Joshua Moerman
10 years ago
23 changed files with 193 additions and 266 deletions
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\input{notes/Algebra} |
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\input{notes/Free_CDGA} |
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\input{notes/Model_Categories} |
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\input{notes/Calculations} |
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\input{notes/Further_Topics} |
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\input{notes/Basics} |
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\input{notes/Serre} |
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\input{notes/Rationalization} |
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We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should remember the topological $n$-simplex defined as convex span. |
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\Definition{apl}{ |
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For all $n \in \N$ define the following cdga: |
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$$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n d x_i)}, $$ |
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where $\deg{x_i} = 0$. So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n d x_i = 0$. |
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} |
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Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $d x_i$). This fact will be used throughout. Also note that we have already seen the dual unit interval $\Lambda(t, dt)$ which is isomorphic to $\Apl_1$. |
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These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$): |
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$$ d_i(x_j) = \begin{cases} |
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x_{j-1}, &\text{ if } i < j \\ |
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0, &\text{ if } i = j \\ |
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x_j, &\text{ if } i > j |
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\end{cases} \qquad d_i : \Apl_n \to \Apl_{n-1} $$ |
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$$ s_i(x_j) = \begin{cases} |
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x_{j+1}, &\text{ if } i < j \\ |
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x_j + x_{j+1}, &\text{ if } i = j \\ |
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x_j, &\text{ if } i > j |
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\end{cases} \qquad s_i : \Apl_n \to \Apl_{n+1} $$ |
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One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k$, this is a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$. |
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\pagebreak |
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\Lemma{apl-contractible}{ |
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$\Apl^k$ is contractible. |
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} |
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\Proof{ |
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We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. In the more geometric context of topological $n$-simplices we would achieve this by dividing by $1-x_0$. However, since this algebra consists of polynomials only, this cannot be done. Instead, we will multiply everything by $(1-x_0)^2$, so that we can divide by $1-x_0$. Define for $i = 1, \ldots, n$: |
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\begin{align*} |
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s(1) &= (1-x_0)^2 \\ |
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s(x_i) &= (1-x_0) \cdot x_{i+1} |
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\end{align*} |
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Extend on the differentials and multiplicatively on $\Apl_n$. As $s(1) \neq 1$ this map is not an algebra map, however it well-defined as a map of cochain complexes. In particular when restricted to degree $k$ we get a linear map: |
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$$ s: \Apl^k_n \to \Apl^k_{n+1}. $$ |
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Proving the necessary properties of an extra degeneracy is fairly easy. For $n \geq 1$ we get (on generators): |
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\begin{align*} |
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d_0 s(1) &= d_0 (1 - x_0)^2 = (1 - 0) \cdot (1 - 0) = 1 \\ |
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d_0 s(x_i) &= d_0((1-x_0)x_{i+1}) = d_0(1-x_0) \cdot x_i \\ |
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&= (1-0) \cdot x_i = x_i |
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\end{align*} |
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So $d_0 s = \id$. |
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\begin{align*} |
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d_{i+1} s(1) &= d_{i+1} (1 - x_0)^2 = d_{i+1} (\sum_{j=1}^n x_j)^2 \\ |
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&= (\sum_{j=1}^{n-1} x_j)^2 = (1-x_0)^2 = s d_i(1) \\ |
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d_{i+1} s(x_j) &= d_{i+1}(1-x_0) d_{i+1}(x_j) = (1-x_0) d_i(x_{j+1}) = s d_i (x_j) |
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\end{align*} |
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So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$. |
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So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl^k \to \ast$ is a weak equivalence. |
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} |
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\Lemma{apl-kan-complex}{ |
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$\Apl^k$ is a Kan complex. |
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} |
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\Proof{ |
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By the simple fact that $\Apl^k$ is a simplicial group, it is a Kan complex \cite{goerss}. |
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} |
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Combining these two lemmas gives us the following. |
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\Corollary{apl-extendable}{ |
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$\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$. |
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} |
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Besides the simplicial structure of $\Apl$, there is also the structure of a cochain complex. |
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\Lemma{apl-acyclic}{ |
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$\Apl_n$ is acyclic, i.e. $H(\Apl_n) = \k \cdot [1]$. |
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} |
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\Proof{ |
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This is clear for $\Apl_0 = \k \cdot 1$. For $\Apl_1$ we see that $\Apl_1 = \Lambda(x_1, d x_1) \iso \Lambda D(0)$, which we proved to be acyclic in the previous section. |
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For general $n$ we can identify $\Apl_n \iso \bigtensor_{i=1}^n \Lambda(x_i, d x_i)$, because $\Lambda$ is left adjoint and hence preserves coproducts. By the Künneth theorem \TheoremRef{kunneth} we conclude $H(\Apl_n) \iso \bigtensor_{i=1}^n H \Lambda(x_i, d x_i) \iso \bigtensor_{i=1}^n H \Lambda D(0) \iso \k \cdot [1]$. |
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So indeed $\Apl_n$ is acyclic for all $n$. |
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} |
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\section{The free cdga} |
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\label{sec:free-cdga} |
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Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation. |
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\begin{definition} |
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The \emph{tensor algebra} of a graded module $M$ is defined as |
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$$ T(M) = \bigoplus_{n\in\N} M^{\tensor n}, $$ |
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where $M^{\tensor 0} = \k$. An element $m = m_1 \tensor \ldots \tensor m_n$ has a \emph{word length} of $n$ and its degree is $\deg{m} = \sum_{i=i}^n \deg{m_i}$. The multiplication is given by the tensor product (note that the bilinearity follows immediately). |
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\end{definition} |
