Before 1948 general practice was a cottage industry. Most GPs worked independently usually from a consulting room in their own house. There were few if any staff with the duties of answering the phone usually falling to the GP's wife (most doctors were at that time male) or in some cases a housekeeper like the indomitable Janet of Dr Finlay's casebook.
When the NHS was created General Practitioners feared the idea of becoming salaried employees of the state and held out against it … Eventually a compromise deal was reached in which GPs would work for the NHS but as Independent Contractors rather than employees.
Much of that independence was however illusory. Whilst GPs remained free in some ways to organise their own practices the work they did for the NHS was controlled by a very tightly defined contract.
A framework already existed; Lloyd George's National Insurance Act of 1911 provided for limited health cover for working men (but not their families). The scheme was administered by Local Insurance Committees covering counties and large conurbations which held a list or panel of doctors prepared to work under the scheme. The panel doctors were subject to "Terms of Service" which were later lifted directly into the NHS GP contract.
The NHS also adopted the medical cards issued by Local Insurance Committees. and the record system consisting of cards contained in a card "Lloyd George envelope" which were still in use up till a few years ago.
This meant that each GP was personally responsible for the care of his or her patients 24 hours a day 7 days a week and 365 days a year. Any failure in that provision could result in being hauled before the dreaded Medical Service Committee and "withholding of remuneration" or fines.
For single handed doctors this meant being on call to patients literally round the clock with no respite unless they could find a locum which was not easy. Even for those in partnership it meant a gruelling on duty rota. As well as the doctor himself the family were involved as at nights and weekends GPs were on call from home with no other provision for answering the telephone.
By the early 1960s working conditions were such that there was a major recruitment crisis in general practice with newly qualified doctors preferring to work in Australia or Canada to the NHS. The then Health Secretary agreed to the 1966 Doctor's Charter which provided amongst other things access to funding for new surgery buildings and partial reimbursement of the wages of ancillary staff. It also included a "Group Practice Allowance" to encourage GPs to work together in practices. Despite this individual contracts and personal round the clock responsibility remained.
Group Practices provided some respite as it was possible for the doctors to take it in turns to cover the Out of Hours period when the surgery was closed. However since the average partnership size was only 4 or 5 doctors this still meant around one night a week and one weekend a month and more if a partner was away on holiday This was in addition to daytime work in the surgery.
In some large cities commercial deputising services were set up employing doctors to cover the out of hour's period but these just weren't viable in much of the country. Their use was also often limited by the local Family Practitioner Committee and frowned on by politicians who felt that some GPs were being paid for staying in bed. The fact that these doctors in had to give up part of their own income to pay the deputising company was either not understood or glossed over.
My own experience was that I entered General Practice in 1972 choosing deliberately to go to an area which wasn't covered by a deputising service as I felt I wanted to give complete care to the patients in my practice. At first as I was only 28 and came from working a hospital rota of 1 in 2 the work seemed manageable. Sometimes if I had been up in the night there was even an hour between morning and evening surgeries to take a nap. I even enjoyed some night calls especially in the summer. Going out and being able to relieve pain or help someone with breathing problems could be a pleasure although it didn't stop one from being tired the next day.
Over the next few years however things began to change. Daytime surgeries got busier and the gap in the middle of the day disappeared. Although most patients were reasonable, an increasing minority regarded it as their right to call the doctor to visit in the evening regardless of whether it was an emergency; some names cropped up so regularly that they were suspected of deliberately waiting till the surgery was closed. Some even thought they were calling a special night service instead of their own doctor who had already been working all day.
Family life had to be planned around the on call rota; my wife who by now had her own job had to be home by 6pm to answer the phone and during weekends couldn't leave the house at all. The children were also restricted; it was only recently that my son, now in his 40s, told me how upset he was by my weekends on call. I was on call and unable to visit my wife in hospital after our second child was born and was also on call when my older son had a serious accident at home. My memory is kneeling beside him trying to staunch bleeding with one hand whilst holding the phone in the other trying to find someone to cover my duty.
Social life also suffered as non-medical friends just couldn't understand why we couldn't join in all of their activities. Even when it was possible to go, I soon established a reputation for going to sleep at the dinner table due to having been up all night before.
In my practice weekends on duty began at 9 am on Friday and lasted until Monday evening. During the week before I would become increasingly tense and on the Monday night relaxation came only from a bottle.
One small additional factor that no one seemed to consider is that GPs answering night calls to patient's homes had to drive themselves often whilst in an exhausted state. As I turned 40 the thought of continuing to work like this for the next 25 years filled me with horror but there seemed little likelihood of change and I remember being treated as an oddity by some senior colleagues at a local Royal College of GPs meeting when I said that I thought we should work to change the contract.
Gradually it became apparent that I was not alone amongst doctors and there were also other forces at work. There were an increasing number of women entering general practice who often needed to juggle family commitments around on call duties. What childcare facilities there were in the NHS ceased to operate outside "normal" working hours. Spouses of both male and female GPs had careers of their own and couldn't be at home as an unpaid telephone answerer.
Workload continued to rise and Kenneth Clarkes' imposition of a new and more demanding contract in 1990 made many doctors feel that the trade-off of 24 hour responsibility for independence was increasingly unattractive. A BMA survey showed that an overwhelming majority of GPs would now like to be relieved of the 24 hour contract and in 1992 the BMA began a campaign to get rid of the contractual obligation which made GPs themselves responsible for organising & providing a round the clock service.
We adopted the slogan "tired doctors are not good doctors" Surprisingly considering their recent handling of the issue the Press were largely sympathetic and understood that GPs couldn't go on working all hours. As so often happens, politicians refused to listen to the arguments with ministers claiming that most of the time GPs were on call they were comfortably at home with their families. "They were in any case" it was stated "paid a generous fee for night visits after 10pm which amounted to only a handful each month". This ignored the fact that the greatest burden of out of hours work was in the evenings and weekends. On a single Sunday at around this time I was called out 37 times only 2 of which qualified for these payments which were in any case only payable to GPs working in small rotas of less than 10 doctors. The GP payment system was and still is complex but the total amount earmarked for Out of Hours duties including Night Visit Fees was an average of just under £6000 per GP per year.
Around 1994 the Department of Health organised a meeting of Family Health Service Managers from around the country which was intended to brief them on the government line on maintaining the 24 hour contract. But they were in for a shock as instead many of the managers reported that their local GPs were on their knees and the system was unsustainable. When Stephen Dorrell became Secretary of State for Health he vowed to resolve the dispute. What he produced were some small changes in the wording of the contract which made it clear that GPs did not have to respond to every request for a home visit and that the full night visit fee would be paid to doctors working in larger rotas. In addition there was an Out of Hours Development Fund which would pay out an average of £3000 per GP to support and encourage new ways of working.
The response of GPs was immediate and massive; all over the country groups got together and formed cooperative organisations sharing the out of hours calls between a large number of local GPs. Within months premises were found and staff such as receptionists and drivers employed some employed nurses with enhanced training to triage calls but the medical work still had to be done by local doctors themselves. In some cases this was on a compulsory rota in others it was possible for the doctor to buy out of his or her shifts and pay others to do the work. But the bottom line remained GPs still had individual responsibility for the provision of a service to their patients round the clock. If the cooperative collapsed or not enough doctors were willing to do their shifts then it was back to GPs covering their own out of hours.
Experience was that it needed a minimum of around 40 GPs to set up a successful cooperative which was relatively easy when based on a large town. But against all the odds they also were made to work in more rural areas such as Cornwall.
But not every GP was part of these groups; in remote areas where a cooperative was not possible they continued to be responsible for providing round the clock cover them. And the commercial deputising companies continued to operate in large conurbations.
Patients largely took to the new system where instead of getting a home visit from their own doctor or a partner they would unless bedbound go to be seen at a special out of hour surgery is. In some cases this meant travelling to a nearby town and inevitably there were complaint s about the delay involved which was taken up vociferously by local MPs. At the same time there were some reports by the Health Service Ombudsman of problems mainly involving commercial deputising services in one complaint in which I advised the Ombudsman there were insufficient doctors employed and a 7 hour delay in seeing a seriously ill patient who died.
As a result of these concerns the DH commissioned a team under Dr David Carson to investigate GP out of Hours Services. The Carson Report published in 2000 proposed some stringent quality standards and made it clear that these should apply not only to organisations such as deputising services and OOH cooperatives but equally to individual GPs & practices providing their own cover.
It is clear that Minsters of the time and the Department of Health were enamoured of this idea as a subsequent report of the Health Select committee was to reveal.
By the start of the 2000s for a number of reasons General Practitoners' morale was at an all time low. Recruitment was increasingly difficult; whereas in the 1980s an advertisement for a partner might bring in as many as 80 applicants by 2000 there were rarely more than a handful. Some practices got none at all leaving the existing partners to cope with an ever increasing workload. As stated at the outset the GP Contract had its foundations in a document produced in 1911 and the payment system was arcane.to the point that many GPs themselves couldn't understand it.
As Independent Contractors GPs have never received a salary from the NHS instead there was a complex system of fees and allowances detailed in an increasingly thick volume known as the Red Book which made up the total practice income. The doctors' personal income derived from what was left after paying all the running costs of the surgery including equipment and the staff.
In a national ballot GPs called overwhelmingly for a new contract. At the same time under the influence of Downing Street advisors the government had its own agenda including opening up primary care to a wider variety of providers and introducing a form of performance related pay.
So the clause in the new contract which allowed GPs to opt out of their total 24/7/365 responsibility clearly suited both sides. It must be stressed that this did not abolish the out of hours GP service it merely passed the responsibility for organising it from individual doctors to the Primary Care Trusts As for the cost: much has been made of a subsequent television programme in which 1 of the 8 BMA GP negotiators expressed surprise that it was only £6000.But this figure wasn't plucked out of the air; it equates exactly to that part of GPs' pre 2004 income which was earmarked for out of hours work.
As well as this £6000 per GP Primary care trusts would have access to the £3000 per GP in Out of Hours Development fund. In other word s they had exactly the same amount of money that GPs had used to fund the very successful Out of Hours cooperatives.
In less than 10 years with GP cooperatives had built up a great deal of expertise in organising out of hours cover and their management boards comprised local GPs. It might be supposed that PCTs would automatically want to use this expertise.
But this was not the case. The NHS Confederation (the managers own organisation) told the Health Select Committee that few PCTs were working positively with GP co-operatives, often instead being adversarial and generating conflict. In Buckinghamshire the GP cooperative was forced out of business literally overnight when the PCTs awarded the contract to a large company called Harmoni. In Cornwall the exemplary Kernowdoc was passed over in favour of Serco. Both Harmoni and Serco has since been the subject of major complaints and in particular have been accused of failing to employ sufficient doctors. Ironically these cases mirror the very situation which led to the Carson Report. Despite these experiences a recent bid by a group of East London GPs to take back organisation of Out of Hours care for their patients was passed over in favour of a private company.
It is clear that the blame for any failings and shortcomings in Out of hours GP Services should rest with not with GPs but The Primary Care Trusts which were until April 1st 2013 responsible for arranging the service As long ago as 2004 the Health Select committee recognised the risks and warned that many PCTs had failed to consider fully the risks and the responsibility for organising Out of Hours services was often being handled at a junior level .It is clear that in many cases this warning was not heeded.
With regard to the rest of GPs work there is no doubt that the 2004 Contract delivered a very welcome funding increase to practices and therefore an increase in GPs personal income but in the intervening 9 years this has been steadily eroded as funding levels have been frozen whilst the running costs of surgeries and staff pay have increased. In addition the Department of Health has year on year increased the work that has to be done in order to qualify for payments. Despite being relieved of the round the clock responsibility most full time GPs are working hours well above the national average.
Excellent description of how we have got to where we are.
There is now a confusing array of choices available to patients when they feel that they need urgent attention. Choice is of little value when the patient does not have the knowledge or time to weigh up the alternatives. Their choice can on occasions put life at risk.
The simple answer, instead of creating an array of alternatives, many of which were frankly dangerous, would have been to make the local "A&E" the default when GP surgeries close for the day.
OK – so if most "A&E" departments are too small or understaffed to cope, then the Treasury needs to give the local hospital the finances necessary to enhance their service. Getting the private sector involved was a red herring that played into the hands of politician profiteers.
The term "A&E" needs to be changed to "Emergency Department".
One of the main problems with this delegation of duty is the availability of the patient's "GP" case notes when they bowl up at an alternative service provider. Why has nobody solved this Communications/IT black hole?
Excellent account. I do remember (as a hospital doctor) feeling very sorry for my GP colleagues esp. female GPs who had to work day and still had to provide night cover. One of my GP colleagues who lived near their surgery recalled how some patients would go to his house at night for medical advice, often non-urgent. The public (and many politicians) think GPs can and should provide 24 hour service. There is no way GPs can provide such service.
The other key dynamic driving GP despair over OOH work was the increasing risk of complaints over this time. The risk of complaints from OOH scenarios was much higher than those arising in office hours.
So we had work that was poorly supported, poorly paid, a strain on us and our families, increasing and apparently unrestrained demand, increasing complaints, exhausted doctors at risk of making mistakes. Not a satisfactory combination for any service- either for us as providers- or in terms of safe and effective service to our patients.
The GP OOH co-ops worked well- and not using their knowledge was a mistake by health boards and PCTs.
Somehow or other we need to get to an effective NHS system response to patients whatever time of day…but not one that leaves the workers routinely exhausted and at risk of making mistakes.
How I wish this excellent resume could be publishes in all the national newspapers & our patients be more aware of the history & difficulties GPs encountered, & the new challenges they are faced with now.
Only today the headline letter in The Telegraph is from a lady in London who lays the failure of OOH squarely at the foot of the GP ( with a few negative comments about he access to day time GP too). There is , unsurprisingly, no evidence to back up her assumptions, just plenty of ammunition fired at her nearest & easiest target. Thankfully there is also a letter from the BMA adjacent to hers, but why hasn't the latter been given the headline, rather than some apparently unhappy, prejudiced, female patient ?
What would the general public do, if by valuing us so little, lost us completely & only had a private service to access? Would they be so blaming as they are now?
Thank goodness I am in the latter stages of my career.
No doubt for brevity, Eric has missed out that the General Practice Committee (GPC) urged GPs to accept the new deal in 2004, largely selling it on the removal of responsiblity for organising out of hours.
This meant that there would be an inevitable ratching up of contractual day-time duties with the abiltiy of DoH to impose any change of contract unilaterally after 3 months of 'negotiation'. This has resulted in massive bureacracy, tick-box filling and a sense of loss of control in GPs – plus of course increase in day time work-load.
We were told by the GPC that the government would never dare to impose contractrual changes unilaterally!
The review body system has failed totally to increase remuneration in line with increasing costs – hence the reduction in take home income.
Most GPs had already largely sorted out their personal out of hours care prior to the vote in 2004. A firm rejection and continuation of the status quo would have been better that what current GPs now have.
Even if Reade would not renegotiate, Secretary of States for health revolve pretty quickly and the next incumbent would have met at the table again.
Many GPs warned clearly of these dangers.
An excellent piece from a stalwart proponent of a better working life for GPs. I hope he is enjoying a well-earned retirement.
The final nail in the coffin of a quality ooh service from experienced GPs was the unwise [I'm being charitable here] idea of Gordon Brown for us to provide 'extended hours', an offer few of us could refuse. As a GP in my mid-fifties I was happy to do a few ooh sessions a month, but not this as well. Which to ditch? No-brainer.
I'm so glad I retired on 31.3.13 because of the totally predictable mess following the Health & Social Care Act 2012!
Thanks, Eric, for an excellent review of the problem.
You don't mention that in 2004, the decision to keep responsibility for OOH was a whole practice one – it was no longer possible – as it had been previously – for one or more GPs within a practice to continue OOH cover while others paid for deputising or joined a co-op.
As far as the ability of the government to impose changes in the GP Contract goes, this was established by Kenneth Clarke in 1990: he made it clear that a GP Contract is not a contract in any legal sense – so nothing new there – and at the very least, the 2004 New GMS Contract removed, once and for all, the Red Book system whereby any "overperformance" by GPs in a shrinking workforce simply resulting in the "overpayment" being clawed back the following year – which had to be an improvement!
Good luck to all those remaining at the coal-face: looks like you need it.
When the PCT led service took over in 2004 – although our population coverage doubled our workload quadrupled – and so did our staffing – going up from 4 staff at thew co-op ( Nurse , GPx 2 and driver ) to 2 call handlers , 3 nurses , 3 doctors , 2 drivers and a Nurse practitioner ( my new role in the PCT led OOH service ). We regularly took 80 calls per evening and also because we were now based next to A&E we also did our best to filter primary care problems out of A&E . I can see a whole historical trail of how little the public now fail cope with the simplest of health problems despite there being a whole raft of options such as NHS Direct , pharmacists before needing to call OOH or go to A&E with inappropriate presentations .
Thank you for an excellent summery. My father was a GP who qualified in 1971 and I am in practice today. Your memories about the impact on family life of wives manning the phones and families stuck at home strike a chord with me. As a working mother, GP and married to a GP I would shudder at thought of trying to work in that way today.
I am trying to write a history of NHS General Practice and was thus most interested to come across your article. My own time as a GP began in 1980 and I would love to interview you if you were willing. Please let me know on the email below if you are.
It is going private already. Sadly private companies don't actually like anyone who has the faintest whiff of atual illness/disease, and these will be the people left to be looked after by a salaried GP service.
This is an interesting deconstruction of GP services.
Excellent summary. I wonder what percentage of the UK were still doing their own OOH by the time the new contract started? I am a GP in South Powys, and we did our own right up until the contract change, but not through choice. I suspect, but I would like it confirmed, that most of the UK was using either deputising or a co-op by 2004, so most patients in the U.K. were unlikely to have seen their own doctor at nights even 13 years ago and beyond. I think that the thing that our patients miss most is the Saturday morning surgery which extended the chance to be seen with notes and practice staff and was more like a 'normal' appointment, and in retrospect that should have stayed. Apart from that the contract change shouldn't have had much influence on OOH, unless the trusts interfered with the providers, which they did, of course. As for lack of GP appointments leading to no beds in the hospital , how is that? Are these GP patients seen in a hospital bed?


