
diff -r 675f53735445 -r 1d76e832d32f text/comm_alg.tex
--- a/text/comm_alg.tex	Fri Jun 04 17:00:18 2010 -0700
+++ b/text/comm_alg.tex	Fri Jun 04 17:15:53 2010 -0700
@@ -13,7 +13,10 @@
 The goal of this \nn{subsection?} is to compute
 $\bc_*(M^n, C)$ for various commutative algebras $C$.
 
-Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}).  This possibility was suggested to us by Thomas Tradler.
+Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative 
+algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with 
+coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}).  
+This possibility was suggested to us by Thomas Tradler.
 
 
 \medskip
@@ -108,8 +111,12 @@
 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
 Let us check this directly.
 
-The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
-(See also  \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
+The algebra $k[t]$ has Koszul resolution 
+$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, 
+which has coinvariants $k[t] \xrightarrow{0} k[t]$. 
+This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
+(See also  \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: 
+$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
 
 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
 The fixed points of this flow are the equally spaced configurations.
@@ -152,7 +159,8 @@
 \]
 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$.
 We will content ourselves with the case $k = \z$.
-One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact.
+One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the 
+same color repel each other and points of different colors do not interact.
 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent
 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple
 corresponding to $X$.




