}~\\mathbb{Q}[s^3, s^2t, st^2, t^3]$, and as the ideal of maximal minors of\n", "$\n", "g=\n", "\\begin{bmatrix}\n", "x_0& x_1& x_2\\\\\n", "x_1& x_2& x_3\n", "\\end{bmatrix}\n", "$.\n", "It then checks that the two definitions agree using Macaulay2's `assert`." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o2 = $S$
\n", "o2 : PolynomialRing
\n", "o3 = $R$
\n", "o3 : PolynomialRing
\n", "o4 = $\\texttt{map}{}\\left(R,\\,S,\\,\\left\\{s^{3},\\:s^{2}t,\\:s\\,t^{2},\\:t^{3}\\right\\}\\right)$
\n", "o4 : RingMap $R\\,\\longleftarrow \\,S$
\n", "o5 = $\\left(\\!\\begin{array}{ccc}\r\n", "x_{0}&x_{1}&x_{2}\\\\\r\n", "x_{1}&x_{2}&x_{3}\r\n", "\\end{array}\\!\\right)$
\n", "o5 : Matrix $S^{2}\\,\\longleftarrow \\,S^{3}$
\n", "o6 = $\\texttt{ideal}{}\\left(x_{2}^{2}-x_{1}x_{3},\\,x_{1}x_{2}-x_{0}x_{3},\\,x_{1}^{2}-x_{0}x_{2}\\right)$
\n", "o6 : Ideal of $S$
\n" ], "text/plain": [ "o2 = $S$\n", "\n", "o2 : PolynomialRing\n", "\n", "o3 = $R$\n", "\n", "o3 : PolynomialRing\n", "\n", "o4 =\n", "$\\texttt{map}{}\\left(R,\\,S,\\,\\left\\\\{s^{3},\\:s^{2}t,\\:s\\,t^{2},\\:t^{3}\\right\\\\}\\right)$\n", "\n", "o4 : RingMap $R\\,\\longleftarrow \\,S$\n", "\n", "o5 = $\\left(\\\\!\\begin{array}{ccc} x_{0}&x_{1}&x_{2}\\\\\\ x_{1}&x_{2}&x_{3}\n", "\\end{array}\\\\!\\right)$\n", "\n", "o5 : Matrix $S^{2}\\,\\longleftarrow \\,S^{3}$\n", "\n", "o6 =\n", "$\\texttt{ideal}{}\\left(x_{2}^{2}-x_{1}x_{3},\\,x_{1}x_{2}-x_{0}x_{3},\\,x_{1}^{2}-x_{0}x_{2}\\right)$\n", "\n", "o6 : Ideal of $S$\n", "\n" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S = QQ[x_0..x_3] -- the embedding space\n", "R = QQ[s,t] -- PP^1 \n", "f = map(R,S,{s^3, s^2*t, s*t^2, t^3})\n", "g = matrix({{x_0,x_1,x_2},{x_1,x_2,x_3}})\n", "C = ker f\n", "assert(C == minors(2, g))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that failed assertions do propagate to stdout." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "stdio:8:6:(3): error: assertion failed\n", "
o9 = $-2\\,\\texttt{P}_{0}+3\\,\\texttt{P}_{1}$
\n", "o9 : ProjectiveHilbertPolynomial
\n" ], "text/plain": [ "\n", " stdio:8:6:(3): error: assertion failed\n", "\n", "o9 = $-2\\,\\texttt{P}_{0}+3\\,\\texttt{P}_{1}$\n", "\n", "o9 : ProjectiveHilbertPolynomial\n", "\n" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert(false)\n", "hilbertPolynomial C" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Long input is OK. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o10 = $20\\,s^{501}t^{501}$
\n", "o10 : $R$
\n" ], "text/plain": [ "o10 = $20\\,s^{501}t^{501}$\n", "\n", "o10 : $R$\n", "\n" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q = poly \"s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501+s501t501\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Output can be suppressed as usual." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "\n" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "degree C;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "M2JK implements a set of *magic* commands. The name is borrowed from IPython.\n", "It can be added as `%o12 = $\\underset{\\vphantom{\\Big|}0}{S^{1}}\\,\\xleftarrow{\\left(\\!\\begin{array}{ccc}\r\n", "x_{1}^{2}-x_{0}x_{2}&x_{1}x_{2}-x_{0}x_{3}&x_{2}^{2}-x_{1}x_{3}\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}1}{S^{3}}\\,\\xleftarrow{\\left(\\!\\begin{array}{cc}\r\n", "-x_{2}&x_{3}\\\\\r\n", "x_{1}&-x_{2}\\\\\r\n", "-x_{0}&x_{1}\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}2}{S^{2}}\\,\\xleftarrow{0}\\,\\underset{\\vphantom{\\Big|}3}{0}$
