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\begin{document}

\begin{abstract}
    We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of ${\rm HH}^1(A,A)$.
    We also prove a more general result concerning Iwanaga Gorenstein algebras, using a more general notion of stable equivalences of Morita type. We provide several applications to commutative algebra and modular representation theory.
    The proof exploits in an essential way the $B_\infty$-structure of the Hochschild cochain complex.

    In the appendix we explain how the  well-definedness of the $p$-power operation on Hochschild cohomology follows from some (originally topological) results of May and Cohen, and (on the algebraic side) Turchin. We give complete proofs, using the language of operads.

\end{abstract}

\maketitle

It is a central theme in representation theory to understand what structure possessed by a finite dimensional associative algebra is shared by algebras with equivalent module categories, or derived categories, or stable module categories.

Stable equivalences
are both more frequent
and less understood than derived equivalences.

A famous illustration of this is the Auslander-Reiten  conjecture: \emph{the number of isomorphism classes of non-projective simple modules is invariant under stable equivalences}. For this reason it is desirable to have as many invariants under stable equivalence as possible.  Hochschild cohomology is an import source of examples. In fact,  Liu, Zhou and Zimmermann have rephrased the Auslander-Reiten conjecture (for symmetric algebras with no  semisimple summands) in terms of the dimension of the projective center in Hochschild cohomology  \cite{LZZ}.

To better understand stable equivalences, Brou\'e introduced stable equivalences of Morita type in \cite{B},

and Keller and Vossieck \cite{KV} and Rickard \cite{Ri} proved that derived equivalences give rise to
stable equivalences of Morita type for self-injective algebras.

Hochschild cohomology
possesses a great deal of structure. Perhaps most famously, it is a Gerstenhaber algebra, and this is enriched by a $B_{\infty}$-algebra structure on the Hoschschild cochain complex. In more specific contexts, more structure appears.

The invariance of these structures has been studied by various authors.

For derived equivalences, Keller
showed the invariance of the Gerstanhaber structure \cite{KellerIII},
and the invariance of $B_{\infty}$-structure of the
cochain complex \cite{KellerI}.

For a stable  equivalence of Morita type between two self-injective
algebras $A$ and $B$, Xi showed in \cite{X} that for $n\geq 1$, the
Hochschild cohomology groups $\mathrm{HH}^n(A,A)$ and $\mathrm{HH}^n(B,B)$ are
isomorphic.
Moreover, K\"onig, Liu and Zhou proved in \cite{KLZ}  the invariance of the Batalin–Vilkovisky structure of stable Hochschild cohomology for symmetric algebras.

 Over a field of positive characteristic, Hochschild cohomology
 possesses also the structure of a  restricted graded Lie
 algebra.

In this context Linckelmann asked the following question:

\emph{is the restricted Lie algebra structure of $ {\rm HH}^{>0}(A,A)$ invariant under stable equivalences of Morita type for self-injective algebras?}

The invariance of this structure is delicate because the $p$-power structure is \emph{non-linear}, and
therefore difficult to handle functorially. A partial answer was given by the second author in \cite{RyD} for the subclass of integrable derivations. We answer Linckelmann's question affirmatively:

\begin{theorem}
\label{stableinvariancetheorem}
Let $A$ and $B$ be finite dimensional, self-injective algebras over a field of positive characteristic. If $A$ and $B$ are stably equivalent of Morita type, then the induced transfer map on Hochschild cohomology gives an isomorphism $ {\rm HH}^{>0}(A,A)\cong {\rm HH}^{>0}(B,B)$ of (shifted) \underline{restricted} graded Lie algebras.
\end{theorem}

We prove a more general result that applies to a pair of Iwanaga Gorenstein algebras (see Theorem \ref{generalVersion}).

We introduce a more general notion of stable equivalence of Morita type that extends the classical definition, and Theorem \ref{stableinvariancetheorem} holds in this larger context (see Theorem \ref{keytheorem} for details).

Theorem \ref{stableinvariancetheorem} yields nontrivial information in the problem of distinguishing stable
equivalence classes, because the same graded Lie algebra can admit different restricted
structures (for example, for abelian Lie algebras see \cite[Chapter 3]{StraFar}).

\begin{corollary}\label{resinvscor}
The following are invariant under stable equivalences of Morita type between finite dimensional self-injective algebras:
\begin{itemize}
    \item the restricted Lie algebra ${\rm HH}^1(A,A)$;
    \item the restricted Lie algebra ${\sf Z}({\rm HH}^1(A,A))$;
    \item the restricted universal enveloping algebras of ${\rm HH}^1(A,A)$ and of ${\sf Z}({\rm HH}^1(A,A))$;
    \item the maximal dimension of a $p$-toral subalgebra of ${\rm HH}^1(A,A)$.

\end{itemize}
\end{corollary}

Even though the centre ${\sf Z}({\rm HH}^1(A,A))$ has trivial bracket, its $p$-power operation might be non-trivial. Any other restricted invariant could be added to the list.

By combining Corollary \ref{resinvscor}  with the the idea
that a precise knowledge of the Lie
algebra structure of $\HHF(A)$  should contain significant information
about $A$ (\cite{LiRu1,Chang,LiRu2,ER,RSS,BKL2} to name a few), the  authors,  in joint work with Manuel Saor\'{i}n,
are
trying to uncover new information in the stable equivalence class of finite dimensional algebras.

As well as the $p$-power operation, there is another invariant defined on the even Hochschild cohomology classes which does not seem to have been used in representation theory.  This operation is analogous to the Cohen-Dyer-Lashof operations from topology. We show that the extra structure is also invariant under stable equivalences of Morita type---see Remark \ref{CDLoperation}.

We provide some applications to modular representation
theory and commutative algebra. In section \ref{tamesym} we consider the classification of algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type, as started by Zhou, Zimmermann and  Taillefer in \cite{ZZ}, \cite{ZI} \cite{Taillefer} who employed various   invariants
, including the Lie structure of $\mathrm{HH}^1$,   to distinguish most of the algebras. We compute
the restricted Lie structure of most of the symmetric algebras of semi-dihedral and quaternion type that have not been classified. The restricted structure
fails to distinguish classes, providing some evidence that in the remaining cases derived  and stable classification should  coincide. The second application is based on the work of
\cite{BKL}; as a consequence of Theorem \ref{stableinvariancetheorem} we obtain an
isomorphism as restricted Lie algebras of $\mathrm{HH}^1$ for all non-nilpotent blocks with defect two and
their Brauer correspondents. Lastly, in Theorem \ref{commutativetheorem} we apply our results to
commutative Gorenstein rings. In particular we show that Kn\"orrer periodicity induces an isomorphism of restricted Lie algebras in Hochschild cohomology.

We wish to emphasise that the $p$-power structure on Hochschild cohomology is actually an artifact of the $B_\infty$-structure on the Hochschild cochain complex, after the work of Turchin \cite{Tourtchine}. This perspective does not seem to be used in representation theory, but it is fundamental in this paper. The usefulness of this approach is explained by the functoriality in Theorem \ref{E2restrictedLie}.

Even though it is aimed at modular representation theorists, much of the proof of Theorem \ref{stableinvariancetheorem} consists of connecting together different concepts from various areas, and therefore much of the paper consists of background, especially on $B_\infty$-algebras.

To show that the $p$-power operation is well-defined it seems necessary to involve some operadic machinery. We do this in the appendix, following work of May and Cohen in topology. The results therein serve as general template for building cohomology operations, and so we hope this appendix might be a useful reference, since these arguments do not seem to have been written down before in their algebraic form.

\tableofcontents

\noindent
{\bf Conventions.} Throughout $k$ is a commutative noetherian ring, and everything is taken to be $k$-linear. Undecorated tensor products
are taken over $k$, and many of the objects we will discuss (notably bimodules and Hochschild cohomology) depend implicitly
on $k$. Not much is lost in imagining $k$ to be a field. We use the notation $(\susp M)_i= M_{i-1}$ for the  shift of a graded object $M$.

\medskip

\noindent
{\bf Acknowledgements.}
We are happy to thank Victor Turchin for explaining to us some facts about cohomology operations (see the appendix). We are thankful as well to Bernhard Keller for several helpful correspondences, and for pointing out an inaccurate statement about the existence of stable restriction functors (see Section \ref{MoreFunctorialitySubsec}). Lastly, we thank Markus Linckelmann for his support, his comments and for posing the question about the invariance of the $p$-power map.

\medskip

\noindent
{\bf Funding.}
The second author has been  supported by the Fundaci\'on `S\'eneca' of Murcia
(19880/GERM/15) and by the INdAM postdoctoral research grant 2019-2020.

\section{The Structure and Functoriality of Hochschild Cohomology}\label{structureofHHSection}

We will borrow a argument from Keller \cite{KellerI} and exploit the functoriality of Hochschild cohomology
for fully faithful embeddings of
categories. This is useful even if we start with  algebras, but since it adds
no extra difficulty we may as well work with categories (or ``algebras with many objects" \cite{Mitchell}) from the beginning.

If $A$ is a small $k$-linear
category then we use the notation ${}_b A_a=\Hom_A(a,b)$ for brevity.
A right dg $A$-module $M$ consists of a complex $M_a$ of  $k$-modules for each object $a$ and an action
$M_b\otimes {}_bA_a\to M_a$ for objects $a$ and $b$ of $A$, associative and unital in the natural sense. Left modules, and bimodules,
are defined similarly. In particular, there is for each object $a$ the (representable) right $A$-module ${}_aA$ and
the (corepresentable) left $A$-module $A_a$. These play the role of free modules.  In addition, $A$ is  naturally a
bimodule over itself. See \cite{Toen} for more details on modules over (dg)
categories and their derived categories.

Always $\dcat(A)$ denotes the full derived category of $A$, and $\dcat(A\- B)$ is the derived category of
$A\- B$ dg bimodules (with $k$ acting centrally). In particular $\dcat(A\- A)\simeq \dcat(A^\ev)$ where
$A^\ev=A^\op\otimes A$.

The Hochschild cohomology of $A$ is $\HH(A,A)= {\rm End}_{\dcat(A\- A)}(A)=\Ext_{A\- A}(A,A)$. Note that this presents
$\HH(A,A)$ as a graded $k$-algebra.

\subsection{B\texorpdfstring{$_\infty$}{}-structure} Hochschild cohomology enjoys a great deal more structure  than its associative product alone,
and this structure manifests most clearly at the chain level.

Thus, going back to \cite{Mitchell}
one defines the Hochschild cochain complex of the category $A$ to be
\[
C^n(A,A)=\Hom(A^{\otimes n},A)=\prod_{a_n,...,a_0}\Hom({}_{a_n}A_{a_{n-1}}\otimes \cdots \otimes {}_{a_1}A_{a_{0}},{}_{a_{n}}A_{a_0}),
\]
where $a_i$ for $0\leq i\leq n$ are objects of $A$. The Hochschild differential has its classical formula.

As long as $A$ is projective over $k$, the cohomology of $C^*(A,A)$ canonically computes $\HH(A,A)$.

