From ead8d8244b84505897488f0d1eb3989bc109da80 Mon Sep 17 00:00:00 2001 From: zeramorphic <50671761+zeramorphic@users.noreply.github.com> Date: Thu, 18 Jan 2024 12:25:31 +0000 Subject: [PATCH] Lectures 01A --- iii/gc/01_definitions_and_resolutions.tex | 148 ++++++++++++++++++++++ iii/gc/main.tex | 6 +- 2 files changed, 152 insertions(+), 2 deletions(-) create mode 100644 iii/gc/01_definitions_and_resolutions.tex diff --git a/iii/gc/01_definitions_and_resolutions.tex b/iii/gc/01_definitions_and_resolutions.tex new file mode 100644 index 0000000..ed3d21d --- /dev/null +++ b/iii/gc/01_definitions_and_resolutions.tex @@ -0,0 +1,148 @@ +\subsection{???} +Let \( G \) be a group. +\begin{definition} + The \emph{integral group ring} \( \mathbb Z G \) is the set of formal sums \( \sum n_g g \), where \( n_g \in \mathbb Z \), \( g \in G \), and only finitely many of the \( n_g \) are nonzero. + An addition operation makes this set a free abelian group: + \[ \qty(\sum m_g g) + \qty(\sum n_g g) = \sum (m_g + n_g) g \] + Multiplication is defined by + \[ \qty(\sum_{h \in G} m_h h)(\sum_{k \in G} n_k k) = \sum \qty(\sum_{hk = g} m_h n_k) g \] + The multiplicative identity is \( 1 e \) where \( e \) is the identity of \( G \). + This produces an associative ring, which underlies the integral representation theory of \( G \). +\end{definition} +\begin{definition} + A \emph{(left) \( \mathbb Z G \)-module} \( M \) is an abelian group under addition together with a map \( \mathbb Z G \times M \to M \) denoted \( (r, m) \mapsto rm \), satisfying + \begin{enumerate} + \item \( r(m_1 + m_2) = rm_1 + rm_2 \); + \item \( (r_1 + r_2)m = r_1 m + r_2 m \); + \item \( r_1(r_2 m) = (r_1 r_2) m \); + \item \( 1 m = m \). + \end{enumerate} +\end{definition} +A module is \emph{trivial} if \( gm = m \) for all \( g \in G \) and \( m \in M \). +We call \( \mathbb Z \) \emph{the} trivial module, given by the trivial action \( gn = n \) for all \( n \in \mathbb Z \) and \( g \in G \). + +The \emph{free} \( \mathbb Z G \)-module on a set \( X \) is the module of formal sums \( \sum r_x x \) where \( r_x \in \mathbb Z G \) and \( x \in X \), and only finitely many of the \( r_x \) are nonzero. +This has the obvious \( G \)-action. +This module will be denoted \( \mathbb Z G\qty{X} \). + +We can define submodules, quotient modules, and so on as one would expect. + +\begin{definition} + A \emph{(left) \( \mathbb Z G \)-map} or \emph{morphism} \( \alpha : M_1 \to M_2 \) is a map of abelian groups with \( \alpha(r m) = r \alpha(m) \) for all \( r \in \mathbb Z G \) and \( m \in M_1 \). +\end{definition} +\begin{example} + The \emph{augmentation map} \( \varepsilon : \mathbb Z G \to \mathbb Z \) is the \( \mathbb Z G \)-map between left \( \mathbb Z G \)-modules given by + \[ \sum n_g g \mapsto \sum n_g \] + This is also a right \( \mathbb Z G \)-map, and also a map of rings. +\end{example} +We will write \( \Hom_G(M, N) \) to be the set of \( \mathbb Z G \)-maps \( M \to N \), which is made into an abelian group under pointwise addition. +\begin{example} + Regarding \( \mathbb Z G \) as a left \( \mathbb Z G \)-module, then + \[ \Hom_G(\mathbb Z G, M) \cong M \] + for any left \( \mathbb Z G \)-module \( M \). + This isomorphism is given by \( \varphi \mapsto \varphi(1) \); the \( \mathbb Z G \)-map is determined by the image of \( 1 \). + \[ \varphi(r) = \varphi(r \cdot 1) = r \varphi(1) \] +\end{example} +Note that \( \Hom_G(\mathbb Z G, M) \) can be viewed as a left \( \mathbb