Add results on tensor products and slice categories

Expanded the section on tensor products of algebras with a full proof and corollary for quotient algebras, including the isomorphism between A/IA and A ⊗_R (R/I). In category theory, added results and proofs about limits and colimits in slice categories, initial and terminal objects, completeness and cocompleteness, and a proposition on fibered products and coproducts with explicit isomorphisms and proofs.

Author: hooyuser

associative_algebra.tex

@@ -90,7 +90,7 @@ \subsection{Graded Object}
 \end{example}
 
 \subsection{Tensor Product}
-\begin{definition}{Tensor Product of Algebras}{}
+\begin{definition}{Tensor Product of Algebras}{tensor_product_of_algebras}
     Let $R$ be a commutative ring and $A$, $B$ be $R$-algebras. The \textbf{tensor product of $R$-algebras $A$ and $B$} is defined by the following universal property: for any triple $(C, f_A, f_B)$, where $C$ is an $R$-algebra and $f_A:A\to C$, $f_B:B\to C$ are $R$-algebra homomorphisms which satisfy
     \[
         f_A(a)f_B(b)=f_B(b)f_A(a),\quad \forall a\in A, b\in B,
@@ -196,13 +196,101 @@ \subsection{Tensor Product}
     \end{enumerate}
 \end{proposition}
 