The cup product in this context is also given by the classical formula: if $x\in \Hom({}_{a_n}A_{a_{n-1}}\otimes \cdots \otimes {}_{a_1}A_{a_{0}},{}_{a_{n}}A_{a_0})$ and $y\in\Hom({}_{b_m}A_{b_{m-1}}\otimes \cdots \otimes {}_{b_1}A_{b_{0}},{}_{b_{m}}A_{b_0})$, with $a_0=b_m$, then $
x\smile y $ is the linear map
\[
{}_{a_n}A_{a_{n-1}}\otimes \cdots \otimes {}_{b_1}A_{b_{0}} \xrightarrow{x\otimes y} {}_{a_{n}}A_{a_0}\otimes{}_{b_{m}}A_{b_0}\xrightarrow{\rm comp.}  {}_{a_{n}}A_{b_0}
\]
(the product is zero if $a_0\neq b_m$). Altogether this makes $C^*(A,A)$ into a dg algebra.

The next series of operations may at first seem mysterious, but all we really need from them are the following two facts: firstly, the \emph{restricted} Lie algebra structure on $\HH(A,A)$ is naturally built from them; and secondly, the operations are preserved by the restriction maps  discussed in the next subsection.

The \emph{brace operations} on $C=C^*(A,A)$ are, for each $s\geq 1$, linear maps
\[
(\-)\{\-,\dots,\-\} \colon C\otimes C^{\otimes s}\to C
\]
of degree $-s$. Thus, if $x\in C^n(A,A)$ and $y_i\in C^{n_i}(A,A)$ then
\[
x\{y_1,\dots,y_s\} \in C^{n+n_1+\cdots+ n_s-s}(A,A).
\]
To define these operations, we consider all ways of applying the $y_i$ in parallel: if $m_0+n_1+m_1+\cdots+ m_{s-1}+n_s+m_s= n+n_1+\cdots +n_s-s$, then denote
\[
y^{(\underline{m},\underline{n})}\colon A^{\otimes(n+n_1+\cdots+ n_s-s)}\xrightarrow{1^{\otimes m_0}\otimes y_1 \otimes 1^{\otimes m_1} \cdots 1^{\otimes m_{s-1}}\otimes y_s \otimes 1^{\otimes m_s} } A^{\otimes n}.
\]
Then we take the signed sum of these plugged into $x$, so
\[
x\{y_1,\dots,y_n\} = \sum_{\underline{m}}(-1)^{(m_0+n_1+\cdots +m_{s-1})(n_s+1) + \cdots + (m_0)(n_1+1)} y^{(\underline{m},\underline{n})} x.
\]
(We have suppressed the objects of $A$ here for readability---as in the formula for the cup products we need all
the objects to ``match up'' to get a nonzero component.)

For example, if ${\bf m}\colon A^{\otimes 2}\to A$ is the multiplication (or composition) on $A$, then the
Hochschild differential is given by $\partial(x)={\bf m}\{x\}+(-1)^{|x|}x\{{\bf m}\}$,
and the cup product is given by $x\smile y ={\bf m}\{x,y\}$.

The binary operation $(\-)\{\-\}$ coincides with Gerstenhaber's \emph{circle product} $x\{y\} = x\circ y$.

This is the only brace operation we need in order to discuss the restricted
Lie algebra structure of $\HH(A,A)$.

These brace operations were introduced by Getzler in \cite{Getzler}.

The various compatibility conditions between them can be summed up by saying that they induce a product on the \emph{bar construction} $BC=\bigoplus_i (\susp C)^{\otimes i}$ which makes $BC$ into a dg Hopf algebra. By defintion, this makes $C^*(A,A)$ an example of a  \emph{B$_{\infty}$-algebra}. See  Getzler and Jones \cite[Section 5.2]{GJ} or Gerstenhaber and Voronov \cite[section 3.2]{GV} for details.

As usual, much of this was anticipated by Gerstenhaber long ago: what is roughly the same structure on the Hochschild chain
complex was used in \cite{Gerst} under the name ``composition complex". The cochain algebra of a topological space, and the
cobar construction of a Hopf algebra, were also given as examples of composition complexes. These would both now be
considered standard examples of B$_{\infty}$-algebras.

It has been proven by McClure and Smith that a $B_\infty$-algebra is essentially the same thing as an $E_2$-algebra \cite{McClureSmith}, answering Deligne's question about the  structure of $C^*(A,A)$.

\subsection{Functoriality}\label{FunctorialitySubsec}
If $\phi:A\to B$ is a fully faithful embedding of categories then there is a restriction functor
$R_\phi: \dcat(B)\to \dcat(A)$ which sends a $B$-module $M$ to the $A$-module with components $(R_\phi M)_a=M_{\phi(a)}$.
This has a left adjoint $L_\phi:\dcat(A)\to \dcat(B)$ given by $L_\phi(M)= M\otimes^{\sf L}_BA$. Since
$R_\phi L_\phi\simeq {\rm id}_{\dcat(A)}$ the functor $L_\phi$ is fully-faithful.

From $\phi $ we get a fully faithful embedding $\phi^\ev :A^\ev\to B^\ev$ and hence a restriction
$R_{\phi^\ev}:\dcat(B\- B)\to \dcat(A\- A)$. Since $R_{\phi^\ev}(B)=A$, this induces a homomorphism
${\rm End}_{\dcat(B\- B)}(B)\to {\rm End}_{\dcat(A\- A)}(A)$ of graded rings.
Identifying these rings with the Hochschild cohomology of $A$ and $B$ respectively we have a homomorphism
\[
\phi^*: \HH(B,B)\to \HH(A,A).
\]
This homomorphism has been exploited by Keller, who observed in \cite{KellerI} that one can lift $\phi^*$ to
the Hochschild cochain complex in such a way that it preserves the $B_\infty$-structure.

Indeed, we define $
\phi^*: C^*(B,B)\to C^*(A,A)$ by its components
\[
\Hom({}_{\phi(a_n)}B_{\phi(a_{n-1})}\otimes \cdots \otimes {}_{\phi(a_1)}B_{\phi(a_{0})},{}_{\phi(a_{n})}B_{\phi(a_0)})\quad\quad\quad\quad\quad\quad\quad
\]
\[\quad\quad\quad\quad\quad\quad\quad\xrightarrow{\ \cong\ } \Hom({}_{a_n}A_{a_{n-1}}\otimes \cdots \otimes {}_{a_1}A_{a_{0}},{}_{a_{n}}A_{a_0}).
\]
That is, $\phi^*$ simply restricts each $k$-linear map in $C^*(B,B)$ to those components only involving objects in the image of $A$. One can check that this is a chain map which induces the homomorphism $\phi^*: \HH(B,B)\to \HH(A,A)$ described above.

It is evident from the formulas in the previous subsection that $
\phi^*: C^*(B,B)\to C^*(A,A)$ respects all of the brace operation. This observation of Keller's is extremely useful. It follows that all operations on  Hochschild cohomology which can be written in terms of the brace structure
must be preserved by $\phi^*$.

\subsection{Graded Lie Algebras}

A graded Lie algebra is a graded $k$-module $L$  equipped with a bilinear bracket
$[\- ,\- ]: L\times L\to L$ and with a quadratic \emph{reduced square} operation
defined on odd elements $(\- )^{[2]}: L^{2i+1}\to L^{4i+2}$,  so $(\alpha
u)^{[2]}=\alpha^2 u^{[2]}$ for any  $\alpha$ in $k$. These operations should
satisfy various axioms (compatibility, anti-symmetry and the graded Jacobi
identity) which can be found in \cite[Remark 10.1.2]{Avramov}.
 The reduced square is redundant if $2$ is a unit in $k$.

\begin{example}
\label{ex1}Any graded associative algebra $A$ over $k$ gives rise to a graded Lie algebra
$A^{\sf Lie}$ with the same underlying graded vector space. The bracket is given
by the graded commutator $[x,y]= xy-(-1)^{|x||y|}yx$ while the reduced square is
simply the square $u^{[2]}=u^2$.
 If $A$ has the structure of a graded Hopf algebra, then the space of primitives in
 $A$ is a sub-graded Lie algebra of $A^{\sf Lie}$.
 \end{example}

\begin{example}
\label{ex2}
The degree-shifted Hochschild cochain complex $\susp C^*(A,A)$ is another example. The classical Gerstenhaber bracket is $[x,y]= x\circ y-(-1)^{|x||y|}y\circ x$
and the reduced square is $x^{[2]}=x\circ x$.
These operations descend to cohomology, making  $\susp\HH(A,A)$ into a graded Lie algebra.

In particular, ${\rm HH}^1(A,A)$ is an (ungraded) Lie algebra. This is an important---and often computable---invariant in representation theory.
\end{example}

\subsection{Restricted Graded Lie Algebras}\label{restrictedsubsection} So far none of this depends on the characteristic of $k$.
The next structure belongs to the positive characteristic world. So, from now on $p$ is a fixed, positive prime, and we assume that $k$ contains a field of characteristic $p$.

A \emph{restricted} graded Lie algebra is a graded Lie algebra $L$ equipped with a $p$-power-linear operation defined on even elements $(\- )^{[p]}:L^{2i}\to L^{2ip}$, so $(\alpha u)^{[p]}=\alpha^p u^{[p]}$ for any $\alpha$ in $k$. This restricted structure must satisfy two standard compatibility conditions with the Lie structure, which can be found in \cite[section 4.2]{ZII} or \cite[section 7]{Tourtchine}.

\begin{example}
If $A$ is a graded associative algebra, then $A^{\sf Lie}$ admits a restricted Lie algebra structure by defining $x^{[p]}=x^p$ on even elements.

If, further, $A$ is a graded Hopf algebra, then $p$th powers of primitive, even degree  elements remain primitive because of the formula $(x\otimes 1+1\otimes x)^p = x^p\otimes 1+1\otimes x^p$. Thus the space of primitives in $A$ is also a restricted Lie algebra.
\end{example}

\begin{example}\label{poweropexample}
For us the most important  example is the shifted Hochschild cohomology $\susp {\rm HH}^{*}(A,A)$. On top of the Gerstenhaber bracket there is a $p$-power operation given on $C^*(A,A)$ by
\[
x^{[p]}= x\{x\}\cdots \{x\} = ((x\circ x) \circ \cdots )\circ x
\]
with $x$ appearing $p$ times in either expression, see  \cite[section 3]{Tourtchine}. It is difficult to show that this operation is well-defined on cohomology classes, a proof is given in the appendix.
\end{example}

More classically, this restricted Lie algebra structure is often presented by identifying $C^*(A,A)$ with the space of coderivations on the bar construction $BA$
\[
C^*(A,A)\cong {\rm coDer}(BA,BA).
\]
This isomorphism was discovered independently by Quillen \cite{Quillen} and Stasheff \cite{Stasheff}. The Gerstenhaber bracket under this identification becomes the graded commutator of coderivations, and the $p$-power structure is given by $p$-fold composition. See \cite{KellerIII} or \cite{ZII} for details.