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text:
Pittsburgh Pirates Have the Building Blocks For a Strong Starting Rotation
Aug 28, 2022; Philadelphia, Pennsylvania, USA; Pittsburgh Pirates starting pitcher Roansy Contreras (59) throws a pitch against the Philadelphia Phillies during the fifth inning at Citizens Bank Park. Mandatory Credit: Eric Hartline-USA TODAY Sports /
PITTSBURGH, PA – AUGUST 23: Roansy Contreras #59 of the Pittsburgh Pirates looks on during the game against the Atlanta Braves at PNC Park on August 23, 2022 in Pittsburgh, Pennsylvania. (Photo by Joe Sargent/Getty Images) /
Budding ace Roansy Contreras
When the Pittsburgh Pirates traded Jameson Taillon to the New York Yankees following the 2020 season the best prospect they received in return was Roansy Contreras. Since then, Contreras has developed into one of the best pitching prospects in all of baseball.
Contreras made one MLB start in 2021 before getting his first extended look in 2022. While he had some ups and downs, Contreras, who is just 23-years-old, looked like a budding ace for much of the 2022 season.
In 21 outings with the Pirates, 18 of which were starts, Contreras pitched 95 innings. Contreras posted a 3.79 ERA, 4.38 FIP, 1.23 HR/9, he walked 9.6% of batters faced, and struck batters out at a 21.1% clip.
Contreras flashed a power fastball and absolutely wicked, wipeout slider during his 2022 season as he went on to be worth a 1.2 WAR. There were times it was clear that Contreras still needs to improve in terms of how he attacks hitters and pitch selection, but the stuff to be a top of the rotation arm is there. If he can take another step forward in 2023, then the Pirates may have a true ace on their hands.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