-\begin{proposition}{}{}
+\begin{proposition}{Tensor Product of Quotient Algebras}{tensor_product_of_quotient_algebras}
     Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Let $I_1\subseteq A_1$, $I_2\subseteq A_2$ be two-sided ideals of $A_1$, $A_2$ respectively. Then we have an $R$-algebra isomorphism
+    \begin{align*}
+        (A_1/I_1)\otimes_R (A_2/I_2) &\xlongrightarrow{\sim}(A_1\otimes_R A_2)/(I_1\otimes_R A_2 + A_1\otimes_R I_2)\\
+        \overline{a_1}\otimes \overline{a_2} &\longmapsto \overline{a_1\otimes a_2}
+    \end{align*}
+    Here, given inclusion $i_1: I_1\hookrightarrow A_1$, the $R$-module $I_1\otimes_R A_2$ is identified as the image of $i_1\otimes_R \mathrm{id}_{A_2}:I_1\otimes_R A_2 \to A_1\otimes_R A_2$
     \[
-        (A_1\otimes_R A_2)/(I_1\otimes_R A_2 + A_1\otimes_R I_2) \cong (A_1/I_1)\otimes_R (A_2/I_2).
+        \mathrm{im}\left(i_1\otimes_R \mathrm{id}_{A_2}\right)=\left\{\sum_{n=1}^m x_n\otimes y_n\in A_1\otimes_R A_2\midv m\in\mathbb{Z}_{\ge 1},\,x_n\in I_1,\, y_n\in A_2\right\},
     \]
-    
+    which is a two-sided ideal of $A_1\otimes_R A_2$. The similar identification applies to $A_1\otimes_R I_2$. 
 \end{proposition}
+\begin{prf}
+    Let $J:=I_1\otimes_R A_2 + A_1\otimes_R I_2$. Define
+    \begin{align*}
+        \iota_1: A_1/I_1 &\longrightarrow (A_1\otimes_R A_2)/J\\
+        a_1 + I_1 &\longmapsto (a_1\otimes 1_{A_2}) + J,
+    \end{align*}
+    If $a_1, a_1'\in A_1$ satisfy $a_1-a_1'\in I_1$, then 
+    \[
+    (a_1\otimes 1_{A_2}) - (a_1'\otimes 1_{A_2}) = (a_1 - a_1')\otimes 1_{A_2} \in I_1\otimes_R A_2 \subseteq J\implies (a_1\otimes 1_{A_2}) + J = (a_1'\otimes 1_{A_2}) + J,
+    \]
+    which shows that $\iota_1$ is well-defined. And we can check that $\iota_1$ is an $R$-algebra homomorphism. Similarly, we can define an $R$-algebra homomorphism
+    \begin{align*}
+        \iota_2: A_2/I_2 &\longrightarrow (A_1\otimes_R A_2)/J\\
+        a_2 + I_2 &\longmapsto (1_{A_1}\otimes a_2) + J.
+    \end{align*}
+    Moreover, the images of $\iota_1$ and $\iota_2$ commute in $(A_1\otimes_R A_2)/J$: for any $a_1\in A_1$, $a_2\in A_2$,
+    \begin{align*}
+        \iota_1(a_1 + I_1)\iota_2(a_2 + I_2) &= \left((a_1\otimes 1_{A_2}) + J\right)\left((1_{A_1}\otimes a_2) + J\right)\\
+        &= (a_1\otimes a_2) + J\\
+        &= \left((1_{A_1}\otimes a_2) + J\right)\left((a_1\otimes 1_{A_2}) + J\right) \\
+        &= \iota_2(a_2 + I_2)\iota_1(a_1 + I_1).
+    \end{align*}   
+    Thus by the \hyperref[th:tensor_product_of_algebras]{universal property of tensor product}, there exists a unique $R$-algebra homomorphism
+    \begin{align*}
+        \varphi: (A_1/I_1)\otimes_R (A_2/I_2) &\longrightarrow (A_1\otimes_R A_2)/J\\
+        (a_1 + I_1)\otimes (a_2 + I_2) &\longmapsto (a_1\otimes a_2) + J.
+    \end{align*}
+    Next, we construct the inverse of $\varphi$. Given the quotient maps $\pi_1:A_1\to A_1/I_1$ and $\pi_2:A_2\to A_2/I_2$, we can define an $R$-algebra homomorphism $\psi:=\pi_1\otimes_R \pi_2$ as
+    \begin{align*}
+        \psi: A_1\otimes_R A_2 &\longrightarrow (A_1/I_1)\otimes_R (A_2/I_2)\\
+        a_1\otimes a_2 &\longmapsto (a_1 + I_1)\otimes (a_2 + I_2).
+    \end{align*}
+    Since for any $x\in I_1$, $y\in A_2$, we have
+    \[
+        \psi(x\otimes y)=(0 + I_1)\otimes (y + I_2)=0
+    \]
+    and for any $x\in A_1$, $y\in I_2$, we have
+    \[
+        \psi(x\otimes y)=(x + I_1)\otimes (0 + I_2)=0,
+    \]
+    we have $J\subseteq \ker(\psi)$. Thus, by the universal property of quotient algebra, there exists a unique $R$-algebra homomorphism
+    \begin{align*}
+        \widetilde{\psi}: (A_1\otimes_R A_2)/J &\longrightarrow (A_1/I_1)\otimes_R (A_2/I_2)\\
+        (a_1\otimes a_2) + J &\longmapsto (a_1 + I_1)\otimes (a_2 + I_2).
+    \end{align*}
+    We can check that $\widetilde{\psi}$ is the inverse of $\varphi$:
+    \begin{align*}
+        \widetilde{\psi}\circ \varphi\left((a_1 + I_1)\otimes (a_2 + I_2)\right) &= \widetilde{\psi}\left((a_1\otimes a_2) + J\right) = (a_1 + I_1)\otimes (a_2 + I_2),\\
+        \varphi\circ \widetilde{\psi}\left((a_1\otimes a_2) + J\right) &= \varphi\left((a_1 + I_1)\otimes (a_2 + I_2)\right) = (a_1\otimes a_2) + J.
+    \end{align*}
+    Therefore, we show that $\varphi$ is an $R$-algebra isomorphism.
+\end{prf}
+
+\begin{corollary}{$mod\; I$ Reduction of $R$-Algebras}{}
+    Let $R$ be a commutative ring and $I\subseteq R$ be an ideal of $R$. For any $R$-algebra $A$, there is an isomorphism of $R$-algebras
+    \begin{align*}
+        A/IA &\xlongrightarrow{\sim} A\otimes_R (R/I) \\
+        \overline{a} &\longmapsto  a\otimes \overline{1_R}\\
+        \overline{ra}&\longmapsfrom a\otimes \overline{r}
+    \end{align*}
+    where
+    \[
+    IA:= \left\{ r a \in A\midv r\in I,\, a\in A\right\}
+    \]
+    is the two-sided ideal of $A$ generated by $I$.
+\end{corollary}
+\begin{prf}
+    Apply \Cref{th:tensor_product_of_quotient_algebras} with $A_1=A$, $A_2=R$, $I_1=\{0\}$, $I_2=I$. We obtain an $R$-algebra isomorphism
+    \begin{align*}
+        A \otimes_R (R/I) &\xlongrightarrow{\sim} (A\otimes_R R)/( 0\otimes_R R+A\otimes_R I)\\
+        a \otimes (r+I) &\longmapsto (a\otimes r) + (A\otimes_R I).
+    \end{align*}
+    Under the canonical isomorphism 
+    \begin{align*}
+        \phi:  A\otimes_R R &\xlongrightarrow{\sim} A\\
+        a\otimes r &\longmapsto r a,
+    \end{align*}
+    $A \otimes_R I$ is mapped to $IA$. Thus we have an $R$-algebra isomorphism
+    \begin{align*}
+        A \otimes_R (R/I) &\xlongrightarrow{\sim} A/IA\\
+        a \otimes (r+I) &\longmapsto r a + IA.
+    \end{align*}
+\end{prf}
 