The following fact deserves to be more well-known in algebra. It is one reason that restricted Lie algebras often
occur in homotopical situations, and it gives a convenient way to handle functoriality. The (originally
topological) result is due to Cohen  \cite{Cohen}.

\begin{theorem}\label{E2restrictedLie}
If $C$ is a $B_\infty$-algebra then $\susp{\rm H_*}(C)$ is naturally a restricted graded Lie algebra. A map of
$B_\infty$-algebras induces a map of restricted graded Lie algebras in homology.

In particular, if $\phi: A\to B$ is a fully faithful embedding of dg categories then $\phi^*:\HH(B,B)\to \HH(A,A)$
is (after shifting) a homomorphism of restricted Lie algebras.
\end{theorem}

In other words, the homology of any $B_\infty$-algebra is a ``restricted Gerstenhaber algebra''.

\begin{proof}

It is well-known that the cohomology of a $B_\infty$-algebra is a Gerstenhaber algebra \cite{GV}, so we only need to explain the $p$-power operation. This is given by the same formula as in example \ref{poweropexample}. The proof that this operation is well-defined turns out to be subtle, so it is dealt with in the appendix.

The statement about Hochschild cohomology follows from Keller's observation \cite{KellerI} that the restriction map $C^*(B,B)\to C^*(A,A)$ is one of $B_\infty$-algebras.
\end{proof}

\begin{remark}
\label{CDLoperation}
The $p$-power map is (when $p$ is odd) just one of the two Cohen-Dyer-Lashof operations which exists on
the homology of any $B_\infty$-algebra, see \cite{Cohen,Tourtchine}. The other operation is defined on
odd cycles by the formula
 $\zeta(x)= \sum_{i+j=p-1}\frac{(-1)^i}{i}x^{[i]}\smile x^{[j]}$ as in \cite[theorem 3.1]{Tourtchine}. It
 doesn't seem that this structure has been exploited by algebraists. For example, using the methods of
 this paper the induced $\zeta$ operation on Hochschild cohomology could conceivably be used to
 distinguish algebras up to stable equivalence of Morita type.

\end{remark}

\begin{corollary}\label{restrictedderivedinv}
If $A$ and $B$ are two dg algebras which are derived equivalent, then there is an isomorphism $\susp {\rm HH}^{>0}(A,A)\cong\susp {\rm HH}^{>0}(B,B)$ of restricted graded Lie algebras.
\end{corollary}

This was originally stated by Zimmerman \cite[proposition 4.4]{ZII}.
The proof in \cite{ZII} seems to be incomplete, because some of the maps used (between complexes of coderivations) are not well-defined at the chain level.

Using Theorem \ref{E2restrictedLie}, Corollary \ref{restrictedderivedinv} now follows from a result of Keller \cite{KellerI}: if $A$ and $B$ are derived equivalent then $C^*(A,A)$ and $C^*(B,B)$ are connected by a zig-zag of quasi-isomorphisms of $B_\infty$-algebras. The main result of this paper is the generalisation to stable equivalences of Morita type.

As another application, the next corollary follows in the same way from Theorem \ref{E2restrictedLie} and \cite[theorem 3.5]{KellerI}.

\begin{corollary}
If $A$ and $B$ are Koszul Dual then

$\susp {\rm HH}^{>0}(A,A)\cong\susp {\rm HH}^{>0}(B,B)$ as restricted graded Lie algebras.
\end{corollary}

\subsection{Singularity Categories and Stable Hochschild Cohomology}\label{stableHHSection}

Let $\dbcat(A)$ be the homotopy category of bounded below complexes of finitely generated projectives that have bounded homology. The singularity category $\sing(A)$ of an algebra (or category) $A$
is defined as the Verdier quotient of  $\dbcat(A)$ by its subcategory ${\rm perf}(A)$ of perfect complexes, that is, bounded complexes of projective modules (cf.~\cite{Buc} or \cite{Orl}).

Following Buchweitz \cite{Buc}, the \emph{stable} (or \emph{singular}, or \emph{Tate}) Hochschild cohomology of $A$ is the graded endomorphism algebra
\[
\sHH(A,A)=\sExt_{A\- A}(A,A)={\rm End}_{\sing(A^\ev)}(A).
\]
For symmetric algebras the positive part of $\sHH(A,A)$
coincides with $\HH(A,A)$ (Corollary 6.4.1 in \cite{Buc}).

This definition only works if $A$ is an object of $\dbcat(A^\ev)$. This is automatic if $A$ is finite over $k$, but in some contexts this is a strong condition on $A$. To avoid worrying about this we can use a large version of the singularity category  from  \cite{Orl}, as Keller has done in \cite{KellerII}.

Let $\cdbcat(A)$ denote the full subcategory of $\dcat(A)$ consisting of dg modules with bounded homology,
and let $\flat(A)$ denote the thick subcategory of $\dcat(A)$ consisting of bounded complexes of flat modules.

By definition $\csing(A)$ is the Verdier quotient $\cdbcat(A)/\flat(A)$. With this large version of the singularity category we may make the general definition
\[
\sHH(A,A)=\sExt_{A\- A}(A,A)={\rm End}_{\csing(A^\ev)}(A).
\]
Through the projection functor $\cdbcat(A\- A)\to\csing(A\-A)$ we have a natural stabilization homomorphism $\HH(A,A)\to\sHH(A,A)$. We must justify that this is consistent with our earlier definition:
\begin{proposition}[{\cite[proposition 1.13]{Orl}} or {\cite[lemma 2.2]{KellerII}}]
The natural functor
$
\sing(A)\longrightarrow\csing(A)
$
is fully-faithful.
\end{proposition}

The statement here is slightly stronger than that of \cite{Orl} or \cite{KellerII}, but the same proof works (keeping in mind that flat modules are the filtered colimit of their projective submodules).

\subsection{More On Functoriality}\label{MoreFunctorialitySubsec} If  $\phi:A\to B$ is a fully faithful embedding, then $R_\phi:\dcat(B)\to \dcat(A)$ always preserves the bounded derived categories. If further each of the right $A$-modules  $R_\phi ({}_{b}B)$ have finite flat dimension, then this functor descends to a stable restriction functor $\underline{R}_\phi:\csing(B)\to \csing(A)$.

On the other hand, $L_\phi:\dcat(A)\to \dcat(B)$ restricts always to perfect objects, and to $\flat(A)$. If we assume that each of the left $A$-modules $R_\phi( B_b)$ have finite flat dimension, then $L_\phi$ will restrict to bounded derived categories, and therefore induce a stable extension of scalars $\underline{L}_\phi:\csing(A)\to\csing(B)$.

When $\phi$ has finite flat dimension on both sides (i.e. all of the restrictions ${}_bB$ and $B_b$ have finite flat dimension), both functors $\underline{L}_\phi$ and $\underline{R}_\phi$ exist and are left and right adjoint to each other respectively
\[
\begin{tikzcd}
 \underline{L}_\phi:\csing(A)\ar[r, shift left = 1mm ]& \ar[l, shift left = 1mm ]\csing(B):\underline{R}_\phi
\end{tikzcd}
\]
If we had assumed that the restrictions ${}_bB$ and $B_b$ were all \emph{perfect}, we would have obtained an adjunction between the small singularity categories $\sing(A)$ and $\sing(B)$. Even in good situations when the small singularity categories are available, the existence of $\underline{L}_\phi$ and $\underline{R}_\phi$ in this greater generality makes $\csing(A)$ useful.

\section{Stable equivalence of Morita type}\label{SEMTsection}

The setup in this section is that we have two flat
$k$-algebras (or categories) $A$ and $B$, and an $A\- B$ bimodule $M$ which has finite flat dimension on both sides (i.e. as a left $A$ and a right $B$-module separately).
We'll always use $a$ for an arbitrary object of $A$, and $b$ for an arbitrary object of $B$.

We borrow a construction from Keller \cite{KellerI}.

Let $T$ the category whose objects are the disjoint union of those from $A$ and $B$, and whose morphisms are given by ${}_aT_a={}_aA_a$, ${}_bT_b={}_bB_b$, ${}_aT_b={}_aM_b$ and ${}_bT_a=0$.
Composition in $T$ is given by composition in $A$ and $B$ and by the action of $A$ and $B$ on $M$.
Diagramatically, $T$ looks like
\[
\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"A"'] & \ar[l,"M"'] b \ar[loop , looseness=4,"B"']
\end{tikzcd}
\]

and $T$ comes with two fully faithful embeddings $A\to T$ and $B\to T$.

We also have two fully faithful embeddings $A^\ev\to T^\ev$ and $B^\ev\to T^\ev$.
The assumption that $M$ has finite flat dimension on both sides, and the assumption that $A$ and $B$ are flat over $k$, together imply that $T^\ev$ has finite flat dimension when restricted to either $A^\ev$ or $B^\ev$ (because the tensor product of two flat modules is flat). As explained in section \ref{MoreFunctorialitySubsec}, this implies that we have stable restriction functors
\[
\underline{R}_A:\csing(A\- A)\to \csing(T\- T)\ \text{ and }\ \underline{R}_B:\csing(B\- B)\to \csing(T\- T).
\]

Since $R_A(T)=A$ and $R_B(T)=B$ we obtain a commutative diagram of homomorphisms
\begin{equation}\label{StabilisationDiagram}
\begin{tikzcd}
    \Ext_{A\- A}(A,A) \ar[d] & \Ext_{T\- T}(T,T) \ar[d] \ar[l,"R_A"'] \ar[r,"R_B"] & \Ext_{B\- B}(B,B) \ar[d]\\
    \sExt_{A\- A}(A,A) & \sExt_{T\- T}(T,T)  \ar[l,"\underline{R}_A"'] \ar[r,"\underline{R}_B"]& \sExt_{B\- B}(B,B).
\end{tikzcd}
\end{equation}

The key to the proof of Theorem \ref{stableinvariancetheorem} is to use this diagram with the following result. This is the technical core of the paper.

\begin{theorem}\label{keytheorem}
If $\-\otimes_A^{\sf L}M:\csing(A\- A)\to \csing(A\- B)$ and $M\otimes_B^{\sf L}\-:\csing(B\- B)\to \csing(A\- B)$ are both fully faithful then
\[
\underline{R}_A:\sExt_{T\- T}(T,T)\to \sExt_{A\- A}(A,A)\quad\text{and}\quad
\underline{R}_B: \sExt_{T\- T}(T,T)\to\sExt_{B\- B}(B,B)
\]
are both isomorphims.
\end{theorem}

The hypotheses of this theorem are what we mean by stable equivalence of Morita type. Proposition \ref{SEMTprop} below connects this with the usual notion.

\begin{proof}
Associated to the embeddings of $A^\ev$, $B^\ev$ and $A^\op\otimes B$ into $T^\ev$ we have the left adjoints (from sections \ref{FunctorialitySubsec} and \ref{MoreFunctorialitySubsec}) which embed $\cdbcat(A\- A)$, $\cdbcat(B\- B)$ and $\cdbcat(A\- B)$ into $\cdbcat(T\- T)$, respectively. To simplify notation we denote all of these functors by $L$ (the domain should always be clear from context).

