Fix typos Typos and Possible Improvements #18

Author: EricWay1024

ha/5-cc.typ

@@ -12,13 +12,13 @@ Let $cA$ be an abelian category.
   The *$n$-cycles* of $Ccx$ are defined as $  Z_n (C) := Ker d_n $ and
   the *$n$-boundaries* are defined as $ B_n (C) := IM d_(n+1). $
 
-  Since $d_n oo d_(n-1) = 0$, we have $ B_n (C) arrow.hook Z_n (C) arrow.hook C_n $ (as subobjects) for all $n$.
+  Since $d_(n-1) oo d_(n) = 0$, we have $ B_n (C) arrow.hook Z_n (C) arrow.hook C_n $ (as subobjects) for all $n$.
 
   The *$n$-th homology* is defined as $ H_n (C) := Coker(B_n (C) arrow.hook Z_n (C)). $
 ]
 
 #notation[
-  We often omit the subscript in $d_n$ and simply write $d$, so $d_n oo d_(n-1) = 0$ becomes $d^2 = 0$. To emphasise that $d$ belongs to the chain complex $C_cx$, we would write either $d_C$, or $d^((C))_n$ if we also need to explicitly specify the index. We sometimes also omit the dot in $Ccx$ and simply write $C$. We might write $Z_n = Z_n (C)$ and $B_n = B_n (C)$.
+  We often omit the subscript in $d_n$ and simply write $d$, so $d_(n-1) oo d_(n) = 0$ becomes $d^2 = 0$. To emphasise that $d$ belongs to the chain complex $C_cx$, we would write either $d_C$, or $d^((C))_n$ if we also need to explicitly specify the index. We sometimes also omit the dot in $Ccx$ and simply write $C$. We might write $Z_n = Z_n (C)$ and $B_n = B_n (C)$.
 ]
 
 
@@ -112,7 +112,7 @@ Let $cA$ be an abelian category.
 // ]
 
 #definition[
-  A *cochain complex* $Ccx$ in $cA$ is a family ${C^n}_(n in ZZ)$ of objects in $cA$ with morphisms $d^n : C^n -> C^(n+1)$ such that $d^n oo d^(n+1) = 0$, where $d^n$ are called *differentials*. The *$n$-cocycles* of $C^cx$ are $ Z^n (C) := Ker d^n $ and the *$n$-coboundaries* are $ B^n (C) := IM d^(n-1). $
+  A *cochain complex* $Ccx$ in $cA$ is a family ${C^n}_(n in ZZ)$ of objects in $cA$ with morphisms $d^n : C^n -> C^(n+1)$ such that $d^(n+1) oo d^(n) = 0$, where $d^n$ are called *differentials*. The *$n$-cocycles* of $C^cx$ are $ Z^n (C) := Ker d^n $ and the *$n$-coboundaries* are $ B^n (C) := IM d^(n-1). $
 
   We have $ B^n ( C) arrow.hook Z^n (C) arrow.hook C^n $ (as subobjects) for all $n$.
 

ha/a-kc.typ

@@ -199,7 +199,7 @@ cohomology* to be
 
 Now recall that if $A$ is an $R$-module and $r in R$, then $r$ is a *zero-divisor* on $A$ if there exists non-zero $a in A$ such that $r a  = 0$. Therefore, $r$ is a *non-zero-divisor* on $A$ #iff the multiplication $A ->^r A$ is injective.
 #definition[
-  If $A$ is an $R$-module, a *regular sequence* on $A$ is a sequence of elements $(x_1, ..., x_n)$ where each $x_i in R$ such that $x_1$ is a non-zero-divisor on $A$ and each $x_i$ is a non-zero-divisor on $A over (x_1, ..., x_(i-1)) A$.
+  If $A$ is an $R$-module, a *regular sequence* on $A$ is a sequence of elements $(x_1, ..., x_n)$ where each $x_i in R$, $x_1$ is a non-zero-divisor on $A$, and for $i >=2$, each $x_i$ is a non-zero-divisor on $A over (x_1, ..., x_(i-1)) A$ .
 ]
 
 #lemma[