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Note that this construction is functorial and it is free in the following sense. |
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\begin{lemma} |
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Let $M$ be a graded module and $A$ a graded algebra. |
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\begin{itemize} |
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\item A graded map $f: M \to A$ of degree $0$ extends uniquely to an algebra map $\overline{f} : TM \to A$. |
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\item A differential $d: M \to M$ extends uniquely to a derivation $d: TM \to TM$. |
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\end{itemize} |
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\end{lemma} |
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\begin{corollary} |
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Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair: |
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$$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$ |
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Moreover it extends and restricts to |
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$$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$ |
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$$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$ |
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\end{corollary} |
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As with the symmetric algebra and exterior algebra of a vector space, we can turn this graded tensor algebra in a commutative graded algebra. |
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\begin{definition} |
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Let $A$ be a graded algebra and define |
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$$ I = \langle ab - (-1)^{\deg{a}\deg{b}}b a \I a,b \in A \rangle $$ |
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Then $A / I$ is a commutative graded algebra. |
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For a graded module $M$ we define the \emph{free commutative graded algebra} as |
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$$ \Lambda(M) = TM / I $$ |
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\end{definition} |
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Again this extends to differential graded modules (i.e. the ideal is preserved by the derivative) and restricts to cochain complexes. |
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\begin{lemma} |
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We have the following adjunctions: |
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$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$ |
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$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$ |
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$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$ |
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\end{lemma} |
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We can now easily construct cdga's by specifying generators and their differentials. Note that a free algebra has a natural augmentation, defined as $\counit(v) = 0$ for every generator $v$ and $\counit(1) = 1$. |
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\paragraph{Preliminaries and Notation} |
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We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is also assumed, but the reader may review some facts on homological algebra in Appendix \ref{sec:algebra} and facts on model categories in Appendix \ref{sec:model_categories}. |
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We will fix the following notations and categories. |
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\begin{itemize} |
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\item $\k$ will denote a field of characteristic zero. Modules, tensor products, \dots\, are understood as $\k$-vector spaces, tensor products over $\k$, \dots. |
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\item $\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out. |
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\item $\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories. |
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\item $\Ab$: category of abelian groups and group homomorphisms. |
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\item $\sSet$: category of simplicial sets and simplicial maps. More generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$. We have the homotopy equivalence $|-| : \sSet \leftadj \Top : S$ to switch between topological spaces and simplicial sets. |
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\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (as defined in the appendix) and graded algebra maps. As a shorthand we will refer to such an object as \emph{dga}. Furthermore $\CDGA_\k$ is the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s). |
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\end{itemize} |
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% lesser margins |
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% \usepackage{geometry} |
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% \geometry{a4paper} |
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% \geometry{twoside=false} |
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% no indent, but vertical spacing |
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% \usepackage[parfill]{parskip} |
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% \setlength{\marginparwidth}{2cm} |
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\documentclass[a4paper,12pt,footinclude=true,headinclude=true,oneside,dottedtoc]{scrbook} |
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\usepackage{amsmath, amsthm} |
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\usepackage[T1]{fontenc} |
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% use english, does not work with classicthesis |
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% \usepackage{polyglossia} |
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% \setmainlanguage[variant=british]{english} |
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\usepackage[parts,drafting,eulerchapternumbers]{classicthesis} |
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\setcounter{tocdepth}{0} % parts, chapters |
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\input{preamble} |
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\begin{document} |
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% \pagenumbering{none} |
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\[ f \quad \frac{1}{2}f \quad \frac{1}{6}f \quad S^1_\Q = \quad X \cdots \] |
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\[ \xymatrix{ A \ar[rr] & & B } \] |
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\end{document} |
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#!/bin/bash |
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set -e |
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file=$1 |
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sed "s|__INPUT__|$file|" test_diagram.tex | xelatex -file-line-error -output-directory=build |
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mv build/texput.pdf test_diagram.pdf |
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#exo-open test_diagram.pdf |
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rm build/texput.* |
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\documentclass[a4paper, 12pt]{amsart} |
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\input{style} |
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\input{preamble} |
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\begin{document} |
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Your diagram: |
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\cdiagrambase{__INPUT__} |
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For reference, some default latex stuff: |
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$$ A \cof B \we X \fib Y $$ |
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\newcommand{\newk}{\mathrm{I\!k}} |
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$\newk$-linear, stuf over $\newk$ |
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\[ \k \newk \] |
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\[ \CDGA_\k \CDGA_\newk \] |
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\end{document} |
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