We start by showing that in $\cdbcat(T\- T)$ there is a natural triangle
\begin{equation}\label{basicTriangle}
   LM\to LA\oplus LB \to T\to \ .
\end{equation}

The adjunction $\Ext_{A\- A}(A,A)=\Ext_{A\- A}(A,RT)\cong \Ext_{T\- T}(LA,T)$ gives the map $LA\to T$ in this triangle. Similarly for $LB\to T$.

 We need to resolve $T$ in order to compute these left adjoints. By truncating a flat  $A^\op\otimes B$ resolution of $M$ somewhere above the flat dimensions of $M$ as a left or right $A$ or $B$-module, we obtain a quasi-isomorphism $\widetilde{M}\to M$ from a complex of bimodules which are all flat as left $A$ and right $B$-modules seperately (this uses flatness of $A$ and $B$ over $k$).

There is then a natural $T$ bimodule (in fact, dg category) $\widetilde{T}$ given by
\[
\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"A"'] & \ar[l,"\widetilde{M}"'] b \ar[loop , looseness=4,"B"']
\end{tikzcd}
\]
and $\widetilde{T}$ admits a quasi-isomorphism to $T$. One can think of $\widetilde{T}$ as a flat left $A$-module resolution of $T$ (with a compatible right $T$ action). In turn, $\widetilde{T}\otimes\widetilde{T}$ is a flat left $A^\ev$-module resolution of $T\otimes T$  (with a compatible right $T^\ev$ action). This means $LA=A\otimes_{A^\ev}^{\sf L}(T\otimes T)\simeq A\otimes_{A^\ev}(\widetilde{T}\otimes\widetilde{T})$. This tensor product looks like
\[
\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"A"'] & \ar[l,"\widetilde{M}"'] b \ar[loop , looseness=4,"0"']
\end{tikzcd} \simeq
\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"A"'] & \ar[l,"{M}"'] b \ar[loop , looseness=4,"0"']
\end{tikzcd}
\]
i.e. ${}_a(LA)_a \simeq A$, ${}_a(LA)_b \simeq \widetilde{M}\simeq M$ and ${}_b(LA)_b=0$. The obvious inclusion into $T$ is the map in the triangle (\ref{basicTriangle}).

Similarly $LB$ is given by
\[
\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"0"'] & \ar[l,"{M}"'] b \ar[loop , looseness=4,"B"']
\end{tikzcd}
\]
and $LM$ is given by
\[
\!\!\!\!\begin{tikzcd}[column sep = 15mm]
    a \ar[loop, looseness=5,"0"'] & \ar[l,"{M}"'] b \ar[loop , looseness=4,"0"']
\end{tikzcd}\!\!\!\!\!\!\!.
\]
The map $LM\to LA\oplus LB$ is given by $M\to M\oplus M,\ m\mapsto (m,-m)$ in the ${}_a(LM)_b\to {}_a(LA\oplus LB)_b$ component, and (necesserily) zero at all other objects. This in fact produces a short exact sequence $0\to LM\to LA\oplus LB \to T\to 0$ of $T\-T$ bimodules, giving the desired triangle.

Since the components of $T$ are perfect as left and right modules over $A$ and $B$ respectively, all the the functors named $L$ descend to functors $\underline{L}$ on singularity categories (this was explained in section \ref{stableHHSection}). Therefore, when we stabilize the triangle (\ref{basicTriangle}) by applying the quotient $\cdbcat(T\- T)\to \csing(T\- T)$, we get
\[
\underline{L}M\to \underline{L}A\oplus \underline{L}B \to T\to \ .
\]
Applying $\sExt_{T\- T}(-,T)$ produces the (Meyer-Vietoris) exact sequence
\[
\cdots\to \sExt_{T\- T}(T,T) \to \sExt_{T\- T}(\underline{L}A,T) \oplus
    \sExt_{T\- T}(\underline{L}B,T) \to \sExt_{T\- T}(\underline{L}M,T) \to \cdots
\]
By the adjunction between $\underline{L}$ and $\underline{R}$ (where $\underline{R}$ is the appropriate stabilized restriction map), and the fact that $\underline{R}\underline{L}$ is  naturally isomorphic to the identity, this long exact sequence is isomorphic to
\[
\cdots\to \sExt_{T\- T}(T,T) \to \sExt_{A\- A}(A,A) \oplus
    \sExt_{B\- B}(B,B) \to \sExt_{A\- B}(M,M) \to \cdots\ .
\]

In this sequence the two components of $\sExt_{T\- T}(T,T) \to \sExt_{A\- A}(A,A) \oplus
    \sExt_{B\- B}(B,B)$ are precisely the stabilized restrictions  $\underline{R}_A$ and $\underline{R}_B$ in the statement of the theorem.

To show that these are isomorphisms it is equivalent to show that the two components of $ \sExt_{A\- A}(A,A) \oplus
    \sExt_{B\- B}(B,B) \to \sExt_{A\- B}(M,M) $ are isomorphisms.  It is sufficient to show that they are each given by tensoring against $M$ (on the appropriate side), and therefore are isomorphisms by hypothesis. Before stabilizing, this means we must check that the following diagram commutes
\[
\begin{tikzcd}[column sep=15mm, row sep=5mm]
    \Ext_{T\-T}(LA,T) \ar[r]\ar[d,"\cong"] & \Ext_{T\-T}(LM,T) \ar[d,"\cong"]\\
    \Ext_{A\-A}(A,A) \ar[r,"-\otimes^{\sf L}_A M"] & \Ext_{A\-B}(M,M)
\end{tikzcd}
\]
One checks this by starting in the lower-left corner. The point is that the adjunction $\Ext_{A\-A}(A,A) \cong \Ext_{T\-T}(LA,T)$ by definition extends a map $\xi:A\to A$ to $LA\to T$  along the tensor product ${}_a(LA)_b=A\otimes^{\sf L}_{A^\ev}(A\otimes M) \simeq M \to M = {}_a T_b$. Since ${}_a(LA)_b\simeq {}_a(LM)_b$ the top arrow makes no difference after restricting to the lower-right corner. (One should really check this diagram commutes using projective bimodule resolutions of $A$ and $M$, but again this makes little difference.) After stablising the diagram the lower arrow becomes an isomorphism by hypothesis.

The verification for $B$ is very similar. This finishes the proof.
\end{proof}

Let us point out that the hypotheses of Theorem \ref{keytheorem} already imply
that $M$ is one half of a stable equivalence of Morita type. This proposition is presumably well-known (at some level of generality).

\begin{proposition}\label{SEMTprop}
Let $M$ be an $A\- B$ dg bimodule
which is perfect on both sides. If $\-\otimes_A^{\sf L}M:\sing(A\- A)\to \sing(A\- B)$ and $M\otimes_B^{\sf L}\-:\sing(B\- B)\to \sing(A\- B)$ are both fully faithful then there is a $B\- A$ dg bimodule $N$, perfect on both sides, such that $M\otimes_B^{\sf L} N\simeq A$ and $N\otimes_A^{\sf L} M \simeq B$ in $\sing(A\- A)$ and $\sing(B\- B)$ respectively.
\end{proposition}

If $A$ and $B$ are Gorenstein algebras then $M$ and $N$ could be replaced with  $B\- A$ bimodules (rather than a complexes of such), by taking syzygies. More classically, if $A$ and $B$ are self-injective then this is easily seen to coincide with the usual definition.

\begin{proof}
Note that ${}^\vee M=\RHom_{A^\op}(M,A)$ is naturally a dg $B\-A$-bimodule. Since ${}_AM$ is perfect there is a natural quasi-isomorphism $\-\otimes_A^{\sf L} M\to \RHom_{A}({}^\vee M,\-)$. It follows that there are natural isomorphisms
\[
\sHom_{A\-A}(A,\-)\xrightarrow{\-\otimes_A^{\sf L} M}\sHom_{A\-B}(M,\-\otimes_A^{\sf L} M)\xrightarrow{\cong}\sHom_{A\-B}(M,\RHom_{A}({}^\vee M,\-))
\]
\[
\hspace{75mm}\cong \sHom_{A\-A}(M\otimes_A^{\sf L} {}^\vee M,\-).
\]
So by the Yoneda lemma $M\otimes_B^{\sf L} {}^\vee M\simeq A$ in $\sing(A\- A)$. Similarly if we set $M^\vee=\RHom_B(M,B)$ then $M^\vee\otimes_A^{\sf L} M\simeq B$ in $\sing(B\- B)$. It follows that ${}^\vee M\simeq M^\vee$, by the usual argument for uniqueness of inverses.
\end{proof}

\section{Stable Invariance of the restricted Structure}

 A ring is called \emph{Iwanaga Gorenstein} if it is two-sided noetherian, and if it has finite injective dimension over itself as both a left and right module (Buchweitz calls these rings strongly Gorenstein \cite{Buc}). By a result of Zaks, this implies that the two injective dimensions $\idim {{}_AA}$ and $\idim A_A$ coincide \cite{Zaks}.

\begin{lemma}\label{Gorlemma}
Assume that $k$ is Gorenstein, and that $A$ is finite and projective over $k$ with finite self-injective dimension $d$ on both sides. Then $A^\ev$ has self-injective dimension at most $2d$ on either side, and
\[
\Ext^i_{A^\ev}(A,A)\to \sExt^i_{A^\ev}(A,A)
\]
is an isomorphism for $i> d$.
\end{lemma}

If would be enough to assume that $A$ is perfect over $k$ here (i.e. having a finite resolution by finitely generated projectives), but this would involve some additional dg algebra techniques.

\begin{proof}
Denote $(\-)^*=\RHom_k(\-,k)$. Since $k$ is Gorenstein $(M^*)^*\simeq M$ whenever $M$ is in $\dbcat(k)$ (e.g.~ \cite{Buc}).

If $M$ is a finitely generated $A^\ev$-module then
\begin{align*}
\RHom_{A^\ev}(M,A^{\ev})& \simeq \RHom_{A^\ev}(M,\Hom(A^*,A))
\\
& \simeq \RHom_{A}(A^*\otimes^{\sf L}_AM,A)\\ &\simeq \RHom_{A}(A^*,\RHom_A(M,A))\\
&\simeq  \RHom_{A^\op}(\RHom_A(M,A)^*,A) .
\end{align*}
The last quasi-isomorphism follows from the Gorenstein duality mention above (which applies since $A$ and $\RHom_A(M,A)$ are in $\dbcat(k)$). The other quasi-isomorphisms are standard adjunctions.

By inspection the last term in this string of isomorphisms has cohomology concentrated in degrees  at most $\idim {{}_AA}+\idim A_A = 2d$. This implies that $\idim A^\ev_{A^\ev}\leq 2d$. The same argument shows that $\idim {{}_{A^\ev}A^\ev}\leq 2d$.