\n", "o12 : ChainComplex
\n", "o13 = $\\left(\\!\\begin{array}{ccc}\r\n", "x_{0}&x_{1}&x_{2}\\\\\r\n", "x_{1}&x_{2}&x_{3}\r\n", "\\end{array}\\!\\right)$
\n", "o13 : Matrix $S^{2}\\,\\longleftarrow \\,S^{3}$
\n" ], "text/plain": [ "o12 =\n", "$\\underset{\\vphantom{\\Big|}0}{S^{1}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{ccc}\n", "x_{1}^{2}-x_{0}x_{2}&x_{1}x_{2}-x_{0}x_{3}&x_{2}^{2}-x_{1}x_{3}\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}1}{S^{3}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{cc}\n", "-x_{2}&x_{3}\\\\\\ x_{1}&-x_{2}\\\\\\ -x_{0}&x_{1}\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}2}{S^{2}}\\,\\xleftarrow{0}\\,\\underset{\\vphantom{\\Big|}3}{0}$\n", "\n", "o12 : ChainComplex\n", "\n", "o13 = $\\left(\\\\!\\begin{array}{ccc} x_{0}&x_{1}&x_{2}\\\\\\ x_{1}&x_{2}&x_{3}\n", "\\end{array}\\\\!\\right)$\n", "\n", "o13 : Matrix $S^{2}\\,\\longleftarrow \\,S^{3}$\n", "\n" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%mode webapp\n", "res C\n", "g" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n" ], "text/plain": [ "o14 = S1<-S3<-S2<-S00123\n", "\n", "o14 : ChainComplex\n", "\n", "o15 = (x0x1x2x1x2x3)\n", "\n", "o15 : Matrix S2<-S3\n", "\n" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%mode texmacs\n", "res C\n", "g" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\r\n", " 1 3 2\r\n", "o16 = S <-- S <-- S <-- 0\r\n", " \r\n", " 0 1 2 3\r\n", "\r\n", "o16 : ChainComplex\r\n", "\r\n", "\r\n", "o17 = | x_0 x_1 x_2 |\r\n", " | x_1 x_2 x_3 |\r\n", "\r\n", " 2 3\r\n", "o17 : Matrix S <-- S\r\n", "\r\n" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%mode standard\n", "res C\n", "g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can preset `mode`, as well as the M2-executable path.\n", "This is done by creating an INI file like so:\n", "```\n", "[magic]\n", "execpath = .../usr-dist/x86_64-Darwin-MacOS-10.14.1/bin/M2-binary\n", "mode = standard\n", "```\n", "You can then tell M2JK to use it by setting the environment variable `$M2JK_CONFIG`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Many more magic command are available. See https://github.com/calysto/metakernel/blob/main/metakernel/magics/README.md." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, more complex language tools are available in the same way they are in an interactive session.\n", "Note, however, that there are limits to the number of character you can transmit.\n", "Additionally, printing to file (including STDOUT, STDERR) is an extremely costly operation,\n", "so timeouts may occur." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\r\n", "o18 = SchurRings\r\n", "\r\n", "o18 : Package\r\n", "\r\n", "\r\n", "o19 = S\r\n", "\r\n", "o19 : SchurRing\r\n", "\r\n", "\r\n", "o20 = s\r\n", " 2,1,1\r\n", "\r\n", "o20 : S\r\n", "\r\n", "\r\n", "\r\n", "\r\n", "o23 = /usr/share/Macaulay2/Core/option.m2:15:14-18:8: --source code:\r\n", " (opts,f) -> args -> (\r\n", " -- Common code for functions created with >> to process options and arguments.