Z G \)-module, given by +\[ (s \varphi)(r) = \varphi(rs);\quad s \in \mathbb Z G \] +Note that the isomorphism +\[ \Hom_G(\mathbb Z G, \mathbb Z G) \cong \mathbb Z G;\quad \varphi \mapsto \varphi(1) \] +satisfies \( \varphi(r) = r \varphi(1) \) and so \( \varphi \) corresponds to multiplication on the right by \( \varphi(1) \). +\begin{remark} + \( G \) may not be abelian, and so we must carefully distinguish left and right actions. +\end{remark} +\begin{definition} + If \( f : M_1 \to M_2 \) is a \( \mathbb Z G \)-map, its \emph{dual maps} \( f^\star \) are \( \mathbb Z G \)-maps \( \Hom_G(M_2, N) \to \Hom_G(M_1, N) \) for each \( \mathbb Z G \)-module \( N \), given by composition on the right with \( f \). + If \( f : N_1 \to N_2 \), its \emph{induced maps} \( f_\star \) are \( \Hom_G(M, N_1) \to \Hom_G(M, N_2) \) given by composition on the left with \( f \). + These are maps of abelian groups. +\end{definition} +We will now present a prototypical example. +\begin{example} + Let \( G = \langle t \rangle \) be an infinite cyclic group. + Consider the graph whose vertices are \( v_i \) for \( i \in \mathbb Z \), where \( v_i \) is joined to \( v_{i+1} \) and \( v_{i-1} \). + Let \( V \) be its set of vertices, and \( E \) be its set of edges. + \( G \) acts by translations on this graph, where \( t \) maps \( v_i \) to \( v_{i+1} \). + The formal sums \( \mathbb Z V \) and \( \mathbb Z E \) can be regarded as \( \mathbb Z G \)-modules. + They are free: \( \mathbb Z V = \mathbb Z G \qty{v_0} \), and \( \mathbb Z E = \mathbb Z G \qty{e} \) where \( e \) is the edge connecting \( v_0 \) and \( v_1 \). + The boundary map is a \( \mathbb Z G \)-map \( d : \mathbb Z E \to \mathbb Z V \) given by \( e \mapsto v_1 - v_0 \). + There is also a \( \mathbb Z G \)-map \( \mathbb Z V \to \mathbb Z \) given by \( v_0 \mapsto 1 \); this corresponds to the augmentation map. +\end{example} +\begin{definition} + A \emph{chain complex} of \( \mathbb Z G \)-modules is a sequence + % https://q.uiver.app/#q=WzAsNSxbMCwwLCJNX3MiXSxbMSwwLCJNX3tzLTF9Il0sWzIsMCwiTV97cy0yfSJdLFszLDAsIlxcY2RvdHMiXSxbNCwwLCJNX3QiXSxbMCwxLCJkX3MiXSxbMSwyLCJkX3tzLTF9Il0sWzIsM10sWzMsNCwiZF97dCsxfSJdXQ== + \[\begin{tikzcd} + {M_s} & {M_{s-1}} & {M_{s-2}} & \cdots & {M_t} + \arrow["{d_s}", from=1-1, to=1-2] + \arrow["{d_{s-1}}", from=1-2, to=1-3] + \arrow[from=1-3, to=1-4] + \arrow["{d_{t+1}}", from=1-4, to=1-5] + \end{tikzcd}\] + such that for every \( t < n < s \), we have \( d_n d_{n+1} = 0 \), and so \( \im d_{n+1} \subseteq \ker d_n \). + We will refer to the entire sequence as \( M_\bullet = (M_n, d_n)_{t \leq n \leq s} \). +\end{definition} +We say that \( M_\bullet \) is \emph{exact} at \( M_n \) if \( \im d_{n+1} = \ker d_n \), and we say it is \emph{exact} if it is exact at all \( M_n \) for \( t < n < s \). +The \emph{homology} of this chain complex is +\[ H_s(M_\bullet) = \ker d_s;\quad H_n(M_\bullet) = \faktor{\ker d_n}{\im d_{n+1}};\quad H_t(M_\bullet) = \coker d_{t-1} = \faktor{M_t}{\im d_{t-1}} \] +A \emph{short exact sequence} is an exact chain complex of the form +% https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiTV8xIl0sWzIsMCwiTV8yIl0sWzMsMCwiTV8zIl0sWzQsMCwiMCJdLFswLDFdLFsxLDIsIlxcYWxwaGEiXSxbMiwzLCJcXGJldGEiXSxbMyw0XV0= +\[\begin{tikzcd} +0 & {M_1} & {M_2} & {M_3} & 0 +\arrow[from=1-1, to=1-2] +\arrow["\alpha", from=1-2, to=1-3] +\arrow["\beta", from=1-3, to=1-4] +\arrow[from=1-4, to=1-5] +\end{tikzcd}\] +That is, \( \alpha \) is injective, \( \beta \) is surjective, and \( \im \alpha = \ker \beta \). +\begin{example} + In our example above, we have the short exact sequence + % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiXFxtYXRoYmIgWiBFIl0sWzIsMCwiXFxtYXRoYmIgWiBWIl0sWzMsMCwiXFxtYXRoYmIgWiJdLFs0LDAsIjAiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XV0= +\[\begin{tikzcd} + 0 & {\mathbb Z E} & {\mathbb Z V} & {\mathbb Z} & 0 + \arrow[from=1-1, to=1-2] + \arrow[from=1-2, to=1-3] + \arrow[from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] +\end{tikzcd}\] + This corresponds to a short exact sequence + % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiXFxtYXRoYmIgWiBHIl0sWzIsMCwiXFxtYXRoYmIgWiBHIl0sWzMsMCwiXFxtYXRoYmIgWiJdLFs0LDAsIjAiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XV0= +\[\begin{tikzcd} + 0 & {\mathbb Z G} & {\mathbb Z G} & {\mathbb Z} & 0 + \arrow[from=1-1, to=1-2] + \arrow[from=1-2, to=1-3] + \arrow[from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] +\end{tikzcd}\] + where \( G = \langle t \rangle \) is an infinite cyclic group, and the map \( \mathbb Z G \to \mathbb Z G \) is given by multiplication on the right by \( t - 1 \). +\end{example} +\begin{definition} + A \( \mathbb Z G \)-module \( P \) is \emph{projective} if, for every surjective \( \mathbb Z G \)-map \( \alpha : M_1 \to M_2 \) and every \( \mathbb Z G \)-map \( \beta : P \to M_2 \), there is a map \( \overline\beta : P \to M_1 \) such that \( \alpha \circ \overline \beta = \beta \). + % https://q.uiver.app/#q=WzAsNCxbMSwwLCJQIl0sWzEsMSwiTV8yIl0sWzAsMSwiTV8xIl0sWzIsMSwiMCJdLFswLDEsIlxcYmV0YSJdLFswLDIsIlxcb3ZlcmxpbmVcXGJldGEiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiwxLCJcXGFscGhhIiwyXSxbMSwzXV0= +\[\begin{tikzcd} + & P \\ + {M_1} & {M_2} & 0 + \arrow["\beta", from=1-2, to=2-2] + \arrow["\overline\beta"', dashed, from=1-2, to=2-1] + \arrow["\alpha"', from=2-1, to=2-2] + \arrow[from=2-2, to=2-3] +\end{tikzcd}\] +\end{definition} +Given any short exact sequence +% https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiTiJdLFsyLDAsIk1fMSJdLFszLDAsIk1fMiJdLFs0LDAsIjAiXSxbMCwxXSxbMSwyLCJmIl0sWzIsMywiXFxhbHBoYSJdLFszLDRdXQ== +\[\begin{tikzcd} + 0 & N & {M_1} & {M_2} & 0 + \arrow[from=1-1, to=1-2] + \arrow["f", from=1-2, to=1-3] + \arrow["\alpha", from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] +\end{tikzcd}\] +we can consider +% https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiXFxIb21fRyhQLCBOKSJdLFsyLDAsIlxcSG9tX0coUCwgTV8xKSJdLFszLDAsIlxcSG9tX0coUCwgTV8yKSJdLFs0LDAsIjAiXSxbMCwxXSxbMSwyLCJmX1xcc3RhciJdLFsyLDMsIlxcYWxwaGFfXFxzdGFyIl0sWzMsNF1d +\[\begin{tikzcd} + 0 & {\Hom_G(P, N)} & {\Hom_G(P, M_1)} & {\Hom_G(P, M_2)} & 0 + \arrow[from=1-1, to=1-2] + \arrow["{f_\star}", from=1-2, to=1-3] + \arrow["{\alpha_\star}", from=1-3, to=1-4] + \arrow[from=1-4, to=1-5] +\end{tikzcd}\] +We could have defined projectivity by saying that this new sequence is exact. +Note that this sequence is always a chain complex regardless if \( P \) is projective, and we always have exactness except possibly at \( \Hom_G(P, M_2) \). diff --git a/iii/gc/main.tex b/iii/gc/main.tex index 35318c9..9f991fc 100644 --- a/iii/gc/main.tex +++ b/iii/gc/main.tex @@ -10,7 +10,9 @@ \tableofcontentsnewpage{} -% \section{Definitions and examples} -% \input{01_definitions_and_examples.tex} +\section{Definitions and resolutions} +\input{01_definitions_and_resolutions.tex} + +% ยง5 from https://www.dpmms.cam.ac.uk/~grw46/LectureNotes2021.pdf may be useful \end{document}