Clearly $A^\ev$ is two-sided noetherian, therefore we've shown that it is Iwanaga Gorenstein. Because of this we can use the long exact sequence
\[
\cdots\to \sExt^{i-1}_{A^\ev}(A,A)\to {\rm Tor}^{A^\ev}_{-i}(A,A^\vee)\to \Ext^i_{A^\ev}(Y,X)\to\sExt^i_{A^\ev}(A,A)\to \cdots
\]
from \cite[theorem 6.2.5]{Buc}, where $A^\vee=\RHom_{A^\ev}(A,A^{\ev})$. Using the isomorphism above with $M=A$ we see that $
A^\vee\simeq \RHom_{A}(A^*,A)
$
has cohomology in degrees $d$ and below. This means that ${\rm Tor}^{A^\ev}_{-i}(A,A^\vee)=0$ for $i>d$, and the statement follows from the long exact sequence.
\end{proof}

\begin{theorem}\label{generalVersion}
Assume that $k$ is Gorenstein, and that $A$ and $B$ are finite and projective as $k$-modules, and that both have (left and right) self-injective dimension less than or equal to $d$.

If $A$ and $B$ are connected by a stable equivalence of Morita type then there is an isomorphism
\[
 {\rm HH}^{>d}(A,A)\cong {\rm HH}^{>d}(B,B)
\]
which preserves the cup product, and (after shifting) the \underline{restricted} graded Lie-algebra structure.
\end{theorem}

\begin{proof}
This follows from observing the commutative diagram (\ref{StabilisationDiagram}) above. The two lower horizontal maps are isomorphisms by Theorem \ref{keytheorem}. The outside vertical maps are isomorphisms in degrees $d$ and above by lemma \ref{Gorlemma}. The upper horizontal maps preserve all the required structure by Theorem \ref{E2restrictedLie}. Combining these facts finishes the proof.
\end{proof}

Our main interest, coming from the field of modular representation theory, is the case of self-injective algebras over fields. Specialising Theorem \ref{generalVersion} to this context yields Theorem \ref{stableinvariancetheorem} from the introduction, with one caveat to be explained in the next subsection.

\subsection{Transfer maps}
\label{transfersection}
To finish the proof of Theorem
\ref{stableinvariancetheorem} from the introduction, we need to explain why the isomorphism of Theorem \ref{generalVersion} is given by a transfer map.

 Transfer maps were introduced by Bouc  \cite{Bouc}  for Hochschild homology, and  defined for the Hochschild cohomology of symmetric algebras by Linckelmann  \cite{Linc}. They are the classical
 way of comparing the Hochschild cohomology of two algebras
 connected by a stable equivalence of Morita type.

 Since the definition of stable equivalence of Morita type we use here is more general than usual, we need to explain what we mean by a transfer map.

If $M$ and $N$ are $A\-B$ and $B\-A$ bimodules respectively, we have a map
\[
\Ext_{A\-A}(A,A)\to \Ext_{B\-B}(N\otimes^{\sf L}_AM,N\otimes^{\sf L}_AM),\quad f\mapsto N\otimes^{\sf L}_A f\otimes^{\sf L}_AM.
\]
Assuming $M$ and $N$ are each of finite flat dimension on both sides, this descends to a map
\[
\sExt_{A\-A}(A,A)\to \sExt_{B\-B}(N\otimes^{\sf L}_AM,N\otimes^{\sf L}_AM).
\]
Suppose we are given an isomorphism $N\otimes^{\sf L}_AM\simeq B$ in $\csing(B\-B)$. The \emph{transfer map} is by definition the composition
\[
{}_N{\rm tr}_M : \sExt_{A\-A}(A,A)\to \sExt_{B\-B}(N\otimes^{\sf L}_AM,N\otimes^{\sf L}_AM)\xrightarrow{\cong} \sExt_{B\-B}(B,B).
\]
In the case of an algebra over a field, this agrees with the more concrete definition given by Linckelmann (where one usually uses $N=M^*$). In this context the transfer map is manifestly an algebra homomorphism, but it turns out \emph{not} to respect the restricted Lie algebra structure of $\sExt_{A\-A}(A,A)$ and  $\sExt_{B\-B}(B,B)$ in general, see the counter-example in \cite[section 6]{RyD}.

If moreover $M\otimes^{\sf L}_AN\simeq A$ then
${}_N{\rm tr}_M$ and ${}_M{\rm tr}_N$ are inverse isomorphisms. Surprisingly, it \emph{is} true in this case that these isomorphisms respect the restricted Lie algebra structure; this follows from Theorem \ref{generalVersion} and the following.

\begin{proposition}\label{transferprop}
The isomorphism $\sExt_{A\-A}(A,A)\cong\sExt_{B\-B}(B,B)$ of Theorem \ref{keytheorem} coincides with ${}_N{\rm tr}_M$.
\end{proposition}
\begin{proof}[Sketch of proof.]
One can check that the following square commutes:
\[
\begin{tikzcd}
    \sExt_{T\-T}(T,T) \ar[r,"\underline{R}_B"]\ar[d,"\underline{R}
    _A"'] & \sExt_{B\-B}(B,B) \ar[d,"M \otimes_B^{\sf L} \-"] \\
    \sExt_{A\-A}(A,A) \ar[r, "\-\otimes_A^{\sf L} M"] \ar[ur, dashed]& \sExt_{A\-B}(M,M).
\end{tikzcd}
\]
The square consists of isomorphisms, which gives us the dashed arrow $\underline{R}_B(\underline{R}_A)^{-1}$. This is the isomorphism constructed in the proof of Theorem \ref{keytheorem}. Since $N\otimes_B^{\sf L} \- $ is the  inverse of $M \otimes_B^{\sf L} \-$, the dashed arrow is also given by $N \otimes_B^{\sf L} \-\otimes_B^{\sf L} M = {}_N{\rm tr}_M$.
\end{proof}

\begin{remark}[Generalisation to all of stable Hochschild cohomology]
The proof of our main theorem goes through stable Hochschild cohomology, but we do not make serious use of its structure, and we are not significantly interested in it as an invariant by itself (only because ordinary Hochschild cohomology is more computatable and more widely used). Nevertheless, we quickly point out now that one can generalize our main theorem to all of $\sHH(A,A)$ (so, into negative degrees).

Wang has studied the structure of stable Hochschild cohomology \cite{Wang}. He constructs a stable version of the Hochschild cochain complex and
establishes an analogue of the Deligne conjecture for it. It follows from this and from Theorem \ref{E2restrictedLie} that $\susp \sHH(A,A)$ is naturally a
restricted graded Lie algebra as well. By inspecting Wang's stable cochain
complex, the arguments above can be used to show that this structure is invariant
under stable equivalence of Morita type.

\end{remark}

\section{Applications in modular representation theory}
\subsection{The classification of tame symmetric algebras}
\label{tamesym}

Since we now know the restricted Lie structure of ${\rm HH}^1(A,A)$ to be an
invariant under stable equivalence of Morita type, we can hope to use it in
classification problems.

We contribute to improve the (as yet incomplete) classification of algebras of dihedral, semi-dihedral and quaternion type in characteristic $2$ up to stable equivalence of Morita type, after the
work of Zhou and Zimmermann \cite{ZZ}, and  Taillefer \cite{Taillefer}. The Lie brackets of ${\rm HH}^1(A,A)$ for some of these algebras have been computed in \cite{Taillefer}. In \cite{ER} the authors prove the solvability of ${\rm HH}^1(A,A)$
when the Ext-quiver of  algebras  of dihedral, semi-dihedral and quaternion type  does not have loops. In \cite{RSS} the authors prove the solvability of  ${\rm HH}^1(A,A)$
for all algebras of
dihedral, semi-dihedral and
quaternion type excluding blocks of group algebras with Klein defect group. Thus, we
 compute the restricted Lie structure for most of the remaining cases. The dihedral case has been
completely classified in \cite {ZZ} and the two remaining families are considered in \cite{Taillefer}.
Our calculations failed to distinguish any algebras which are currently not known
to be in different stable Morita equivalence classes. This provides some evidence that most these classes are in fact stably equivalent.
In order to prevent others
from having to repeat our calculations, we outline them below.
In what follows we use
the notation of \cite{Taillefer}.

\begin{itemize}
    \item The algebras $SD(1\mathcal{A})^k_2(c,1)$ and $SD(1\mathcal{A})^k_1(c{'},
    1)$, for $c \neq c'$ non-zero, are
    derived equivalent and their first  Hochschild cohomologies are isomorphic as
    Lie algebras. Using the same isomorphism
     of \cite[proposition 4.9]{Taillefer},  ${\rm HH}^1(SD(1\mathcal{A})^k_2(c,1))$ and ${\rm
    HH}^1(SD(1\mathcal{A})^k_1(c{'}, 1))$
    are isomorphic as restricted Lie algebras.

    Therefore we are still not able to say whether or not these algebras are
    stably Morita equivalent.

    \item In the case of $Q(1\mathcal{A})^k_2(0,d)$ and
    $Q(1\mathcal{A})^k_2(0,d')$ for $k$ odd and $d\neq d{'}$ we have an isomorphism
    of restricted Lie algebras, but one needs to slightly change the isomorphism of \cite[Corollary 5.6]{Taillefer}. More precisely, the adjusted isomorphism
    acts as: $\psi\mapsto\lambda\psi'$, $\chi\mapsto\tilde{\mu}\chi'$ and
    $\theta_{-2} \mapsto \mu \theta_{-2}'$,
    where $\lambda^2=(\frac{d}{d'})^3$, $\tilde{\mu}=\frac{\lambda'}{d}$
    and $\mu=\frac{d}{d'\tilde{\mu}}$, in the notation of \cite{Taillefer}.

    \item In the case ${c, d} \neq {c', d'}$ with $cd \neq 0$ and $c{'}d{'} \neq 0$ we couldn't establish an isomorphism of restricted Lie algebras. Therefore we still do not know if $Q(1\mathcal{A})^k_2 (c, d)$ and $Q(1\mathcal{A})^k_2(c{'}, d{'})$ for  are stably equivalent of Morita type or not.

\end{itemize}

\subsection{Blocks of group algebras of defect \texorpdfstring{$2$}{}  and quantum complete intersections}
Let $p$ be an odd prime. Let $B$ be a block of a group algebra with defect group
$D$ of order $p^2$ and let $C$ the  Brauer correspondent of $B$
(a block of $kN_G(D)$). Then by \cite{Rou} there is a
stable equivalence of Morita type between $B$ and $C$.
In addition, if $C$ has one simple module, then by \cite{KL} $B$ has
one simple module. If $B$ is a nilpotent block, then $B$ is Morita
equivalent to $kD$ and therefore $\rm{HH}^1(B,B)$ is a Jacobson-Witt algebra. Otherwise,
following the same arguments of the proof of
\cite[Corollary 1.4]{BKL}, we have that $D$ is elementary
abelian of rank 2 and
a basic algebra of $C$ is a quantum
complete intersection $A$ of rank 2 with quantum parameter $q$
which is a root of unity. In \cite{BKL} the authors have calculated
the restricted structure of $\rm{HH}^1(A,A)$. Then we have

\begin{proposition}
Let $B$ a block of a group algebra $kG$ with defect group of order $p^2$ and with Brauer correspondent $C$. Assume that $C$ has one simple module and that $B$ is not nilpotent. Then $\rm{HH}^1(B)=\mathcal{H}\oplus \mathcal{S}$, where $\mathcal{H}$ is a maximal p-toral subalgebra and $\mathcal{S}$
is the derived subalgebra of $\rm{HH}^1(B)$.
\end{proposition}

Note that the validity of Brou\'e's abelian defect group conjecture
would imply
that $B$ is derived equivalent to $C$ and by a result of Roggenkamp and Zimmermann  \cite[Proposition 6.7.4]{ZR} we would have a Morita equivalence between $B$ and $C$. Kessar proves in  \cite{Kess} that the Morita equivalence holds for $p=3$.