Let $p$ be a prime and ${\mathcal{O}}$ a complete local principal ideal domain
with residue field $k=$ ${\mathcal{O}}/J({\mathcal{O}})$ of characteristic $p$; we allow
the case ${\mathcal{O}}=$ $k$.
Let $G$ be a finite group. The blocks of $\OG$ are the primitive
idempotents in $Z(\OG)$. By a result of Yoshida in \cite{YoshidaII},
the category $\coMack(\OG)$ of cohomological Mackey functors of $G$
with coefficients in $\mathrm{mod}({\mathcal{O}})$ is equivalent to the category of
finitely generated modules over the algebra
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{\OG}(\oplus_H\ {\mathcal{O}} G/H)^\mathrm{op}$, where $H$ runs over the subgroups
of $G$. This description implies that the category $\coMack(\OG)$
decomposes as a direct sum of the categories $\coMack(G,b)$, with $b$
running over the blocks of $\OG$, such that each $\coMack(G,b)$ is
equivalent to the category of finitely generated modules over the
algebra $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{\OG}(\oplus_H\ b\cdot {\mathcal{O}} G/H)^\mathrm{op}$, where $H$ runs as
before over the subgroups of $G$. By results of Th\'evenaz and Webb in
\cite[\S 17]{TW}, this is the block decomposition of $\coMack(\OG)$.
We use Yoshida's result to describe $\coMack(G,b)$ in an analogous way
at the source algebra level, obtaining as a consequence an alternative
proof of a result of Rognerud in \cite{Rogn}, stating that blocks with
isomorphic source algebras have equivalent categories of cohomological
Mackey functors. A {\it defect group} of a block $b$ of
$\OG$ is a maximal $p$-subgroup $P$ of $G$ such that $\OP$ is isomorphic
to a direct summand of $\OGb$ as an $\OP$-$\OP$-bimodule. A {\it source
idempotent} of $b$ is a primitive idempotent $i$ in the fixed point
algebra $(\OGb)^P$ with respect to the conjugation action of $P$ on
$\OGb$ such that $\OP$ is isomorphic to a direct summand of $i\OG i$ as
an $\OP$-$\OP$-bimodule. The algebra $A=$ $i\OG i$ with its canonical
image $Pi$ of $P$ in $A^\times$ is called a {\it source algebra of} $b$.

\begin{Theorem}\label{coMacksourcealgebra}
Let $G$ be a finite group, $b$ a block of $\OG$ with a defect group $P$
and let $A$ be a source algebra of $b$. Set $N=$
$\oplus_{Q}\ A\tenOQ {\mathcal{O}}$, where in the direct sum $Q$ runs over the
subgroups of $P$. Set $E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(N)$.
There is an equivalence of categories $\coMack(G,b) \cong \mathrm{mod}(E^\mathrm{op})$.
In particular, blocks with isomorphic source algebras have equivalent
categories of cohomological Mackey functors.
\end{Theorem}

By a result of Rognerud \cite[Proposition 4.5]{Rogn}, the last
statement holds more generally for {\it permeable Morita equivalences};
that is, Morita equivalences preserving $p$-permutation modules (and the
proof presented below shows this as well).
It is not necessary to take the direct sum over all subgroups of $P$ in
the statement of \ref{coMacksourcealgebra}.
Indeed, by \cite[Proposition 6.1]{Liperm}, every indecomposable direct
summand of  $A\tenOQ O$ for some subgroup $Q$ of $P$ is isomorphic to a
direct summand of $A\tenOR O$ for some fully centralised subgroup $R$
with respect to the fusion system ${\mathcal{F}}}    \def\OS{{\mathcal{O}S}$ determined by $A$ on $P$.
Thus we obtain a Morita equivalent endomorphism algebra if we
take the direct sum over a set of representatives of the
${\mathcal{F}}}    \def\OS{{\mathcal{O}S}$-isomorphism classes of fully ${\mathcal{F}}}    \def\OS{{\mathcal{O}S}$-centralised subgroups of $P$.

\medskip
At the source algebra level, a splendid derived equivalence
between two source algebras $A$, $A'$ of blocks with a common
defect group $P$ is a derived equivalence induced by a {\it splendid
Rickard complex of $A$-$A'$-bimodules $X$}; that is, $X$ is a bounded
chain complex of $A$-$A$'-bimodules whose terms are direct sums of
summands of the bimodules $A\tenOQ A'$, where $Q$ runs over the
subgroups of $P$, such that we have homotopy equivalences
$X\ten_{A'}X^*\simeq$ $A$ and $X^*\tenA X\simeq$ $A'$ as complexes of
$A$-$A$-bimodules and of $A'$-$A'$-bimodules, respectively. Here $X^*$
denotes the ${\mathcal{O}}$-dual of $X$, rewritten as a chain complex. Rognerud
showed in \cite{Rogn} that a splendid Rickard equivalence between two
blocks induces a derived equivalence between the associated
categories of cohomological Mackey functors. We construct
two-sided complexes which realise this derived equivalence.

\begin{Theorem} \label{coMackderived}
Let $A$, $A'$ be source algebras of blocks $b$, $b'$ of finite group
algebras $\OG$, $\OG'$, respectively, with a common defect group $P$.
Set $N=$ $\oplus_Q\ A\tenOQ {\mathcal{O}}$ and $N'=$ $\oplus_Q\ A'\tenOQ {\mathcal{O}}$,
where $Q$ runs over the subgroups of $P$. Set
$E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(N)$ and $E'=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{A'}(N')$.
Let $X$ be a splendid Rickard complex of $A$-$A'$-bimodules.
Set $Y=$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(X\ten_{A'} N', N)$ and
$Y'=$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_{A'}(X^*\ten_{A} N, N')$.
Then $Y$ is a complex of $E$-$E'$-bimodules, $Y'$ is a complex
of $E'$-$E$-bimodules, the terms of $Y$ and $Y'$ are finitely
generated projective as left and as right modules, and we have
homotopy  equivalences
$$Y \ten_{E'} Y' \simeq E$$
$$Y'\ten_{E} Y \simeq E'$$
as complexes of $E$-$E$-bimodules and $E'$-$E'$-bimodules,
respectively. In particular, if $\OGb$ and ${\mathcal{O}} G'b'$ are
splendidly Rickard equivalent, then the categories
$\coMack(G,b)$ and $\coMack(G',b')$ are derived equivalent.
\end{Theorem}

As mentioned above, the last statement of \ref{coMackderived} is due to
Rognerud \cite[Theorem 5.8]{Rogn}, where - as before - this is shown to
hold more generally for {\it permeable derived equivalences} (that is,
derived equivalences induced by Rickard complexes which preserve
complexes of $p$-permutation modules).  The first statement of
\ref{coMackderived} is a special case of a more general construction of
derived equivalences for endomorphism algebras, which we will describe
in the next section (and which also allows for the extra generality
regarding permeable derived equivalences).

\medskip
There are versions of the above results for stable endomorphism algebras
and stable equivalences. Let $A$ be an ${\mathcal{O}}$-algebra. An $A$-module
$U$ is {\it relatively ${\mathcal{O}}$-projective} if $U$ is isomorphic to
a direct summand of $A\tenO W$ for some ${\mathcal{O}}$-module $W$.
For any two $A$-modules $U$, $U'$ we denote by $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A^\mathrm{pr}(U,U')$ the
subspace of $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,U')$ consisting of all $A$-homomorphisms which
factor through a relatively ${\mathcal{O}}$-projective $A$-module.
We set $\underline{\mathrm{Hom}}_A(U,U')=$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,U')/\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A^\mathrm{pr}(U,U')$.
We denote by $\stmod(A)$ the ${\mathcal{O}}$-linear category having the
finitely generated $A$-modules as objects, with homomorphism spaces
$\underline{\mathrm{Hom}}_A(U,U')$ for any two finitely generated $A$-modules $U$, $U'$,
such that the composition of morphisms in $\stmod(A)$ is induced
by the composition of $A$-homomorphisms (one checks that this is
well-defined). The resulting category is no longer abelian, but
if $A$ is symmetric, then this category can be shown to be triangulated
(a fact we will not need in this note).
For $G$ a finite group and $b$ a block of $\OG$, we denote by
$\coMackbar(G,b)$ the full subcategory of $\coMack(G,b)$
consisting of all Mackey functors in $\coMack(G,b)$ whose evaluation at
the trivial subgroup $\{1_G\}$ of $G$ is zero.

\begin{Theorem} \label{coMackstable}
Let $G$ be a finite group, $b$ a block of $\OG$ with a defect group $P$
and let $A$ be a source algebra of $b$.
Set $N=$ $\oplus_Q\ A\tenOQ {\mathcal{O}}$, where $Q$ runs over the
subgroups of $P$. Set $\underline{E} =$ $\underline{\mathrm{End}}_A(N)$.
There is an equivalence of categories $\coMackbar(G,b) \cong
\mathrm{mod}(\underline{E}^\mathrm{op}))$.
\end{Theorem}

\begin{Corollary} \label{recollementCor}
With the notation above, there is a recollement of abelian categories
$$\xymatrix{\coMackbar(G,b) \ar[r] & \ar@<5pt>[l] \ar@<-5pt>[l]
\coMack(G,b) \ar[r] & \ar@<5pt>[l] \ar@<-5pt>[l] \mathrm{mod}(\OGb) }$$
which is isomorphic to the idempotent recollement of the algebra $E$
with the idempotent $\tau$, where $\tau$ is the canonical
projection of $N$ onto its summand $A=$ $A\tenO {\mathcal{O}}$.
\end{Corollary}

Let $A$, $A'$ be symmetric ${\mathcal{O}}$-algebras. Following Brou\'e
\cite[\S 5]{BroueEq} an $A$-$A'$-bimodule $M$ is said to {\it induce a
stable equivalence of Morita type} if $M$ is finitely generated
projective as a left $A$-module, as a right $A'$-module, and if
$M\ten_{A'} M^*\cong$ $A\oplus U$ for some projective $A$-$A$-bimodule
and $M^*\tenA M\cong$ $A'\oplus U'$ for some projective
$A'$-$A'$-bimodule $U'$. In that case, the functors $M\ten_{A'}-$ and
$M^*\tenA-$ induce inverse equivalences $\underline{\mathrm{mod}}(A)\cong$ $\underline{\mathrm{mod}}(A')$.
Stable equivalences of Morita type can be composed by tensoring the
relevant bimodules together. In particular, a stable equivalence of
Morita type between two block algebras induces a stable equivalence of
Morita type between any two source algebras of these blocks.