 \subsection{Tensor Algebra}
 \begin{definition}{Tensor Algebra $T^{\bullet}(M)$}{}

category_theory.tex

@@ -462,7 +462,7 @@ \subsection{Slice Category}
     Suppose $\mathsf{C}$ is a category with terminal object $\bullet$. The coslice category $\left(\bullet / \mathsf{C}\right)$ is called the \textbf{category of pointed objects} of $\mathsf{C}$ and is denoted by $\mathsf{C}_\bullet$.
 \end{definition}
 
-\begin{proposition}{}{}
+\begin{proposition}{Limits and Colimits in Slice Categories}{limits_and_colimits_in_slice_categories}
     Let $\mathsf{C}$ be a category. Fix some object $X\in\mathrm{Ob}\left(\mathsf{C}\right)$. Then
     \begin{enumerate}[(i)]
         \item The forgetful functor $U: \mathsf{C}/X \to \mathsf{C}$ \hyperref[th:preserve_reflect_create_limits]{strictly creates} all colimits. If $\varinjlim (U\circ F)$ exists for some diagram $F:\mathsf{J}\to X/\mathsf{C}$, then $\varinjlim F$ exists and we have natural isomorphism
@@ -473,6 +473,14 @@ \subsection{Slice Category}
         \[
             U\left( \varprojlim F\right) \cong \varprojlim (U\circ F).
         \]
+        \item If 
+        \[
+        \left(\varprojlim U\circ F\right)\times_{\varprojlim_{j\in{\mathsf{J}}}X} X
+        \]
+        exists for some diagram $F:\mathsf{J}\to \mathsf{C}/X$, then $\varprojlim F$ exists and can be given as 
+        \[
+             \left(\varprojlim U \circ F\right)\times_{\varprojlim_{j\in{\mathsf{J}}}X} X \longrightarrow X.
+        \]
     \end{enumerate}
 \end{proposition}
 
@@ -625,6 +633,26 @@ \subsection{Slice Category}
     \end{enumerate}
 \end{prf}
 
+\begin{corollary}{Initial and Terminal Objects in Slice Categories}{}
+      Let $\mathsf{C}$ be a category and fix some object $X\in\mathrm{Ob}\left(\mathsf{C}\right)$. 
+     \begin{itemize}
+        \item If $\mathsf{C}$ has an initial object $\bot$, then the slice category $\mathsf{C}/X$ has an initial object $\left(\bot \to X\right)$. 
+        \item The slice category $\mathsf{C}/X$ has a terminal object $\mathrm{id}_{X}:X \to X$.
+        \item If $\mathsf{C}$ has a terminal object $\top$, then we have a natural isomorphism of categories $\mathsf{C}/\top \cong \mathsf{C}$.
+    \end{itemize}
+\end{corollary}
+
+
+\begin{corollary}{}{}
+    Let $\mathsf{C}$ be a category and fix some object $X\in\mathrm{Ob}\left(\mathsf{C}\right)$. 
+    \begin{itemize}
+        \item If $\mathsf{C}$ is complete, then the slice category $\mathsf{C}/X$ is also complete. 
+        \item  If $\mathsf{C}$ is cocomplete, then the slice category $\mathsf{C}/X$ is also cocomplete.
+    \end{itemize}
+\end{corollary}
+\begin{prf}
+    According to \Cref{th:limits_and_colimits_in_slice_categories}, the equalizers and products in $\mathsf{C}/X$ can be constructed from limits in $\mathsf{C}$. By \Cref{th:existance_theorem_for_limits}, $\mathsf{C}/X$ is also complete. The dual argument shows that if $\mathsf{C}$ is cocomplete, then $\mathsf{C}/X$ is also cocomplete.
+\end{prf}
 