\section{Applications in commutative algebra}

Suppose $A$ is a commutative Gorenstein ring essentially of finite type over a
field $k$, with Krull dimension $d$. Then $A^\ev$ is  Gorenstein of
Krull dimension $2d$, and   $\Ext_{A^\ev}^i(A,A^{\ev})=0$ for $i>d$ (by
\cite[Theorem 2.1 (iv)]{AI} for example). Knowing this, the same proof as for
theorem \ref{generalVersion} gives:

\begin{theorem}\label{commutativetheorem}
Assume that $A$ and $B$ are commutative Gorenstein rings essentially of finite type over a field $k$, both of Krull dimension less than or equal to $d$.
If $A$ and $B$ are connected by a stable equivalence of Morita type then there is an isomorphism $
 {\rm HH}^{>d}(A,A)\cong {\rm HH}^{>d}(B,B)
 $
which preserves the cup product, and the restricted graded Lie-algebra structure.
\end{theorem}

\begin{example}\label{KnorrerEx}
A well-known family of stable equivalences of Morita type is given by Kn\"orrer's periodicity theorem  \cite{Knorrer}---but this is usually not phrased in these terms.

Let $S=k[x_1,...,x_n]$ and let $f$ be a nonzero element of $S$. We set $A=S/(f)$ and $B=S[u,v]/(f+uv)$. Theses rings are Gorenstein of dimension $n-1$ and $n+1$ respectively. We consider $U=A[u]=B/(v)$ and $V=A[v]=B/(u)$ as $A\-B$ and $B\-A$ bimodules respectively. One can check that $U\otimes^{\sf L}_B V\simeq A$ in $\csing(A^\ev)$ and $V\otimes^{\sf L}_A U\simeq B$ in $\csing(B^\ev)$. Hence $A$ and $B$ are stably equivalent of Morita type (no assumptions on the field $k$ are needed). It follows from Theorem \ref{commutativetheorem} that  there is an isomorphism ${\rm HH}^{>n+1}(A,A)\cong {\rm HH}^{>n+1}(B,B)$ preserving the product and the restricted Lie structure.
\end{example}

For hypersurface rings
stable Hochschild cohomology is always $2$-periodic, and so ordinary Hochschild cohomology  becomes $2$-periodic above the dimension of the ring. In the case of Kn\"orrer periodicity, example \ref{KnorrerEx} says that ${\rm HH}^*(A,A)$ and $ {\rm HH}^*(B,B)$ have the same periodic part.

\appendix
\section{Well-definedness of the \texorpdfstring{$p$}{}-power operation}

Here we will explain briefly some ideas due to May and Cohen which can be used to exhibit the $p$-power operation on Hochschild cohomology.  These ideas are not well-known outside of the topology community, so it seems worthwhile to sketch them here. We have relegated the arguments to an appendix because we need to assume some knowledge of operads.  Our thanks are due to Victor Turchin for explaining Theorem \ref{Maythm} to us.

Our goal here is to finish the proof of Theorem \ref{E2restrictedLie} above: we need to show that the $p$-power operation is well-defined on the homology of a $B_\infty$-algebra.
But let us point out that it is quite easy to see that the $p$-power operation on ${\rm HH}^1(A,A)$ is well-defined
; this appendix is only needed for classes in ${\rm HH}^{>1}(A,A)$.  Even though our main interest is ${\rm HH}^1(A,A)$, we include this appendix for the sake of completeness, and because these arguments are difficult to find in the literature.

As explained in section \ref{restrictedsubsection}, one can use the model $\coder(BA,BA)$ to exhibit the restricted structure on Hochschild cohomology. It is stated in \cite[Lemma 4.2]{ZII} that this approach yields a well-defined $p$-power operation, but the proof there is  incomplete. To establish well-definedness, one must show that for every odd cycle $x\in \coder^i(BA,BA)$ and every $y\in \coder^{i+1}(BA,BA)$ there is an element $w$ such that $(x+\partial(y))^p = x^p +\partial(w)$. It is not checked in \cite{ZII} that $w$ is a coderivation, and unfortunately the choice used there is not. It seems to be \emph{extremely} complicated to construct directly a suitable $w$, and hence a nonconstructive method is needed.

Here we assume knowledge of operads (specifically, $\Sigma$-split dg operads \cite{Hinich}). We work with the brace operad $\mathcal{B}$ \cite{GV}, which is an example of a $\Sigma$-split $E_2$-operad. It seems to be necessary to involve this machinery to establish that the $p$-power operation is well defined. The following two theorems together settle this matter.

The first is a result of May \cite{May};  it is a general template for building cohomology operations. Since  \cite{May} appeared just before operads were first defined, and since it was written in topological terms, it seems worthwhile to give an algebraic argument here.

\begin{theorem}[May]\label{Maythm}
Let $\O$ be a $\Sigma$-split dg operad, and let $A$ be a dg $\O$-algebra. Suppose that $\xi \in \O(n)^j$ is such that $\partial(\xi)$ becomes zero in $\O(n)\otimes_{\Sigma_n} ({\rm sign})$. Then, for $i$ odd, the operation
\[
{\rm H}^i(A)\to {\rm H}^{ni+j}(A),\quad a\mapsto \xi(a,...,a)
\]
is well-defined on cohomology classes.

Similarly, if $\partial(\xi)=0$ in $\O(n)\otimes_{\Sigma_n} ({\rm triv})$ then $\xi$ induces a well-defined operation on even cohomology classes.
\end{theorem}

The next theorem originates in topology, in the work Cohen \cite{Cohen}. The arguments of \cite{Cohen} are purely topological calculations concerning double loops spaces. The algebraic version here is essentially due to Turchin.

\begin{theorem}[Cohen, Turchin]\label{Cohenthm}
Let $p$ be a prime and let $\B$ be the brace operad over $\mathbb{F}_p$. Denote by
$\xi = (\-)\{\-\}\cdots\{\-\}$ the $p$-fold iterated brace operation (in Gerstenhaber's notation $\xi = ((\-\circ \-)\circ \cdots \circ \- ) \circ \-$).  Then $\partial(\xi)$ becomes zero in $\B(p)\otimes_{\Sigma_p} ({\rm sign})$. Hence $\xi$ induces a well-defined, $p$-power linear operation on the homology of any $\B$-algebra.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{Maythm}.]
We only prove the first statement, involving the sign representation. The statement about the trivial representation is proven the same way.

So assume that $\partial(\xi)=0$ in $\O(n)\otimes_{\Sigma_n} ({\rm sign})$, and take two cycles $a,a'\in A^i$ and $b\in A^{i-1}$ with $\partial(b)=a-a'$. Our goal is to show that $\xi(a,...,a)$ and $\xi(a',...,a')$ are cohomologous cycles.

Let $I$ be a one-dimensional vector space concentrated in degree $i$. We identify $a$ and $a'$ with the corresponding maps $I\to A$, and we consider the induced maps
\[
\begin{tikzcd}
 \O(n)\otimes_{\Sigma_n}(I^{\otimes n}) \ar[r, shift left=3pt, "\O(n)\otimes_{\Sigma_n}(a^{\otimes n})"] \ar[r, shift right=3pt, "\O(n)\otimes_{\Sigma_n}(a'^{\otimes n})"']  &[10mm] \O(n)\otimes_{\Sigma_n} A^{\otimes n} \ar[r,"\mu"] & A.
\end{tikzcd}
\]
Here $\mu$ is the given $\O$-action on $A$. Since $i$ is odd $I^{\otimes n}\cong({\rm sign})$ as left $\Sigma_n$-modules, so by  assumption $\xi$ represents a class in $\O(n)\otimes_{\Sigma_n}(I^{\otimes n})$. By construction
\[
\mu (\O(n)\otimes_{\Sigma_n}a^{\otimes n})(\xi) = (-1
)^{nj}\xi(a,...,a)\ \text{ and }\ \mu (\O(n)\otimes_{\Sigma_n}a'^{\otimes n})(\xi) = (-1
)^{nj}\xi(a',...,a').
\]
Hence it suffices to show that $\O(n)\otimes_{\Sigma_n}(a^{\otimes n})$ and $\O(n)\otimes_{\Sigma_n}(a'^{\otimes n})$ induce the same map in cohomology.

Let $V$ be the linear span in $A$ of $a,a'$ and $b$. This is a subcomplex of $A$ (with dimension $3$ or less). Both $a,a'\colon I\rightrightarrows V$ are quasi-isomorphisms, and both are split by the retraction $r:V\to I$ with $r(b)=0$ and $r(a)=r(a')=$ the generator of $I$. Since $r$ is a quasi-isomorphism, so is $r^{\otimes n}: V^{\otimes n}\to I^{\otimes n}$. Since $\O$ is $\Sigma$-split, $\O(n)\otimes_{\Sigma_n}(r^{\otimes n})$ is also a quasi-isomorphism. Finally, since $\O(n)\otimes_{\Sigma_n}(a^{\otimes n})$ and $\O(n)\otimes_{\Sigma_n}(a'^{\otimes n})$ (or strictly speaking, the induced maps into $\O(n)\otimes_{\Sigma_n}(V^{\otimes n})$) are both split by the same quasi-isomorphism $\O(n)\otimes_{\Sigma_n}(r^{\otimes n})$, they must induce the same map in cohomology.
\end{proof}

In the next proof we will need certain permutations $\tau_i\in \Sigma_n$ given by
\[
\tau_i(j)=\begin{cases} j & \text{ if } j< n-i\\
n & \text{ if } n-i\leq j< n\\
n-i & \text{ if } j=n
\end{cases}
\]
And we will use the fact that ${\rm sign}(\tau_i)=(-1)^i$. We also need to record some relations which hold in the brace operad:

\begin{equation}\label{rel1}
\partial(\-\{\-\})= (\-\smile\-) - (\-\smile\-)\tau_1
\end{equation}
\begin{equation}\label{rel2}
(\-\smile \-)\{ \-\}= (\-\smile( \- \{\-\})) + ((\-\{\-\})\smile\-)\tau_1,
\end{equation}
These can be found in \cite{GV} (they both go back to Gerstenhaber's earlier work \cite{Gerst2}).

\begin{proof}[Proof of Theorem \ref{Cohenthm}.]
We denote $\xi^{(n)}=(\-)\{\-\}\cdots\{\-\}$ with $n$ inputs. In particular $\xi^{(1)}$ is the identity and $\xi^{(p)}$ is $\xi$ itself. Note that $\xi^{(n)}$ has degree $1-n$.