\medskip
A stable equivalence of Morita type between two block algebras $\OGb$,
${\mathcal{O}} G'b'$ is called {\it splendid} if it is induced by a
$p$-permutation bimodule. In that case  the defect groups of $b$ and
$b'$ are isomorphic, and for some identification of a defect group $P$
of $b$ with a defect group of $b'$ there is a choice of source algebras
$A$ and $A'$ such that the induced stable equivalence of Morita type
between $A$ and $A'$ is given by an $A$-$A'$-bimodule which is an
indecomposable direct summand of $A\tenOP A'$.  By
\cite[7.6.6]{Puigbook} the fusion systems of $A$ and $A'$ are then
equal (see also \cite[3.1]{Lisplendid} for a proof of this fact).

\begin{Theorem} \label{coMackstableeq}
Let $G$, $G'$ be finite groups, $b$ a block of $\OG$, and
$b'$ a block of ${\mathcal{O}} G'$.
A splendid stable equivalence of Morita type
between $\OGb$ and ${\mathcal{O}} Gb'$ induces an equivalence of categories
$$\coMackbar(G,b) \cong \coMackbar(G',b')\ .$$
\end{Theorem}

Let ${\mathcal{A}}$ be an abelian category with enough projective objects.
The {\it projective dimension} $\mathrm{pdim}(U)$ of an object $U$ in ${\mathcal{A}}$
is the smallest nonnegative integer $n$ for which there exists a
projective resolution which is zero in all degrees larger than $n$,
provided that $U$ has a bounded projective resolution, and
$\mathrm{pdim}(U)=$ $\infty$ otherwise. The {\it finitistic dimension of ${\mathcal{A}}$}
is the supremum of the set of integers $\mathrm{pdim}(U)$, with $U$ running
over all objects in ${\mathcal{A}}$ having a finite projective dimension
(and with the convention that the finitistic dimension is $\infty$ if
this supremum is unbounded). The left or right finitistic dimension
of a finite-dimensional $k$-algebra is the finitistic dimension of its
category of finitely generated left or right modules, respectively.
The {\it $p$-sectional rank} $s(G)$ of a finite group $G$ is the
maximum of the ranks of $p$-elementary abelian subquotients of $G$.
Tambara showed in \cite{Tam} that $1+s(G)$ is equal to the finitistic
dimension of $\coMack(kG)$. We show here, essentially by
adapting Tambara's proof, that there is an analogous statement for
block algebras.

\begin{Theorem} \label{coMackfin}
Let $G$ be a finite group, $b$ a block of $kG$ and $P$ a defect
group of $b$. The finitistic dimension of $\coMack(G,b)$ is equal
to $1+s(P)$.
\end{Theorem}

\begin{Remark}
By results of Th\'evenaz and Webb in \cite[\S 17]{TW}, the category
$\Mack(kG)$ of all Mackey functors on $G$ over $k$ has a block
decomposition which corresponds bijectively to the block decomposition
of $kG$. It seems however unknown whether the blocks of $\Mack(kG)$
admit a description in terms of the source algebras of the corresponding
blocks of $kG$. In contrast to Theorem \ref{coMackfin}, Bouc, Stancu
and Webb \cite[Theorem 1.2]{BSW} showed that the finitistic dimension
of $\Mack(kG)$ is zero.
\end{Remark}

The proofs of the above theorems will be given in the
last section. If not stated otherwise, modules are unital left modules.

\section{Derived equivalences and endomorphism algebras}

An ${\mathcal{O}}$-algebra $A$ is called {\it symmetric} if $A$ is isomorphic
to its ${\mathcal{O}}$-dual $A^*$ as an $A$-$A$-bimodule. If $s\in$ $A^*$ is the
image of $1_A$ under some bimodule isomorphism $A\cong$ $A^*$, then
$s$ is called a {\it symmetrising form on} $A$. Matrix algebras,
finite group algebras over ${\mathcal{O}}$, their block algebras and source
algebras are symmetric.

For $A$ an ${\mathcal{O}}$-algebra, $(U,\delta)$ a complex of right
$A$-modules and $(V,\epsilon)$ a complex of left $A$-modules we denote
by $U\tenA V$ the complex of ${\mathcal{O}}$-modules which in degree $n$
is equal to $\oplus_{(i,j)}\ U_i\tenA V_j$, with $(i,j)$
running over the set of pairs of integers satisfying
$i+j=$ $n$. The differential of $U\tenA V$ is the sum of the
maps $\mathrm{Id}}             \def\tenA{\otimes_A_{U_i}\ten \epsilon_j$ and $(-1)^i\delta_i\ten \mathrm{Id}}             \def\tenA{\otimes_A_{V_j}$ from
$U_i\tenA V_j$ to $U_i\tenA V_{j-1}$ and to $U_{i-1}\tenA V_j$.
If $U$, $V$ are bimodule complexes, then $U\tenA V$ inherits those
bimodule structures.

For $(V,\delta)$ and $(W,\zeta)$ two complexes of $A$-modules,
we denote by $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V,W)$ the complex of ${\mathcal{O}}$-modules
which in degree $n$ is equal to $\oplus_{(i,j)}\ \mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V_i,W_j)$,
with $(i,j)$ runnig over the pairs of integers satisfying
$j-i=$ $n$. The differential is the sum of the maps from
$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V_i,W_j)$ to $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V_i,W_{j-1})$ and to
$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V_{i+1}, W_{j})$, where the first is given by
composition with $\zeta_j$, and the second by precomposition
with $(-1)^{n+1}\epsilon_{i+1}$. As before, if $V$, $W$ have
bimodule structures, then these induce bimodule structure on
the complex $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V,W)$. Note that this sign convention implies
that the $k$-dual $V^*$ of the complex $V$ is the complex which
in degree $n$ is equal to $V^*_{-n}$, with differential
$V^*_{-n}\to$ $V^*_{-n+1}$ given by precomposing with
$(-1)^{n+1}\epsilon_{-n+1} : V_{-n+1}\to$ $V_{-n}$.
If $A$ is symmetric and $s\in$ $A^*$ is a symmetrising form of $A$,
then composition with $s$ induces an isomorphism of complexes
$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(V,A)\cong$ $V^*$.

For $A$, $A'$ symmetric $k$-algebras, a {\it Rickard complex
of $A$-$A'$-bimodules} is a bounded complex $X$ of $A$-$A'$-bimodules
whose terms are finitely generated projective as left $A$-modules
and as right $A'$-modules, such that we have homotopy
equivalences $X\ten_{A'} X^*\simeq$ $A$ and $X^*\tenA X\simeq$
$A'$ as complexes of $A$-$A$-bimodules and $A'$-$A'$-bimodules,
respectively.

\begin{Theorem} \label{endoderived}
Let $A$, $A'$ be symmetric $k$-algebras. Let $N$ be a finitely
generated $A$-module, and let $N'$ be a finitely generated
$A'$-module. Set $E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(N)$ and $E'=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{A'}(N')$.
Let $X$ be a Rickard complex of $A$-$A'$-bimodules. Suppose that
every indecomposable direct summand of every term of $X\ten_{A'} N'$
is isomorphic to a direct summand of $N$ and that every
indecomposable direct summand of every term of $X^*\tenA N$ is
isomorphic to a direct summand of $N'$.
Set $Y=$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_{A}(X\ten_{A'} N', N)$ and
$Y'=$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_{A'}(X^*\tenA N, N')$.
Then $Y$ is a complex of $E$-$E'$-bimodules, $Y'$ is a complex
of $E'$-$E$-bimodules, the terms of $Y$ and $Y'$ are finitely
generated projective as left and as right modules, and we have
homotopy  equivalences
$$Y\ten_{E'} Y' \simeq E$$
$$Y'\ten_{E} Y \simeq E'$$
as complexes of $E$-$E$-bimodules and $E'$-$E'$-bimodules,
respectively. In particular, $E$ and $E'$ are
derived equivalent. Moreover, if $X$ and $X^*$ induce a Morita
equivalence between $A$ and $A'$, then $Y$ and $Y'$ induce
a Morita equivalence between $E$ and $E'$.
\end{Theorem}

\begin{proof}
If $i$, $j$ are idempotents in $A$, then we have a canonical
isomorphism $iA\tenA Aj\cong$ $iAj$, induced by multiplication in $A$.
We translate this to endomorphism algebras. Let  $U$, $V$ be
direct summands of $N$. Let $\tau$ be a projection of $N$ onto $U$,
viewed as an idempotent in $E$, and let $\sigma$ be a projection of
$N$ onto $V$.
Then $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,N)\cong$ $E\circ\tau$ is a projective left $E$-module
via composition of homomorphisms. Similarly, $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(N,V)\cong$
$\sigma\circ E$ is a projective right $E$-module via precomposition by
homomorphisms in $E$.
Thus the first observation applied to $E$ instead of $A$
implies that composition of homomorphisms induces an isomorphism
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(N, V)\ten_E\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,N) \cong \mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,V)$$
which is natural and additive in $U$ and $V$. Thus this isomorphism holds
for $U$ and $V$ finite direct sums of direct summands of $N$.
Similar statements hold for $A'$ and $N'$.
By the assumptions, the indecomposable direct summands of the terms of
$X^*\ten_{A} N$ are isomorphic to direct summands of $N'$. Thus the above
isomorphism, applied with $U=$ $V=$ $X^*\ten_{A} N$ and $N'$ instead of $N$,
shows that composition of homomorphisms yields an isomorphism
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_{A'}(N', X^*\ten_{A} N) \ten_{E'} \mathrm{Hom}}           \def\tenkS{\otimes_{kS}_{A'}(X^*\ten_{A} N, N')
\cong \mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{A'}(X^*\ten_A N)$$
as graded $E$-$E$-bimodules. A straightforward verification
(checking signs) shows that this isomorphism commutes with the
differentials, so this is an isomorphism of complexes of
$E$-$E$-bimodules.
By a standard adjunction, the complex on the left side is isomorphic
to $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(X\ten_{A'} N', N) = Y$.
By another standard adjunction, the right side is isomorphic to the
complex
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(N, X\ten_{A'} X^* \tenA N)$$
Since $X$ is a Rickard complex, we have
$X\ten_{A'} X^* \ten_A N\simeq N$, and hence
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(N, X\ten_{A'} X^* \ten_{A} N)\simeq \mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(N,N)=E$$
This shows that $Y\ten_{E'} Y'\simeq$ $E$. A similar argument shows
that $Y'\ten_{E} Y \simeq E'$, which completes
the proof.
\end{proof}