\r\n", " uncurry(f, override (opts,args))\r\n", " )\r\n", " | symbol class value location of symbol\r\n", " | ------ ----- ----- ------------------ \r\n", " | f FunctionClosure FunctionClosure[/usr/share/Macaulay2/Core/methods.m2:1. /usr/share/Macaulay2/Core/option.m2:15:8-15:9\r\n", " | opts OptionTable OptionTable{EHPVariables => (e, h, p)} /usr/share/Macaulay2/Core/option.m2:15:3-15:7\r\n", " | GroupActing => GL \r\n", " | SVariable => s \r\n", " | -- function f:\r\n", " | /usr/share/Macaulay2/Core/methods.m2:154:67-154:105: --source code:\r\n", " | methodFunction := new MethodFunctionWithOptions from (opts >> o -> arg -> innerMethodFunction(o,arg));\r\n", " | | symbol class value location of symbol\r\n", " | | ------ ----- ----- ------------------ \r\n", " | | opts OptionTable OptionTable{EHPVariables => (e, h, p)} /usr/share/Macaulay2/Core/methods.m2:151:37-151:41\r\n", " | | GroupActing => GL \r\n", " | | SVariable => s \r\n", " | | innerMethodFunction CompiledFunctionClosure CompiledFunctionClosure[] /usr/share/Macaulay2/Core/methods.m2:153:11-153:30\r\n", " | | outputs List {false, false, false, false} /usr/share/Macaulay2/Core/methods.m2:151:42-151:49\r\n", " | | methopts OptionTable OptionTable{Binary => false . /usr/share/Macaulay2/Core/methods.m2:151:28-151:36\r\n", " | | Dispatch => {Thing, Thing, Thing, Thing} . \r\n", " | | Options => {EHPVariables => (e, h, p), SVa. \r\n", " | | TypicalValue => Thing . \r\n", " | | methodFunction MethodFunctionWithOptions schurRing /usr/share/Macaulay2/Core/methods.m2:154:5-154:19 \r\n", " | -- option table opts:\r\n", " | OptionTable{EHPVariables => (e, h, p)}\r\n", " | GroupActing => GL\r\n", " | SVariable => s\r\n", "\r\n" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "loadPackage(\"SchurRings\")\n", "S = schurRing(QQ,s,4)\n", "p = S_(2,1,1)\n", "debug Core\n", "assert(rawCompare(raw p, raw p)==0)\n", "code schurRing" ] } ], "metadata": { "kernelspec": { "display_name": "Macaulay2", "language": "", "name": "m2" }, "language_info": { "file_extension": ".m2", "mimetype": "text/x-macaulay2", "name": "Macaulay2" } }, "nbformat": 4, "nbformat_minor": 4 } Macaulay2-Jupyter-Kernel-0.8.2/demo/m2book.ipynb 0000664 0000000 0000000 00000027056 15216244364 0021416 0 ustar 00root root 0000000 0000000 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "*Macaulay2* is a software system for research in algebraic geometry by Daniel R. Grayson and Michael E. Stillman.\n", "It is available online in source code form and compiled for various architectures at https://macaulay2.com/.\n", "\n", "What follows is a reproduction of the code (and some text) from the preface to\n", "\n", "> “Computations in algebraic geometry with *Macaulay 2*”,\n", "David Eisenbud,\n", "Daniel R. Grayson,\n", "Michael Stillman, and\n", "Bernd Sturmfels (Eds.) (2002) (DOI: 10.1007/978-3-662-04851-1)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o1 = $\\frac{68}{55}$
\n", "o1 : ${\\mathbb Q}$
\n" ], "text/plain": [ "o1 = $\\frac{68}{55}$\n", "\n", "o1 : ${\\mathbb Q}$\n", "\n" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "3/5 + 7/11" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o2 = $93\\,326\\,215\\,443\\,944\\,152\\,681\\,699\\,238\\,856\\,266\\,700\\,490\\,715\\,968\\,264\\,381\\,621\\,468\\,592\\,963\\,895\\,217\\,599\\,993\\,229\\,915\\,608\\,941\\,463\\,976\\,156\\,518\\,286\\,253\\,697\\,920\\,827\\,223\\,758\\,251\\,185\\,210\\,916\\,864\\,000\\,000\\,000\\,000\\,000\\,000\\,000\\,000$
\n" ], "text/plain": [ "o2 =\n", "$93\\,326\\,215\\,443\\,944\\,152\\,681\\,699\\,238\\,856\\,266\\,700\\,490\\,715\\,968\\,264\\,381\\,621\\,468\\,592\\,963\\,895\\,217\\,599\\,993\\,229\\,915\\,608\\,941\\,463\\,976\\,156\\,518\\,286\\,253\\,697\\,920\\,827\\,223\\,758\\,251\\,185\\,210\\,916\\,864\\,000\\,000\\,000\\,000\\,000\\,000\\,000\\,000$\n", "\n" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "100!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let’s give a hint of the breadth of types of mathematical objects available in *Macaulay2* with some examples. In *Macaulay2* one defines a quotient ring of a polynomial ring $R$ over the rational numbers by entering a command such as the one below." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o3 = $R$