We will prove by induction on $n$ that in $\B(n)\otimes_{\Sigma_n} ({\rm sign})$ we have
\begin{equation}\label{indhyp}
\partial(\xi^{(n)})=\sum_{i=1}^{n-1}(-1)^{ni}\binom{n}{i}\xi^{(i)}\smile \xi^{(n-i)}.
\end{equation}
This formula was discovered by Turchin (with different signs), see \cite[Proposition 11.1]{TourtchineII}. For completeness we include a detailed proof.  The base case uses the relation (\ref{rel1}):
\[
\partial(\xi^{(2)})=(\-\smile\-) - (\-\smile\-)\tau  = 2 \xi^{(1)}\smile \xi^{(1)}.
\]
For the induction step we use the Leibniz rule
\begin{equation*}
\partial(\xi^{(n+1)})  = \partial(\xi^{(2)}\circ_1\xi^{(n)}) = \partial(\xi^{(2)})\circ_1\xi^{(n)} - \xi^{(2)}\circ_1\partial(\xi^{(n)}).
\end{equation*}
We expand the first summand here using (\ref{rel1}):
\begin{align*}
\partial(\xi^{(2)})\circ_1\xi^{(n)} & = (\-\smile\-)\circ_1\xi^{(n)}-((\-\smile\-)\tau_1)\circ_1\xi^{(n)}\\
& = (\-\smile\-)\circ_1\xi^{(n)}-(\-\smile\-)\circ_2\xi^{(n)} \tau_n\nonumber\\
& = \xi^{(n)}\smile \xi^{(1)}+(-1)^{n+1}\xi^{(1)}\smile \xi^{(n)}.\nonumber
\end{align*}
And also the second summand:
\begin{align*}
 \xi^{(2)}\circ_1\partial(\xi^{(n)})
    & =  \sum_{i=1}^{n-1}(-1)^{ni}\binom{n}{i}\xi^{(2)}\circ_1\left[\xi^{(i)}\smile \xi^{(n-i)}\right]\\
    & = \sum_{i=1}^{n-1}(-1)^{ni}\binom{n}{i}\left[\xi^{(i+1)}\smile \xi^{(n-i)}\tau_{n-i}+(-1)^{i-1}\xi^{(i)}\smile \xi^{(n-i+1)}\right]\nonumber\\
    & = \sum_{i=1}^{n-1}\binom{n}{i}\left[(-1)^{ni+n-i}\xi^{(i+1)}\smile \xi^{(n-i)}+(-1)^{ni+i-1}\xi^{(i)}\smile \xi^{(n-i+1)}\right].\nonumber
\end{align*}
The first equality is the induction hypothesis; the second one uses (\ref{rel2}) precomposed with $\xi^{(i)}\otimes \xi^{(n-i)}$; the last equality just groups signs together.
If we combine these two summands and group together coefficients we find
\begin{align*}
\partial(\xi^{(2)})\circ_1&\xi^{(n)} - \xi^{(2)}\circ_1\partial(\xi^{(n)})   \\
       & = -\sum_{i=1}^{n}\left[(-1)^{n(i-1)+n-i+1}\binom{n}{i-1}+(-1)^{ni+i-1}\binom{n}{i}\right]\xi^{(i)}\smile \xi^{(n+1-i)}\\
       & = \sum_{i=1}^{n}(-1)^{(n+1)i}\left[\binom{n}{i-1}+\binom{n}{i}\right]\xi^{(i)}\smile \xi^{(n+1-i)}\\
       & = \sum_{i=1}^{n}(-1)^{(n+1)i}\binom{n+1}{i}\xi^{(i)}\smile \xi^{(n+1-i)}.
\end{align*}
This completes the proof by induction that (\ref{indhyp}) holds. Since $\binom{p}{i}=0$ for $i=1,...,p-1$
this completes as well the proof that $\partial(\xi)=0$ in $\B(p)\otimes_{\Sigma_p}({\rm sign})$.

By May's theorem \ref{Maythm} above, $\xi$ induces a well-defined cohomology operation on any $\B$-algebra.
\end{proof}


TITLE: Simplifying expression $\sqrt{6+2\sqrt{5}} - \sqrt{6-2\sqrt{5}}.$
QUESTION [3 upvotes]: I'm stuck. Squaring it will change it's value. Is there any general method of simplifying expressions of the form $$\sqrt{a+b}-\sqrt{a-b} = c \ \ ?$$

REPLY [2 votes]: Here is how you de-nest double layer square roots.  All variables below are assumed rational.
Start with
\begin{equation} \tag{1}
\sqrt{a+ \sqrt{b}}
= \sqrt {x} + \sqrt{y},
\end{equation}
which has the conjugate relation
\begin{equation} \tag{2}
\sqrt{a - \sqrt{b}}
= \sqrt{x} -\sqrt{y},
\end{equation}
where $x\ge y $ wlog.
Multiply these two relations together, and use the difference of squares factorization:
\begin{equation} \tag{3}
\sqrt {a^2-b}
\equiv r
= x-y.
\end{equation}
The quantity $a^2-b$ must be a perfect square if the de-nesting is to succeed.  If so, call its square root $r $.
Now using the uniqueness of quadratic sure, square (1) and equate rational parts, getting:
\begin{equation} \tag{4}
a
=x + y.
\end{equation}
Equations (3) and (4) constitute a linear system for $x$ and $y$, which is easily solved, thus
\begin{equation*}
x = \frac{a + r}{2}
\qquad \text{and} \qquad
y = \frac{a - r}{2}.
\end{equation*}
Plug these results into the given problem  ($a=6, b=20$) and infer that the de-nested roots are:
$$\sqrt {6+2\sqrt {5}} =\sqrt {5}+1
\qquad \text{and} \qquad
\sqrt {6-2\sqrt {5}} =\sqrt {5}-1
$$
with the difference of $2$.
Note:If in this development we allow negative values of $b $ and drop the rationality requirement on $r$, this is also a method to extract square roots of complex numbers.


TITLE: How do I find the cardinality of set with all the infinite binary sequences which consist of infinite "ones" and "zeros"?
QUESTION [2 upvotes]: Set $S$ contains all the infinite binary sequences which must consist of infinite "ones" and infinite "zeros", first of all I am not sure if it is countable or not.. I thought about cantors diagonal but I am not sure about that, hope someone could help me here.
thank you in advance.

REPLY [2 votes]: Let $U = $ the set of all infinite series
$S = $ the set of all infinite series with infinite 1s and 0s
$A = $ the set of all infinite series with finite 0s
$B = $ the set of all infinite series with finite 1s.
$U = S \cup A \cup B$.
Claim 1: U is uncountable.
Claim 2: A and B are countable
Claim 3: the countable union of countable set is countable.
Hence, if $S$ were countable, $U$ would be countable.  It isn't so $S$ is uncountable.
===
Claim 1: This is Cantor's diagonal.  If U were countable we could list all sequences so $U = \{A_i\} = \{\{a_{i_k}\}\}$.  Then the sequence $B = \{1-a_{i_i}\} \not \in U$.  A contradiction so $U$ is uncountable.
Claim 2: If an infinite sequence, {$a_n$} contains a finite number of 0s then there is an $a_m$ that is the last 0 value and all the remaining values are 1.  Consider the finite sequence {$b_n$} that has $m$ terms were $b_i = a_i$.  For every infinite sequence with finite number of 0 there is such a corresponding finite sequence.  And for every finite sequence we can find a corresponding infinite sequence by "sticking" an infinite number of 1s on the end.
So there is a 1-1 correspondence between finite sequences  and infinite sequences with finite 0s.
But the set of finite sequences is countable.  ($t(\{b_n\})=\sum_{i=0}^m b_n*2^i \in \mathbb N$ is a 1-1 mapping from the finite sequences to the natural numbers.)
So there are countably many infinite sequences with finite 0s, and likewise there are countably many infinite sequences with finite 1.
Claim 3: Let $\{S_i\}$ be a class of countable sets so that each  $S_j = \{s_{j,k}\}$ so $\cup_i S_i = \{s_{j,k}\}$ with some possibility of multiple representation. It suffices to show that $\mathbb N \times \mathbb N$ is countable.  Basically the map (0,0)-> 0, (1,0) ->1, (0,1) ->2, (2,0)->3, (1,1) -> 4, (0,2)->5 etc or $(i,j) \rightarrow \sum_{l=0}^{j+k}l + k$ is a 1-1 mapping.


TITLE: Comoving Hubble radius in terms of expansion scale factor
QUESTION [0 upvotes]: I am currently reading Baumann's Cambridge lecture slides on cosmology and am confused on the notion of the Hubble radius. On page 10 it is stated that for a perfect fluid with constant equation of state $w = P/\rho$, the comoving Hubble radius is $(aH)^{-1} \propto a^{0.5(1+3w)}$, but I haven't the faintest idea where this equation came from, as no derivation is given. Can anyone provide one?

REPLY [1 votes]: In Planck units, $M_P=(8\pi G)^{-1}=1$.


TITLE: Let $A=\{0,1\}$. How many strings of length $5$ are in $A^*$ where at least two $1$ are next to each other?
QUESTION [0 upvotes]: Let $A=\{0,1\}$. How many strings of length $5$ where at least two $1$ next to each other are there in $A^*$?

REPLY [0 votes]: Here's an ugly/inconvenient direct count, by breaking them up into instances by the maximum consecutive instances of $1$'s.
Strings where only $11$ appears:
1) A single $11$, no other $1's$: Obviously $11000$ is such a string. We can permute this in ${4 \choose 1}=4$ ways to obtain all strings with exactly one $11$ in it. (+4)
2) A single $11$, with single $1$'s: $11010,11001,01101$ and their mirror images. (+6)
3) Two disjoint $11$'s: The only possible one is $11011$. (+1)
Strings where only $111$ appears:
4) We count the number of reconfigurations of $11100$ with no extra $1$'s, ${3\choose 1}=3$. (+3)
5) With single $1$'s: $11101,10111$. (+2)
Strings where only $1111$ appears:
6) We count the number of reconfigurations of $11110$, ${2\choose 1}=2$. (+2)
Strings where only $11111$ appears:
7) Obviously, 1. (+1)
So there are, $4+6+1+3+2+2+1=19$ strings.