 \section{String Diagram}
 String diagrams are a convenient way to represent the composition of natural transformations. From top to bottom, a string diagram represents a series of vertical compositions of natural transformations. 
@@ -2829,7 +2857,7 @@ \section{Limit and Colimit}
 \end{prf}
 
 
-\begin{theorem}{Existance Theorem for Limits}{}
+\begin{theorem}{Existance Theorem for Limits}{existance_theorem_for_limits}
     Let $\mathsf{C}$ be a category and $F:\mathsf{J}\to\mathsf{C}$ be a functor. If a category $\mathsf{C}$ has equalizers and all products indexed by the classes $\mathrm{Ob}(\mathsf{J})$ and $\mathrm{Hom}(\mathsf{J})$, then $\mathsf{C}$ has all limits of shape $\mathsf{J}$.
 \end{theorem}
 
@@ -3164,7 +3192,7 @@ \section{Limit and Colimit}
 This can be proved by duality. Here we give another proof. 
 \[
 \begin{tikzcd}
-    \left[\mathsf{I},\mathsf{C}\right]  \arrow[d, "\diagfunctor_\mathsf{J}"'] & \mathsf{C}\arrow[d, "\diagfunctor_{\mathsf{I}\times\mathsf{J}}"'] \arrow[r, "\diagfunctor_\mathsf{J}"] \arrow[l, "\diagfunctor_\mathsf{I}"'] & \left[\mathsf{J},\mathsf{C}\right] \arrow[d, "\diagfunctor_\mathsf{I}"] \\
+    \left[\mathsf{I},\mathsf{C}\right]  \arrow[d, "\diagfunctor_\mathsf{J}"'] & \mathsf{C}\arrow[d, "\diagfunctor_{\mathsf{I}\times\mathsf{J}}"'] \arrow[r, "\diagfunctor_\mathsf{J}"] \arrow[l, "\diagfunctor_\mathsf{I}"'] & \left[\mathsf{J},\mathsf{C}\right] \arrow[d, "\diagfunctor_\mathsf{I}"] \\[1em]
     \left[\mathsf{J},\left[\mathsf{I},\mathsf{C}\right]\right] \arrow[r, "\cong"] & \left[\mathsf{I}\times \mathsf{J},\mathsf{C}\right] & \left[\mathsf{I},\left[\mathsf{J},\mathsf{C}\right]\right]\arrow[l, "\cong"']
     \end{tikzcd}
     \]
@@ -3509,6 +3537,61 @@ \subsection{Fibered Product and Fibered Coproduct}
 