\section{On the finitistic dimension of endomorphism algebras}

The result of this section holds regardless of the characteristic
of the field $k$.  Let $A$ be a finite-dimensional $k$-algebra and
$U$ a finitely generated $A$-module. We denote by $\mathrm{add}(U)$ the
full additive subcategory of all $A$-modules which are isomorphic to
finite direct sums of direct summands of $U$.
We use without further reference the following standard facts.
If $V$ belongs to $\mathrm{add}(U)$, then $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,V)$ is a projective right
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$-module, and any finitely generated projective right
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$-module is of this form, up to isomorphism.  Given two
idempotents $i$, $j$ in $A$, every homomorphism of right $A$-modules
$iA\to$ $jA$ is induced by left multiplication with an element in $jAi$.
Multiplication by $i$ is exact; in particular, if $Z$ is a complex of
$A$-modules which is exact or which has homology concentrated in a
single degree, the same is true for finite direct sums of complexes
of the form $iZ$. Translated to endomorphism algebras this implies that
for any two $A$-modules $V$, $W$ in $\mathrm{add}(U)$, any homomorphism of right
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$-modules $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,V)\to$ $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,W)$ is induced by
composition with an $A$-homomorphism $V\to$ $W$. Thus any complex of
finitely generated projective right $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$-modules is isomorphic to
a complex obtained from applying the functor $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,-)$ to a complex of
$A$-modules $Z$ whose terms belong to $\mathrm{add}(U)$. Moreover, if $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U,Z)$ is
exact or has homology concentrated in a single degree, the same is true
for any complex of the form $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U',Z)$, where $U'\in$ $\mathrm{add}(U)$.

Let $A$ and $B$ be symmetric $k$-algebras.
Following \cite{LinHecke}, an $A$-$B$-bimodule $M$ is said
to induce a {\it separable equivalence between $A$ and $B$} if
$M$ is finitely generated projective as a left $A$-module, as a
right $B$-module, and if $A$ is isomorphic to a direct summand of
$M\tenB M^*$ as an $A$-$A$-bimodule and $B$ is isomorphic to
$M^*\tenA M$ as  a $B$-$B$-bimodule. If there is such a bimodule,
we say that $A$ and $B$ are {\it separably equivalent.}
The link between this section and block theory comes from the
fact that a block algebra of a finite group is separably equivalent to
its defect group algebras via a $p$-permutation bimodule. Since the
notion of separable equivalence is transitive, any two block algebras
with isomorphic defect groups are separably equivalent via
$p$-permutation bimodules.

\begin{Theorem} \label{endofin}
Let $A$, $B$ be symmetric $k$-algebras. Let $M$ be an
$A$-$B$-bimodule inducing a separable equivalence between $A$ and $B$.
Let $U$ be a finitely generated $A$-module and $V$ a finitely generated
$B$-module. Suppose that $M^*\tenA U\in$ $\mathrm{add}(V)$ and that
$M\tenB  V\in$ $\mathrm{add}(U)$. Then the right finitistic dimensions of
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ and of $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_B(V)$ are equal.
\end{Theorem}

\begin{proof}
Set $E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ and $F=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_B(V)$. We show first that
$\mathrm{add}(M\tenB V)=$ $\mathrm{add}(U)$ and $\mathrm{add}(M^*\tenA U)=$ $\mathrm{add}(V)$.
By the assumptions, we have $\mathrm{add}(M\tenB V)\subseteq$ $\mathrm{add}(U)$.
Thus $\mathrm{add}(M^*\tenA M \tenB V)\subseteq$ $\mathrm{add}(M^*\tenA U)\subseteq$
$\mathrm{add}(V)$. Since $B$ is isomorphic to a direct summand of the
$B$-$B$-bimodule $M^*\tenA M$, it follows that $V$ is isomorphic
to a direct summand of $M^*\tenA M\tenB V$. Thus the previous
inclusions of additive categories are equalities. The same
argument with reversed roles shows the second equality.
We show next that for any finitely generated right $F$-module
of finite projective dimension there is a finitely generated
right $E$-module with the same pojective dimension.
Let $Y$ be a right $F$-module with a finite projective dimension
$\mathrm{pdim}(Y)=$ $n$. By the remarks at the beginning of this
section, any such resolution is of the form
$$0\to\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_n)\to\cdots\to\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_1)\to\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_0)\to Y\to 0$$
obtained from applying the functor $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,-)$ to a complex
of $B$-modules
$$Z = \cdots\to 0\to V_n\to\cdots\to V_1\to V_0\to 0 \to \cdots$$
whose components $V_i$ are in $\mathrm{add}(V)$, together with a surjective
homomorphism of right $F$-modules $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_0)\to$ $Y$.
Thus the complex $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,Z)$ has homology concentrated in degree $0$,
and the homology in degree $0$ is $Y$.  By the remarks at the beginning
of this section, it follows that for any $B$-module $V'$ in $\mathrm{add}(V)$,
applying the functor $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V',-)$ to the complex of $B$-modules $Z$
yields a complex of $k$-vector spaces
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V',Z)$$
which has nonzero homology only in degree $0$, if any.
By the assumptions, this applies in particular to the $B$-module
$V'=$ $M^*\tenA M\tenB V$. Using a standard adjunction, we obtain
a complex of the form
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(M\tenB V, M\tenB Z)$$
which has homology concentrated in degree $0$.
This is a complex of right $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(M\tenB V)$-modules.
Since $\mathrm{add}(M\tenB V)=$ $\mathrm{add}(U)$, it follows that the complex
$$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U, M\tenB Z)$$
has homology concentrated in degree $0$. This is a complex of right
$E$-modules. Its homology in degree $0$ is therefore a right
$E$-module $X$ with projective dimension at most $n$. We show that
$X$ has projective dimension exactly $n$. If $n=0$, there is nothing
to prove, so assume that $n>0$. The fact that $\mathrm{pdim}(Y)=n$ implies
in particular that the first nonzero differential
$\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_n)\to\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,V_{n-1})$ in the complex $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_B(V,Z)$
is not split injective. Thus the first nonzero differential
$V_n\to$ $V_{n-1}$ in $Z$ is not split injective. But then the first
nonzero differential of $M\tenB Z$ is not split injective either,
for if it were, so would be that of $M^*\tenA M\tenB Z$, which is
not possible, as $Z$ is a direct summand, as a complex of $B$-modules,
of $M^*\tenA M\tenB Z$. This in turn implies that the first nonzero
differential of $\mathrm{Hom}}           \def\tenkS{\otimes_{kS}_A(U, M\tenB Z)$ is not split injective as a right
$E$-module homomorphism.  This shows that $\mathrm{pdim}(X)=$ $\mathrm{pdim}(Y)$.
Thus the right finitistic dimension of $E$ is at least as large as
that of $F$. Exchanging the roles of $E$ and $F$ shows the equality
of right finitistic dimensions as stated.
\end{proof}

\section{Proofs}

Let $A$ be an ${\mathcal{O}}$-algebra which is finitely generated as an ${\mathcal{O}}$-module.
Let $U$, $U'$ be finitely generated $A$-modules. An idempotent in
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ corresponds to a projection of $U$ onto a direct summand
of $U$. The Krull-Schmidt theorem implies that this correspondence
induces a bijection between conjugacy classes of primitive idempotents
in $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ and isomorphism classes of indecomposable direct
summands of $U$. Thus if $U$ and $U'$ have the same isomorphism classes
of indecomposable direct summands, or equivalently, if $\mathrm{add}(U)=$
$\mathrm{add}(U')$, then $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ and $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U')$
are Morita equivalent. We will use this well-known fact without
further reference.