TITLE: Is it true that $\lim \limits_{x \to a} [l(x)-m(x)] = 0 \implies l(x)$ has the same sign of $m(x)$ for $x$ sufficiently close to $a$?
QUESTION [3 upvotes]: This was present in a demonstration of a theorem, however, it didn't seem obvious to me. Is this a general result?
Here is the exact context where this appeared (from Spivak's Calculus book):
If f is function for which $f(a) =f'(a) = f''(a) = ... = f^{n-1}(a) = 0$ and $f^{n}(a) > 0$, then $f$ will have a local minimum at $x=a$ if $n$ is even.
The demonstration uses a Taylor polynomial $$P_{n,a}(x) = \frac {f^{n}(a)}{n!}(x-a)^n$$
as $$\lim\limits_{x \to a} \frac{f(x)-P_{n,a}(x)}{(x-a)^n} =0$$
We would have
$$\lim\limits_{x \to a} \left [ \frac{f(x)}{(x-a)^n} - \frac{f^{(n)}(a)}{n!} \right ] = 0$$
Which implies that $\frac{f(x)}{(x-a)^n}$ has the same sign as $\frac{f^{(n)}(a)}{n!}$, for $x$ sufficiently close to $a$.
As a consequence, $f(x)>0=f(a)$, because $f^{(n)}(a)>0$ and $(x-a)^n>0$, so there's a local minimum at $a$.

REPLY [2 votes]: First I will answer the question given as the title of your post.
If $\lim\limits_{x \to a}\{l(x) - m(x)\} = 0$ then it does not necessarily mean that $l(x)$ and $m(x)$ have the same sign as $x \to a$.
As as example let $ l(x) = x, m(x) = -x$ and $a = 0$ then clearly $l(x), m(x)$ have opposite signs as $x \to 0$.
However the context related to the analysis of maxima/minima via repeated derivatives is bit different from the title of your post. In the maxima/minima analysis you are given that $f^{(n)}(a) \neq 0$ and
$$\lim\limits_{x \to a} \left [ \frac{f(x)}{(x-a)^n} - \frac{f^{(n)}(a)}{n!} \right ] = 0$$ So in this case $l(x) = f(x)/(x - a)^{n}$ and $m(x) = f^{(n)}(a)/n!$ and the thing to note is that here $m(x)$ is a non-zero constant.
Thus we can rewrite the question as follows:

If $m$ is a non-zero constant and $\lim_{x \to a}\{l(x) - m\} = 0$ then does it mean that $l(x)$ is of the same sign as that of $m$ when $x \to a$?

Clearly the answer is Yes. Because this means that $\lim\limits_{x \to a}l(x) = m$ and thus as $x \to a$, $l(x)$ is very near $m$ and since $m \neq 0$ then $l(x)$ has same sign as that of $m$.
To see this it is important to forget the $\epsilon, \delta$ (because they tend to complicate simple stuff about inequalities and destroy so much the joy of learning calculus).
Let's assume that $m$ is positive and picture $m$ on number line as follows:
-------------------------$0$---------------$m$-----------------------
As $x \to a$ the values of $l(x)$ will come close to $m$ and in the definition of limit we are guaranteed that values of $l(x)$ will come as close to $m$ as we want and hence for values of $x$ sufficiently near $a$ we should ultimately have one of the following pictures
-------------------------$0$-------$l(x)$--------$m$-----------------------
-------------------------$0$---------------$m$-------$l(x)$----------------
In both cases we can see that $l(x)$ is positive. A more concrete argument is like this: suppose that $l(x)$ remains negative (or zero) so that it is to the left of $0$ on number line. In that case the gap between $l(x)$ and $m$ will always be greater than (or equal to) $m$ and hence values of $l(x)$ can't come as close to $m$ as we want. This contradiction shows that values of $l(x)$ must be ultimately positive as $x \to a$.
If $m < 0$ we can modify the above figures for number line in an obvious manner and all the action takes place to the left of $0$ on number line.


TITLE: Given a polynomial $f(z)\in\mathbb{C}[z]$, there exists only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots?
QUESTION [2 upvotes]: Given a polynomial $f(z)\in\mathbb{C}[z]$, $\exists$ only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots?
Is above true in general?
Is it true for polynomials of the form $f(z) = (z-z_1)\cdot ... \cdot (z - z_n)$ where $z_1, ... , z_n \in \mathbb{C}$are distinct?

REPLY [3 votes]: A multiple root of $f(z)-c$ is at the same time a root of $f'(z)$. As the polynomial $f'$ has only finitely many roots, the claim follows. The only special case is when $f'\equiv 0$, and indeed then $f$ is constant and for one specific value of $c$ has roots at all.


TITLE: Dual of a Banach space is isometric to a subspace of a Lipschitz dual
QUESTION [3 upvotes]: Suppose $E$ is a Banach space. Denote $E^*$ as a dual space of $E$ and $Lip(E)$ as the Lipschitz dual of $E$. In other words, $Lip(E) = \{ f:E \rightarrow \mathbb{R} | f(0)=0 \}$ and all $f$ is Lipschitz.
In the book Geometric Nonlinear Functional Analysis, volume $1$, page $173$, there is this sentence before Proposition $7.5$:

$E^*$ is naturally isometric to a subspace of $Lip(E)$.

Question: How to show the statement above?
Can give any hint?

REPLY [3 votes]: Recall that continuous linear functionals are Lipschitz, so every $e^* \in E^*$ is a Lipschitz function, that is we have $E^* \subseteq \def\L{\operatorname{Lip}}\L(E)$. To show the isometry: We have for $e^* \in E^*$ that
\begin{align*}\def\norm#1{\left\|#1\right\|}
  \norm{e^*}_{\L(E)} &= \sup_{e \ne f} \frac{\def\abs#1{\left|#1\right|}\abs{e^*(e) - e^*(f)}}{\norm{e-f}_E}\\
        &= \sup_{e\ne f}\frac{\abs{e^*(e-f)}}{\norm{e-f}_E}\\
        &= \sup_{e'\ne 0}\frac{\abs{e^*(e')}}{\norm{e'}_E}\\
        &=\norm{e^*}_{E^*}
\end{align*}


TITLE: Homomorphism preserves normality
QUESTION [1 upvotes]: let $\phi:G\rightarrow G'$ be a homomorphism, and let N' be a normal subgroup of G'. I want to show that $\phi^{-1}[N']$ is also normal subgroup of G.
My work :
since homomorphism preserves subgroup, I only need to show that the normality of $\phi^{-1}[N']$.
let $N=\phi^{-1}[N']$.
first, I will show that $gN$ is a subset of $Ng$.
take $gn$ in $gN$ s.t $g$ is in $G$, $n$ is in $N$.
then $\phi(gn)=\phi(g)\phi(n)$ is in $\phi(g)N'$
and since $N'$ is normal, for some $n'$ in $N'$, $\phi(g)\phi(n)=n'\phi(g)$
and since $N=\phi^{-1}[N']$, there exists some $n_{0}$ in $N$ s.t $\phi(n_{0})=n'$.
then  $\phi(gn)=\phi(g)\phi(n)=n'\phi(g)=\phi(n_{0})\phi(g)=\phi(n_{0}g)$
and at here, I want to say that $gn=n_{0}g$ and finish, but I can't guarantee the homomorphism is injective. and it seems I have to do this without the injectiveness. any advice?

REPLY [1 votes]: idea: $\phi^{-1}(N')$ contains the kernel of $\phi$
You might as well note that $\phi^{-1}(N')$ is the kernel of $G\rightarrow G' \rightarrow G'/N'$, and kernels are normal.


TITLE: How to write $H = \{x: \cos(x) > 0\}$ as the union of the intervals?
QUESTION [0 upvotes]: I have $\dfrac {1}{2}\left( 4\pi k\pm k \right)  ,k\in \mathbb{Z}$. But I don't understand how to represent it as the union of the intervals.

REPLY [2 votes]: The function $\cos $ is $2\pi$ periodic and positive on the interval $\left(-\frac\pi2,\frac\pi2\right)$ so
$$H=\bigcup_{k\in\Bbb Z} \left(-\frac\pi2+2k\pi,\frac\pi2+2k\pi\right)$$


TITLE: If $u = 2 + 3i$ and $v = -5 + 4i$, then $u + v =$?
QUESTION [5 upvotes]: I have an exercise, it says:
If $u = 2 + 3i$ and $v = -5 + 4i$, then $u + v =$ ?
My answer: $$u + v = (2 - 5) + (3 + 4)i = -3 + 7i$$
but the correct answer is $7 - 3i$.
What is wrong with my solution?

REPLY [2 votes]: We sum real and imaginary parts thus
$$2+3i-5+4i = -3+7i$$
There must be a typo in your study material.


TITLE: root of $g$ is smaller than that of $f$.
QUESTION [0 upvotes]: For a fixed natural no. $n\ge4$, consider $$f(x)=x^3-(n+2)x^2+2nx-2,$$ $$g(x)=x^3-(n+3)x^2+2(n+1)x-2,$$ It seems that smallest root of $g$ is smaller than that of $f$. Can someone show how to prove it.
Note: All roots of $f$ and $g$ are positive.

REPLY [1 votes]: I chose notation $f_n(x):=x^3-(n+2)x^2+2nx-2 \ $ (your $f(x)$).
In this way, $g(x) \equiv f_{n+1}(x)$.
(see curves ($C_n$) below for $n=4,5,6$ and $7$ in green, purple, orange and red resp. ; $x$ axis ranges from $0$ to $7.5$ ; I have used Desmos).
We observe that all curves ($C_n$) have in common two points $(0,-2)$ and $(2,-2)$ (this can be easily proved).
Let us remark that
$$f_{n+1}(x)=f_{n}(x)+\varphi(x) \ \ (*) \ \ \ \text{with} \ \ \varphi(x) := -x^2+2x = x(2-x).$$
Now, let us bracket the roots of $f_n$, using values $x=0,1,2,n,n+1$:
$$f_n(0)=-2 \  \ ;  \  \ f_n(1)=n-3  \  \  ;  \  \  f_n(2)=-2  \  \  ;  \  \  f_n(n)=-2   \  \  ;  \  \  f_n(n+1)=n^2-3.$$
Therefore, for all $n \geq 4$:
$$f_n(0)<0  \  \  ;  \  \  f_n(1)>0  \  \  ;  \  \  f_n(2)<0  \  \  ;  \  \  f_n(n)<0  \  \  ;  \  \  f_n(n+1)>0$$
Thus, as $f_n$ is continuous, for $n \geq 4$, $f_n$ possesses always three real roots :

$\alpha_n \in (0,1)$
$\beta_n \in (1,2)$
$\gamma_n \in (n,n+1)$

It remains to show that $\alpha_{n+1} < \alpha_{n}$.
As $\varphi(x)>0$ on $(0,2)$, and $\alpha_n \in (0,1)$, we have $a_n:=\varphi(\alpha_n)>0$. Using relationship (*):
$$f_{n+1}(\alpha_n)=f_{n}(\alpha_n)+\varphi(\alpha_n)=0+a_n>0$$
Thus $f_{n+1}(x)$, being $<0$ for $x=0$ and $>0$ for $x=\alpha_n < 1$, and having a unique root in $(0,1)$, has this (leftmost) root $\alpha_{n+1}$ such that $\alpha_{n+1}<\alpha_{n}$.
Edit:
It is not uninteresting to remark that,

$(\alpha_n)_n$ is a decreasing sequence converging to $0_+$,
$(\beta_n)_n$ is an increasing sequence converging to $2_-$, and, as a consequence of Vieta's formulas,
$|\gamma_n-n|\rightarrow 0$ as $n \rightarrow \infty$.