\n", "o3 : QuotientRing
\n" ], "text/plain": [ "o3 = $R$\n", "\n", "o3 : QuotientRing\n", "\n" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = QQ[x,y,z]/(x^3-y^3-z^3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Having done that, we can compute in the ring." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o4 = $3\\,x^{2}y+3\\,x\\,y^{2}+2\\,y^{3}+3\\,x^{2}z+6\\,x\\,y\\,z+3\\,y^{2}z+3\\,x\\,z^{2}+3\\,y\\,z^{2}+2\\,z^{3}$
\n", "o4 : $R$
\n" ], "text/plain": [ "o4 =\n", "$3\\,x^{2}y+3\\,x\\,y^{2}+2\\,y^{3}+3\\,x^{2}z+6\\,x\\,y\\,z+3\\,y^{2}z+3\\,x\\,z^{2}+3\\,y\\,z^{2}+2\\,z^{3}$\n", "\n", "o4 : $R$\n", "\n" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x+y+z)^3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make matrices over the ring." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o5 = $\\left(\\!\\begin{array}{ccc}\r\n", "x&y&z\r\n", "\\end{array}\\!\\right)$
\n", "o5 : Matrix $R^{1}\\,\\longleftarrow \\,R^{3}$
\n" ], "text/plain": [ "o5 = $\\left(\\\\!\\begin{array}{ccc} x&y&z \\end{array}\\\\!\\right)$\n", "\n", "o5 : Matrix $R^{1}\\,\\longleftarrow \\,R^{3}$\n", "\n" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = vars R" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o6 = $\\left(\\!\\begin{array}{ccc}\r\n", "x^{2}&y^{2}&z^{2}\r\n", "\\end{array}\\!\\right)$
\n", "o6 : Matrix $R^{1}\\,\\longleftarrow \\,R^{3}$
\n" ], "text/plain": [ "o6 = $\\left(\\\\!\\begin{array}{ccc} x^{2}&y^{2}&z^{2} \\end{array}\\\\!\\right)$\n", "\n", "o6 : Matrix $R^{1}\\,\\longleftarrow \\,R^{3}$\n", "\n" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c = matrix {{x^2,y^2,z^2}}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make modules over the ring." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o7 = $\\texttt{cokernel}\\ \\left(\\!\\begin{array}{ccc}\r\n", "x&y&z\r\n", "\\end{array}\\!\\right)$
\n", "o7 : $R$-module, quotient of $R^{1}$
\n" ], "text/plain": [ "o7 = $\\texttt{cokernel}\\ \\left(\\\\!\\begin{array}{ccc} x&y&z\n", "\\end{array}\\\\!\\right)$\n", "\n", "o7 : $R$-module, quotient of $R^{1}$\n", "\n" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = coker b" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "o8 = $\\texttt{image}\\ \\begin{array}{l}\\left\\{2\\right\\}\\vphantom{x-y^{2}0-z^{2}}\\\\\\left\\{2\\right\\}\\vphantom{-yx^{2}-z^{2}0}\\\\\\left\\{2\\right\\}\\vphantom{-z0y^{2}x^{2}}\\end{array}\\left(\\!\\begin{array}{cccc}\r\n", "\\vphantom{\\left\\{2\\right\\}}x&-y^{2}&0&-z^{2}\\\\\r\n", "\\vphantom{\\left\\{2\\right\\}}-y&x^{2}&-z^{2}&0\\\\\r\n", "\\vphantom{\\left\\{2\\right\\}}-z&0&y^{2}&x^{2}\r\n", "\\end{array}\\!\\right)$
\n", "o8 : $R$-module, submodule of $R^{3}$