\begin{proof}[Proof of Theorem \ref{coMacksourcealgebra}]
Let $G$ be a finite group, $b$ a block of $\OG$, $P$ a defect
group of $b$, and $i\in$ $(\OGb)^P$ a source idempotent of $b$.
By \cite[3.5]{Puigpoint}, multiplication with $i$ induces a Morita
equivalence between $kGb$ and the source algebra $A=$ $i\OG i$.
By Yoshida's theorem \cite[4.3]{YoshidaII} mentioned above, the
category $\coMack(G,b)$ is equivalent to
$\mathrm{mod}(\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{\OGb}(\oplus_H\ b\cdot {\mathcal{O}} G/H)^\mathrm{op})$, where $H$ runs over
the subgroups of $G$. Fix a subgroup $H$ of $G$, and let $U$ be an
indecomposable direct summand of $b\cdot {\mathcal{O}} G/H\cong$
$b\cdot\mathrm{Ind}}           \def\tenK{\otimes_K^G_H({\mathcal{O}})$.
Then $U$ has trivial source. By \cite[6.1]{Likleinfour}
there exists a vertex $Q$ of $U$ such that $U$ is isomorphic to
a direct summand of $\OG i\tenkQ {\mathcal{O}}\cong$ $\OG i\ten_A A\tenkQ {\mathcal{O}}$.
Thus $U\cong$ $\OG i\tenA V$ for some indecomposable direct summand $V$
of $A\tenOQ {\mathcal{O}}$. Conversely, if $W$ is an
indecomposable direct summand of $N$, then $\OG i\tenA W$ is isomorphic
to a direct summand of ${\mathcal{O}} G/Q$ for some subgroup $Q$ of $P$.
This implies that $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{\OGb}(\oplus_H\ b\cdot {\mathcal{O}} G/H)^\mathrm{op}$ is Morita
equivalent to $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{\OGb}(\OG i\tenA N)^\mathrm{op}$,
where $H$ runs over the subgroups of $G$.
Since multiplication by $i$ induces a Morita equivalence between
$\OGb$ and $A$ it follows that the last algebra is isomorphic
to $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(N)^\mathrm{op}=$ $E^\mathrm{op}$, whence the equivalence $\coMack(G,b)\cong$
$\mathrm{mod}(E^\mathrm{op})$. Let $A'$ be a source algebra of a block $b'$ of a finite
group $G'$ having $P$ as a defect group. Set $N'=$ $\oplus_Q A'\tenOQ {\mathcal{O}}$,
where $Q$ runs over the subgroups of $P$. By the above, a Morita
equivalence between $A$ and $A'$ which induces a bijection between the
isomorphism classes of indecomposable direct summand of $N$ and of $N'$
induces an equivalence $\coMack(G,b)\cong$ $\coMack(G',b')$.
\end{proof}

\begin{proof}[{Proof of Theorem \ref{coMackderived}}]
We show that this is a special case of \ref{endoderived}.
Let $Q$ be a subgroup of $P$. We need to show that every
indecomposable direct summand of one of the terms of the complex
of $A$-modules $X\ten_{A'} A\tenOQ {\mathcal{O}}\cong$ $X\tenOQ {\mathcal{O}}$ is isomorphic
to a direct summand of $A\tenOR {\mathcal{O}}$ for some subgroup $R$ of $P$.
Since the terms of $X$ are finite direct sums of summands of bimodules
of the form $A\tenOS A'$, with $S$ running over the subgroups of
$P$, it suffices to show that for $S$ a subgroup of $P$, any
indecomposable direct summand $U$ of the left $A$-module
$A\tenOS A' \tenkQ {\mathcal{O}}$ is isomorphic
to a direct summand of $A\tenOR {\mathcal{O}}$ for some subgroup $R$.
Since $A'$ is a permutation $\OP$-$\OP$-module which is finitely
generated projective as a left and right $\OP$-module, it follows
that any indecomposable direct summand of $A'$ as an
$\OQ$-$\OS$-bimodule is isomorphic to $\OS \tenOR (_\psi{\OQ})$
for some injective group homomorphism $\psi : R\to$ $Q$. (See
\cite[Appendix]{Lisplendid} for a review on the use of the
bimodule structure of source algebras, going back to work of
Puig \cite{Puigloc}.) Thus $U$ is isomorphic to a direct summand of a
left $A$-module of the form
$A\tenOS \OS\tenOR (_\psi{\OQ})\tenOQ{\mathcal{O}}\cong$
$A\tenOR{\mathcal{O}}$ as claimed. Exchanging the roles of $A$ and $A'$
completes the proof of the first statement. Let last statement
follows from this, together with the description in
\ref{coMacksourcealgebra} of cohomological Mackey
functors in terms of source algebras.
\end{proof}

\begin{proof}[{Proof of Theorem \ref{coMackstable}}]
Let $G$ be a finite group, $b$ a block of $\OG$, and $P$ a defect
group of $b$. Let $N$ and $\underline{E}$ as in the statement.
Since $N$ is ${\mathcal{O}}$-free, it follows that an
$A$-endomorphism $\varphi$ of $N$ factors through a relatively
${\mathcal{O}}$-projective $A$-module if and only if it factors through a
projective $A$-module. This is in turn equivalent to stating
that $\varphi$ is a sum of endomorphisms which factor
through the summand $A\tenO {\mathcal{O}}$ of $N$. It follows that
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}^\mathrm{pr}_A(N)$ is equal to the ideal in $E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(N)$ generated
by the projection $\tau$ of $N$ onto $A\tenO {\mathcal{O}}$.
Thus an $E$-module $U$ is the inflation to $E$ from
an $\underline{E}$-module if and only if $\tau$ annihilates $U$.
By Yoshida's translation of $E^\mathrm{op}$-modules into cohomological Mackey
functors in \cite{YoshidaII}, the $E^\mathrm{op}$-modules annihilated by $\tau$
correspond exactly to the cohomological Mackey functors which vanish at
the trivial subgroup. This shows the equivalence of categories
$\coMackbar(G,b)\cong$ $\mathrm{mod}(\underline{E}^\mathrm{op})$.
\end{proof}

\begin{proof}[{Proof of Corollary \ref{recollementCor}}]
With the notation of the previous proof, we have $\underline{E} = $
$E/E\circ \tau \circ E$. We also have $\tau\circ E\circ \tau =$
$\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(A)\cong$ $A^\mathrm{op}$. Thus we have a standard recollement
determined by the idempotent $\tau$ in $E$.
The result follows, using the Morita equivalences between $A$,
$\mathrm{mod}(E^\mathrm{op})$, $\mathrm{mod}(\underline{E}^\mathrm{op})$ and $\OGb$, $\coMack(G,b)$,
and $\coMackbar(G,b)$ from \cite[3.5]{Puigpoint},
\ref{coMacksourcealgebra}, \ref{coMackstable}, respectively.
\end{proof}

We will need the following variation of a fact stated at
the beginning of this section: the idempotents in the kernel of the
canonical algebra homomorphism $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)\to$ $\underline{\mathrm{End}}_A(U)$ correspond
exactly to the relatively ${\mathcal{O}}$-projective indecomposable direct
summands of $U$. Thus if two finitely generated $A$-modules $U$ and
$U'$ have the same isomorphism classes of indecomposable
direct summands which are not relatively ${\mathcal{O}}$-projective, then
$\underline{\mathrm{End}}_A(U)$ and $\underline{\mathrm{End}}_A(U')$ are Morita equivalent.

\begin{proof}[{Proof of Theorem \ref{coMackstableeq}}]
Suppose that there is a splendid stable equivalence of Morita
type between $\OGb$ and ${\mathcal{O}} G'b'$. As mentioned in the introduction,
it follows from \cite[7.6.6]{Puigbook} or \cite[3.1]{Lisplendid} that
there is an identification of $P$ with a defect group of $b'$ and that
there are source algebras $A$ of $\OGb$ and $A'$ of ${\mathcal{O}} G'b'$ such
that the induced stable equivalence of Morita type between $A$ and
$A'$ is given by a bimodule $M$ which is an indecomposable direct
summand of $A\tenOP A'$.
Set $N=$ $\oplus_Q\ A\tenOQ {\mathcal{O}}$ and $N'=$ $\oplus_Q\ A'\tenOQ {\mathcal{O}}$,
where in the direct sums $Q$ runs over the subgroups of $P$.
Set $E=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{A}(N)$ and $\underline{E}=$ $\underline{\mathrm{End}}_{A}(N)$. Similarly,
Set $E'=$ $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{A'}(N')$ and $\underline{E}'=$ $\underline{\mathrm{End}}_{A'}(N')$. By
\ref{coMackstable}, in order to show that $\coMackbar(G,b)\cong$
$\coMackbar(G',b')$ it suffices to show that $\mathrm{mod}(\underline{E})\cong$
$\mathrm{mod}(\underline{E}')$. Since $M$ induces a stable equivalence of Morita
type, it follows that we have an algebra isomorphism
$\underline{\mathrm{End}}_A(M\ten_{A'} N') \cong \underline{E}'$.
Thus it suffices to show that the $A$-modules $N$ and $M\ten_{A'} N'$
have the same isomorphism classes of indecomposable nonprojective
direct summands.

Let $V'$ be an indecomposable nonprojective direct summand of $N'$.
Then there is a subgroup $Q$ of $P$ such that $V'$ is an indecomposable
nonprojective direct summand of $A'\tenOQ V'$. It follows that
$M\ten_{A'} V'$ has, up to isomorphism, a unique indecomposable
nonprojective direct summand $V$, which is by construction
isomorphic to a direct summand of $A\tenOP A'\tenOQ {\mathcal{O}}$.
The argument used in the proof of Theorem
\ref{coMackderived} above shows that $V$ is a summand of
$A\tenOR{\mathcal{O}}$ for some subgroup $R$ of $P$, hence $V$ is isomorphic
to a summand of $N$. Exchanging the roles of $N$ and $N'$ as
well as $M$ and $M^*$ implies that the functors $M\ten_{A'}-$
and $M^*\tenA-$ induce a bijection between the isomorphism
classes of indecomposable nonprojective direct summands of $N$ and
$N'$. The result follows.
\end{proof}