 \end{example}
 
+\begin{proposition}{}{}
+    Let $\mathsf{C}$ be a category which admits fibered products. Let $\sigma: S\to T$, $f: X\to S$, and $g:Y\to S$ be morphisms in $\mathsf{C}$. Then there is an  isomorphism
+    \[
+      X \times_{S} Y \cong \left(X \times_{T} Y\right) \times_{(S \times_{T} S)} S.
+    \]
+    Dually, let $\mathsf{C}$ be a category which admits fibered coproducts. Let $\sigma: T\to S$, $f: S\to X$, and $g:S\to Y$ be morphisms in $\mathsf{C}$. Then there is an  isomorphism
+    \[
+      X \sqcup_{S} Y \cong \left(X \sqcup_{T} Y\right) \sqcup_{(S \sqcup_{T} S)} S.
+    \]
+\end{proposition}
+\begin{prf}
+    \[
+        \begin{tikzcd}
+Z \arrow[rrddd, "b"', bend right=49] \arrow[rrrdd, "b", bend left=49] \arrow[rd, "b"] &                                                                                                             &                                               &                       \\[2em]
+                                                                                      & S \arrow[rdd, "\mathrm{id}"', bend right] \arrow[rrd, "\mathrm{id}", bend left] \arrow[rd, "\Delta_\sigma"] &                                               &                       \\[2em]
+                                                                                      &                                                                                                             & S \times_{T} S \arrow[r, "p"] \arrow[d, "p"'] & S \arrow[d, "\sigma"] \\[2em]
+                                                                                      &                                                                                                             & S \arrow[r, "\sigma"']                        & T                    
+\end{tikzcd}\hspace{5em} 
+    \begin{tikzcd}
+Z \arrow[rddd, "a_1"', bend right] \arrow[rrrd, "a_2", bend left] \arrow[rd, "a"] &                                                                         &                                            &                       \\[2em]
+                                                                                  & X\times_T Y \arrow[rr, "\pi_Y"] \arrow[dd, "\pi_X"'] \arrow[rd, "h", dashed] &                                            & Y \arrow[d, "g"]      \\[2em]
+                                                                                  &                                                                         & S\times_T S \arrow[d, "p"'] \arrow[r, "p"] & S \arrow[d, "\sigma"] \\[2em]
+                                                                                  & X \arrow[r, "f"']                                                       & S \arrow[r, "\sigma"']                     & T                    
+\end{tikzcd}
+    \]
+    \[
+    \begin{tikzcd}
+Z \arrow[rd, dashed] \arrow[rdd, "a"', bend right] \arrow[rrd, "b", bend left] &                                                                              &                              \\[2em]
+      &  \left(X \times_{T} Y\right) \times_{(S \times_{T} S)} S \arrow[d] \arrow[r] & S \arrow[d, "\Delta_\sigma"] \\[3em]
+       & X \times_{T} Y \arrow[r, "h"']   & S \times_{T} S              
+\end{tikzcd}
+    \]
+    We have the following isomorphisms natural in $Z$:
+    \begin{align*}
+        &\hspace{1.3em}\mathrm{Hom}(Z,\left(X \times_{T} Y\right) \times_{S \times_{T} S} S)\\
+        &\cong
+\mathrm{Hom}(Z,X \times_{T} Y)\times_{\mathrm{Hom}(Z,S\times_T S)}\mathrm{Hom}(Z,S)\\
+&\cong \left\{(a,b)\midv a\in \mathrm{Hom}(Z,X \times_{T} Y),\, b\in\mathrm{Hom}(Z,S),\, h\circ a=\Delta_\sigma\circ b\right\}\\
+&\cong \left\{(a,b)\midv a\in \mathrm{Hom}(Z,X)\times_{\mathrm{Hom}(Z,T)}\mathrm{Hom}(Z,Y),\, b\in\mathrm{Hom}(Z,S),\, h\circ a=\Delta_\sigma\circ b\right\}\\
+&\cong \left\{(a_1,a_2,b)\midv a_1\in \mathrm{Hom}(Z,X),\, a_2\in \mathrm{Hom}(Z,Y),\, b\in\mathrm{Hom}(Z,S),\, \sigma\circ f\circ a_1=\sigma\circ g\circ a_2 ,\,  f\circ a_1= g\circ a_2 = b\right\}\\
+&\cong \left\{(a_1,a_2)\midv a\in \mathrm{Hom}(Z,X ),\, b\in\mathrm{Hom}(Z,Y),\,  f\circ a_1=  g\circ a_2\right\}\\
+&\cong \mathrm{Hom}(Z,X )\times_{\mathrm{Hom}(Z,S)}\mathrm{Hom}(Z,Y)\\
+&\cong \mathrm{Hom}(Z,X \times_{S} Y)
+    \end{align*}
+    where the forth isomorphism follows from the natural isomorphism
+    \begin{align*}
+        \mathrm{Hom}(Z,S\times_T S)&\xlongrightarrow{\sim} \mathrm{Hom}(Z,S)\times_{\mathrm{Hom}(Z,T)}\mathrm{Hom}(Z,S)\\
+         \phi&\longmapsto \left(p\circ \phi, p\circ \phi\right).
+    \end{align*}
+    By Yoneda lemma, we have the desired isomorphism
+    \[
+      X \times_{S} Y \cong \left(X \times_{T} Y\right) \times_{(S \times_{T} S)} S.
+    \]
+\end{prf}
+
 \begin{definition}{Fibered Coproduct / Pushout}{}
     \begin{center}
         \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]