\n" ], "text/plain": [ "o8 = $\\texttt{image}\\\n", "\\begin{array}{l}\\left\\\\{2\\right\\\\}\\vphantom{x-y^{2}0-z^{2}}\\\\\\\\\\left\\\\{2\\right\\\\}\\vphantom{-yx^{2}-z^{2}0}\\\\\\\\\\left\\\\{2\\right\\\\}\\vphantom{-z0y^{2}x^{2}}\\end{array}\\left(\\\\!\\begin{array}{cccc}\n", "\\vphantom{\\left\\\\{2\\right\\\\}}x&-y^{2}&0&-z^{2}\\\\\\\n", "\\vphantom{\\left\\\\{2\\right\\\\}}-y&x^{2}&-z^{2}&0\\\\\\\n", "\\vphantom{\\left\\\\{2\\right\\\\}}-z&0&y^{2}&x^{2} \\end{array}\\\\!\\right)$\n", "\n", "o8 : $R$-module, submodule of $R^{3}$\n", "\n" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = ker c" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make projective resolutions of modules." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o9 = $\\underset{\\vphantom{\\Big|}0}{R^{1}}\\,\\xleftarrow{\\left(\\!\\begin{array}{ccc}\r\n", "x&y&z\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}1}{R^{3}}\\,\\xleftarrow{\\left(\\!\\begin{array}{cccc}\r\n", "0&-z&-y&x^{2}\\\\\r\n", "-z&0&x&-y^{2}\\\\\r\n", "y&x&0&-z^{2}\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}2}{R^{4}}\\,\\xleftarrow{\\left(\\!\\begin{array}{cccc}\r\n", "x&-y^{2}&z^{2}&0\\\\\r\n", "-y&x^{2}&0&z^{2}\\\\\r\n", "z&0&x^{2}&y^{2}\\\\\r\n", "0&z&y&x\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}3}{R^{4}}\\,\\xleftarrow{\\left(\\!\\begin{array}{cccc}\r\n", "-x^{2}&-y^{2}&z^{2}&0\\\\\r\n", "-y&-x&0&-z^{2}\\\\\r\n", "z&0&-x&-y^{2}\\\\\r\n", "0&z&y&x^{2}\r\n", "\\end{array}\\!\\right)}\\,\\underset{\\vphantom{\\Big|}4}{R^{4}}$
\n", "o9 : ChainComplex
\n" ], "text/plain": [ "o9 =\n", "$\\underset{\\vphantom{\\Big|}0}{R^{1}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{ccc}\n", "x&y&z\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}1}{R^{3}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{cccc}\n", "0&-z&-y&x^{2}\\\\\\ -z&0&x&-y^{2}\\\\\\ y&x&0&-z^{2}\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}2}{R^{4}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{cccc}\n", "x&-y^{2}&z^{2}&0\\\\\\ -y&x^{2}&0&z^{2}\\\\\\ z&0&x^{2}&y^{2}\\\\\\ 0&z&y&x\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}3}{R^{4}}\\,\\xleftarrow{\\left(\\\\!\\begin{array}{cccc}\n", "-x^{2}&-y^{2}&z^{2}&0\\\\\\ -y&-x&0&-z^{2}\\\\\\ z&0&-x&-y^{2}\\\\\\ 0&z&y&x^{2}\n", "\\end{array}\\\\!\\right)}\\,\\underset{\\vphantom{\\Big|}4}{R^{4}}$\n", "\n", "o9 : ChainComplex\n", "\n" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "res M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make projective varieties." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o10 = $X$
\n", "o10 : ProjectiveVariety
\n" ], "text/plain": [ "o10 = $X$\n", "\n", "o10 : ProjectiveVariety\n", "\n" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = Proj R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make coherent sheaves and compute their cohomology." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "o11 = ${\\mathbb Q}^{1}$
\n", "o11 : ${\\mathbb Q}$-module, free
\n" ], "text/plain": [ "o11 = ${\\mathbb Q}^{1}$\n", "\n", "o11 : ${\\mathbb Q}$-module, free\n", "\n" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "HH^1 cotangentSheaf X" ] } ], "metadata": { "kernelspec": { "display_name": "Macaulay2", "language": "", "name": "m2" }, "language_info": { "file_extension": ".m2", "mimetype": "text/x-macaulay2", "name": "Macaulay2" } }, "nbformat": 4, "nbformat_minor": 4 } Macaulay2-Jupyter-Kernel-0.8.2/demo/screenshot1.png 0000664 0000000 0000000 00002723355 15216244364 0022135 0 ustar 00root root 0000000 0000000 PNG IHDR L JiCCPICC Profile HWXS[RIhH RK E*I ĐD.*v*Z Y+ZZ (bʛX{;79sJ;N O*Euȓ#BXRX. &Rv\\42t\uQrs@(A! > ^ @O@XYj\j\I@ 2Ǔe B~ѾD C8/ xT^ %v!+qf sxYX]Jȡb47l\P ;8h"Ydfط[93q$#&b}ߋ*{QHGMr*FAl q$7&Zs!+-p5 glF|ΔqLWiZ CoE)1j89bm( fS, m J"BشLYx^' [,sc4@7Y(a' 哢jCԵcBI^SZ}%͍TanRo0AcqMp'?x9y!u߆ǖ.
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