\begin{proof}[{Proof of Theorem \ref{coMackfin}}]
Denote by $A$ a source algebra of $kGb$ for the defect group $P$.
It is well-known that $A$ and the defect group algebra $kP$
are separably equivalent via the $A$-$kP$-bimodule $A_P$, obtained
from restricting the $A$-$A$-bimodule $A$ on the right side to $kP$
via the canonical injection $kP\to$ $A$ (see e.g. \cite[4.2]{Liperm}).
Since $A$ is symmetric, the dual of $A_P$ is isomorphic to ${_P{A}}$,
obtained from restricting the $A$-$A$-bimodule $A$ to $kP$ on the left
side. By Tambara's main result in \cite{Tam}, the finitistic dimension of
$\coMack(kP)$ is equal to $1+s(P)$. Set $V=$ $\oplus_{Q\leq P}\ kP/Q$
and $U=$ $A\tenkP V=$ $\oplus_{Q\leq P}\ A\tenkQ k$. We have
$({_P{A})\tenA U}=$ $\mathrm{Res}}           \def\tenOR{\otimes_{\mathcal{O}R}_P(U)$.
This is a permutation $kP$-module, because $A$ has a $P\times P$-stable
$k$-basis. Thus the functor $A\tenkP-$ sends $V$ to $\mathrm{add}(U)$, and
the functor ${_P{A}}\tenA - $ sends $U$ to $\mathrm{add}(V)$.  It follows
from Theorem \ref{endofin} that $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{kP}(V)$ and $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$ have
the same right finitistic dimension. Since the categories of
finitely generated right modules over $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_{kP}(V)$ and $\mathrm{End}}           \def\tenkQ{\otimes_{kQ}_A(U)$
are equivalent to $\coMack(kP)$ and $\coMack(G,b)$, respectively,
by Yoshida's theorem \cite[4.3]{YoshidaII} and \ref{coMacksourcealgebra}
above, it follows that the finitistic dimension of $\coMack(G,b)$ is
$1+s(P)$ as stated.
\end{proof}

redpajama_set_name: RedPajamaArXiv


How I wish I could find someone to hypnotize me and send me back in time to recall the first two years of my life. How I wish I could tear out of the deep recesses of my brain images, pictures of grandparents who, so I have been told, adored me. How I wish to be able to remember these very special people from long gone.
I was their first and only grandchild, adored, fussed over, spoiled. According to my parents, they came by daily as long as it was safe, in order to hold me, to play with me. My paternal grandfather's greatest joy, I am told, was to be given the opportunity to take me for a walk, to have me all to himself. He did not expect any other grandchildren. One of his two sons had died in his teens, his daughter could not conceive, and his other son had married quite late in life. So, I was " IT," the eagerly expected and yearned-for descendant, the one to carry on the family traditions, if not the family name. What would I not give were it possible see him and maybe even be able to talk to him for just a few minutes.
According to all accounts, the man was a tower of goodness, patience, tolerance and devotion to his children, to his wife, to his neighbors and to his God. His son, my father, as well as his daughter-in-law, my mother, could not sing his praises enough for the way he behaved in business, for the way he cared for his family, for his generosity to all in need, his gentleness, and the way he would be able to sooth the most heated argument.
I have only one picture of him, one that had been salvaged by a cousin who had fled burning Germany and settled in the New World. He stands there with his wife and daughter, erect, head high, the picture of a grand old gentleman. He has a short black beard, neatly trimmed, is wearing a black coat over a black suit, a black hat, and black tie, and is holding a black umbrella. However, there is nothing sinister about him; maybe I am just imagining it, but his face seems to radiate the gentleness of his soul.
How I wish I could hold him, hug him, see his soft smile. How I would love to feel his hand on my head, to look up into his eyes and tell him all about the wonderful grandson that would be born after the war and who would be named in his honor. I would tell him about his wonderful four great-grandchildren and his eight great-great grandchildren. How proud he would be of them, not only of their accomplishments, but also of the wonderful people they have become. He would radiate with joy at the business sense they inherited from him and at their straight, decent character, their commitment to the traditions he cherished so.
But he was not even granted a normal death or a grave to rest in peace, a place where we could visit and find some solace. He was brutally shot while being loaded like cattle onto a German army truck, shot because his rheumatism did not allow him to move fast enough to please his captors. His crime? He was a Jew, an "old" man of sixty who was useless as a productive slave and of no use to the occupiers.


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text:
Letters to the Dead in Ancient Egypt
In the biblical Book of Luke, the story is told of Lazarus and the Rich Man in which a man of wealth and the poorest beggar both die on the same day. The beggar, Lazarus, finds himself in paradise while the rich man is in torment. He looks up to see Father Abraham with Lazarus beside him and asks if Lazarus might bring him some water, but this is denied; there is a great chasm fixed between those in heaven and those in hell, and none may cross. The rich man then asks if Abraham could send Lazarus back to the world of the living to warn his family because, he says, he has five brothers who are all living the same self-indulgent lifestyle he enjoyed and he does not want them to suffer the same fate. When Abraham answers, saying "They have Moses and the prophets; let them listen to them," the rich man responds that his brothers will not listen to scripture, but if someone should come back from the dead, they would surely listen to him. Abraham then says, "If they do not listen to Moses and the prophets they will not be convinced even if someone rises from the dead" (Luke 16:19-31).
This story has been interpreted in many different ways over the centuries in order to make various theological points, but its theme is timeless: what happens after death? The rich man thought he was living a good life but finds himself in the worst kind of afterlife while Lazarus, who suffered on earth, is welcomed to a reward in heaven. The request of the rich man to send Lazarus back to earth sounds reasonable in that if someone came back from the dead to tell what it was like, people would certainly listen and live their lives differently; Abraham, however, denies the request.
Abraham's response, however disappointing it might seem to the rich man, is an accurate assessment of the situation. In the present day, people's stories of Near Death Experiences are accepted by those who already believe in that kind of afterlife and are denied by those who do not. Even if someone should come back from the dead, if one cannot accept that kind of reality, one will not believe their story and, in this same way, will certainly not accept ancient stories regarding the same kind of event.
Letter to the Dead
by The Trustees of the British Museum (Copyright)
In ancient Egypt, however, the afterlife was a certainty throughout most of the civilization's history. When one died, one's soul went on to another plane, leaving the body behind, and hoped for justification by the gods and an eternal life in paradise. There was no doubt that this afterlife existed, save during the period of the Middle Kingdom (2040-1782 BCE), and even then the literature which expresses cynicism toward the next life could be interpreted as a literary device as easily as a serious theological challenge. The soul of a loved one did not cease to exist at death nor was there any danger of a surprise in the afterlife such as the rich man from Luke experiences.
An exception is in the fictional work from Roman Egypt (30 BCE - 646 CE) known as Setna II, which is the probable basis for the Luke tale. In one part of Setna II, Si-Osire leads his father Setna to the underworld and shows him how a rich man and a poor man experienced the afterlife. Contrary to Setna's earlier understanding that a wealthy man would be happier than the poor, the rich man suffers in the underworld and the poor man is elevated. Si-Osire leads his father to the afterlife to correct his misunderstanding, and their short trip there illustrates the closeness ancient Egyptians felt to the next world. The dead lived on and, if one wanted to, one could even communicate with them. These communications are known today as 'letters to the dead'.
The Egyptian Afterlife & the Dead
It was believed that, after one died and the proper mortuary rituals had been observed, one passed on to judgment before Osiris and his tribunal, and if one had lived a good life, one was justified and passed on to paradise. The question of 'What is a good life?' was answered through the recitation of the Negative Confession before the tribunal of Osiris and the weighing of the heart in the balance against the white feather of truth, but even before one's death, one would have a fairly good idea of one's chances in the Hall of Truth.
The Egyptians did not rely on ancient texts to instruct them on moral behavior but on the principle of ma'at, harmony and balance, which encouraged them to live in peace with the earth and with their neighbors. Certainly, this principle was illustrated in religious stories, embodied in the goddess of the same name, invoked in written works such as medical texts and hymns, but it was a living concept which one could measure one's success in meeting daily. One would not need someone to come back from the dead with a warning; one's actions in life and their consequences would be enough - or should have been - to give a person a fairly good indication of what awaited them after death.
by Cesar Ojeda (CC BY-NC-ND)
The justified dead, now in paradise, had the ear of the gods and could be persuaded to intercede on people's behalf in answering questions, predicting the future, or defending the petitioner against injustice. The gods had created a world of harmony, and all one needed to do in order to reach paradise in the next was to live a life worthy of eternity. If one made each day an exercise in creating a life one would wish to continue forever, founded on the concept of harmony and balance (which of course included consideration and kindness for one's neighbors), one could be confident of entry to paradise after death.
Still, there were supernatural forces at work in the universe which could cause one problems along life's path. Evil demons, angry gods, and the unhappy or vengeful spirits of the dead could all interfere with one's health and happiness at any time and for any reason. Simply because one was favored by a god, like Thoth, in one's life and career did not mean that another, like Set, could not bring one grief. Further, there were simply the natural difficulties of existence which troubled the soul and threw one off balance such as sickness, disappointment, heartbreak, and death of a loved one. When these kinds of troubles, or those more mysterious, came upon a person, there was something direct they could do about it: write a letter to the dead.
Letters to the Dead date from the Old Kingdom (c. 2613 - 2181 BCE) through the Late Period of Ancient Egypt (525-332 BCE), essentially the entirety of Egyptian history. When a tomb was constructed, depending upon one's wealth and status, an offerings chapel was also built so that the soul could receive food and drink offerings on a daily basis. The letters to the dead, often written on an offering bowl, would be delivered to these chapels along with the food and drink and would then be read by the soul of the departed. Egyptologist David P. Silverman notes how "In most cases, however, interaction between the living and the dead would have been more casual, with spoken prayers that have left no trace" (142). It is for this reason that so few letters to the dead exist today but, even so, enough do to understand their intent and importance.
One would write a letter in the same way one wrote to a person still living. Silverman explains:
Whether inscribed on pottery bowls, linen, or papyrus, these documents take the form of standard letters, with notations of addressee and sender and, depending upon the tone of the letter, a salutation: "A communication by Merirtyfy to Nebetiotef: How are you? Is the West taking care of you as you desire?" (142)
The 'west', of course, is a reference to the land of the dead, which was thought to be located in that direction. Osiris was known as the 'First of the Westerners' in his position as Lord of the Dead. As Silverman and others note, a response was expected to these letters since Spell 148 and Spell 190 of The Egyptian Book of the Dead enabled a spirit to let the living know how it was doing in the afterlife.
Egyptian God Osiris
by Ali Kalamchi (Copyright)
Once salutations and pleasantries were expressed, the sender would get to the matter of the message and this was always a request for an intercession of some kind. Often, the writer reminds the recipient of some kindness they performed for them or the life they lived happily together on earth. Egyptologist Gay Robins cites one of these:
A man points out in a letter to his dead wife that he married her 'when I was a young man. I was with you when I was carrying out all sorts of offices. I was with you and I did not divorce you. I did not cause your heart to grieve. I did it when I was a youth and when I was carrying out all sorts of important offices for Pharaoh, life, propsperity, health, without divorcing you, saying "She has always been with me- so said I!"' In other words, as men climbed the bureaucratic ladder, it was probably not unknown for them to divorce the wives of their youth and remarry a woman more appropriate or advantageous to their higher rank. (63-64)
This husband reminds his wife of how faithful and dutiful he was to her prior to making his request for her help with his problem. Egyptologist Rosalie David notes how "requests found in the letters are varied: some sought help against dead or living enemies, particularly in family disputes; others asked for legal assistance in support of a petitioner who had to appear before the divine tribunal at the Day of Judgment; and some pleaded for special blessings or benefits" (282). The most often made requests, however, deal with fertility and birth through appeals for a healthy pregnancy and child, most often a son.
Letters & Responses from the Dead
A writer would receive a response from the dead in a number of different ways. One could hear from the deceased in a dream, receive some message or 'sign' in the course of a day, consult a seer, or simply find one's problem suddenly resolved. The dead, after all, were in the company of the gods, and the gods were known to exist and, further, to mean only the best for human beings. There was no reason to doubt that one's request had been heard and that one would receive an answer.
Osiris was the lord of justice, and it only made sense that a soul in his presence would have greater influence than one still in the body on earth. Should this seem strange or 'archaic' to a modern-day reader, it should be remembered that there are many who observe this same belief today. The souls of the departed, especially those considered holy, are still thought to have more pull with the divine than someone on earth. Silverman comments:
In all cases, the deceased is urged to take action on behalf of the writer, often against malignant spirits who have afflicted the author and his or her family. Such requests frequently refer to the underworld court and the role of the deceased within it: "you must instigate litigation with him since you have witnesses at hand in the same city of the dead". The principle is stated succinctly on a bowl in the Louvre in Paris: "As you were one who was excellent upon earth, so you are one who is in good standing in the necropolis". Despite this legalistic aspect, the letters are never formulaic but vary in content and length. (142)
Clearly, writing to someone in the afterlife was the same as writing to one in another city on earth. There is almost no difference between the two types of correspondence. A letter written in the 2nd century CE from a young woman named Sarapias to her father follows roughly the same model:
Sarapias to Ammonios, her father and lord, many greetings. I constantly pray that you are well and I make obeisance on your behalf before Philotera. I left Myos Hormos quickly after giving birth. I have taken nothing from Myos Hormos...Send me 1 small drinking cup and send your daughter a small pillow. (Bagnall & Cribiore, 166)
One would write a letter to the DEAD in the same way one wrote to a person still living.
The only difference between this letter and one a son writes to his deceased mother (c. First Intermediate Period of Egypt, 2181-2040 BCE) is that Sarapias asks for material objects to be sent while the son requests spiritual intervention. The son begins his letter with a similar salutation and then, just as Sarapias explains how she needs a cup and pillow sent, makes his request for aid. He also reminds his mother of how dutiful a son he was while she lived, writing, "You did say this to your son, 'Bring me quails that I may eat them', and this your son brought to you, seven quails, and you did eat them" (Robins, 107). Letters like this one also make clear to the deceased that the writer has not 'garbled a spell' in performing the necessary rituals. This would be most important in making sure that the soul of the deceased continued to be remembered so it could live well in the afterlife.
Once the soul had read the letter, the writer had only to be patient and wait for a response. If the writer had committed no sins and had performed all the rituals properly, they would receive a positive response in some fashion. After making their requests, the writers frequently promised gifts in return and assurances of good conduct. Robins comments on this:
In a First Intermediate Period letter to the dead, a husband tells his wife: 'I have not garbled a spell before you, while making your name to live upon the earth', and he promises to do more for her if she cures him of his illness: 'I shall lay down offerings for you when the sun's light has risen and I shall establish an altar for you'. The woman's brother also asks for help and he says, 'I have not garbled a spell before you; I have not taken offerings away from you'. (173)
Since the dead person retained their personal identity in the next world, one would write them using the same kinds of touches that had worked in life. If one had gotten their way through threats, then threats were used such as suggesting that, if one did not get one's wish, one would cut off offerings at the tomb. Offerings were made to the gods in their shrines and temples regularly, and the gods clearly heard and responded, and so it was thought the dead did the same. The problem with such threats would be that, if one stopped bringing offerings, one was more likely to be haunted by an angry spirit than have their request granted. Just as the gods frowned on petulant people's impiety in withholding offerings, so did the dead.
Every ancient culture had some concept regarding the afterlife, but Egypt's was the most comprehensive and certainly the most ideal. Egyptologist Jan Assman notes:
The widespread prejudice that theology is the exclusive achievement of biblical, if not Christian, religion is unfounded with regard to ancient Egypt. On the contrary, Egyptian theology is much more elaborate than anything that can be found in the Bible. (2)
The Egyptians left nothing to chance - as can be observed in the technical skill evident in the monuments and temples which still stand - and this was as true of their view of eternity as anything else. Every action in one's life had a consequence not only in the present but for eternity. Life on earth was only one part of an everlasting journey and one's behavior affected one's short-term and long-term future. One could feel assured of what awaited after life by measuring one's actions against the standard of harmonious existence and the example set by the gods and the natural world.
Egyptian Pottery Soul House
The Egyptian version of the story in Luke, though similar, is significantly different. The rich man in Setna II would expect to find punishment in the next life for ignoring the principle of ma'at. The beggar in the story would not have expected, nor been entitled to, a reward simply for suffering. Everyone suffered, after all, at one time or another, and the gods owed no one any special recognition for that.
In Setna II, the rich and poor man are punished and rewarded because their actions on earth either dishonored or honored ma'at, and, while others may have envied or pitied them, they could have expected what awaited them beyond death. In the Christianized version of Setna II which appears in Luke, neither the rich man nor Lazarus has any idea what is waiting for them. The Luke version of the story, in fact, would have probably been confusing to an ancient Egyptian who, if they had a question concerning the afterlife and what waited beyond, could simply write a letter and ask.
Bibliography Related Content Books Cite This Work License
Magic and Theology in Ancient Egypt by Jan AssmanAccessed 27 Apr 2017.
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David, R. Religion and Magic in Ancient Egypt. Penguin Books, 2003.
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New International Version. The Bible. Zondervan, 2010.
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Strudwick, H. The Encyclopedia of Ancient Egypt. Metro Books, 2006.
Wallis Budge, E. A. The Egyptian Book of the Dead. Dover Publications, 1967.
Wilkinson, R. H. The Complete Gods and Goddesses of Ancient Egypt. Thames & Hudson, 2003.
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Teaching Materials2
Egyptian Gods - The Complete List
The gods and goddesses of Ancient Egypt were an integral part of...
Burial of the dead is the act of placing the corpse of a dead person...
The Egyptian Book of the Dead is a collection of spells which enable...
Ghosts in Ancient Egypt
A text known as The Lay of the Harper, dating from the Middle Kingdom...
Interrelations of Kerma and Pharaonic Egypt
The vacillating nature of Ancient Egypt's associations with...
Ancient Egyptian Government
The government of ancient Egypt was a theocratic monarchy...
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Written by Bagnall, Roger & Cribiore, Raffaella, published by University of Michigan Press (2015)
Ancient Egypt by David P.
Written by Anonymous, published by Oxford University Press (2021)
Written by David, Rosalie, published by Penguin Books (2003)
Women in Ancient Egypt
Written by Robins, Gay, published by Harvard University Press (1993)
The Oxford History of Ancient Egypt
Written by Shaw, Ian & Photos, published by Oxford (2021)
Mark, J. J. (2017, June 02). Letters to the Dead in Ancient Egypt. Ancient History Encyclopedia. Retrieved from https://www.ancient.eu/article/1051/
Mark, Joshua J. "Letters to the Dead in Ancient Egypt." Ancient History Encyclopedia. Last modified June 02, 2017. https://www.ancient.eu/article/1051/.
Mark, Joshua J. "Letters to the Dead in Ancient Egypt." Ancient History Encyclopedia. Ancient History Encyclopedia, 02 Jun 2017. Web. 14 Jan 2021.
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The strings are made from (prepare for the science bit!) Ernie Ball's 'maraging' steel hex cores with an Aluminum Bronze wrap wire and whilst we have no idea what 'maraging' means, the steel hex core gives the strings more output when used with pickup systems and the Aluminum Bronze gives the corrosion resistance and brighter, more projected tone.
Ernie Ball provided us with two identical acoustics, fairly generic Far Eastern mid-market guitars, one fitted with a set with Ernie Ball 'Earthwood' acoustic strings and the other with the new 'Aluminum Bronze' set. The 'Earthwood' set are made from a very different set of materials to the Aluminum Bronze, with a tin plated steel core wire wrapped with an 80% copper, 20% zinc alloy. Whilst they sounded great, the Aluminum Bronze set offered an easily perceptible brighter top end and fuller bottom end with a lot more projection and volume. Miked up, the difference was easy to perceive and for listeners in the room the difference in tone and volume was marked. Through a pickup based system the AB strings offered a higher output and more balanced tone that was significantly more pleasing to the ears and picked up more dynamics of a performance.
As ever, the proof is in the listening (and the playing, of course!) so check out our video and hear the difference.
The AB strings are certainly impressive and, whilst slightly more expensive than traditional acoustic sets (around $3 or £2 more than Earthwoods at street prices, we reckon) they certainly offer enough of an improvement to justify the extra cost. If they last as long as Ernie Ball suggests then they represent a definite step forward in acoustic string design and are well worth trying out.
For the record, the new Ernie Ball Aluminum Bronze strings come in four sets: 011-052, 012-054, 013-054 and 010-050 and in case you developed a passion for our testbed review guitars and are dying to know what they are, they're both Woodstock WHW41309 models.