An Introduction to Morita Theory Matt

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 An Introduction to Morita Theory
Matt Booth
October 2015

Nov. 2017: made a few revisions. Thanks to Ning Shan for catching a typo.

My main reference for these notes was Chapter II of Bass’s book Algebraic
K-Theory (1968); you can find a more detailed exposition there.

1 Motivation
When we’re doing representation theory we want to study the structure of the
category Mod-A for some ring A. So we want to know if and when two different
rings A and B give us the same category. With this in mind, two rings are said
to be Morita equivalent when their module categories are equivalent. In many
cases, we often only care about rings up to Morita equivalence. If this is the
case, then given a ring A, we’d like to find some particularly nice representative
of the Morita equivalence class of A.

2 Morita Equivalence
First some notation: Let R be a ring. Write Mod-R for the catgeory of
right R-modules and R-module homomorphisms. Write mod-R for the (full)
subcategory of finitely generated R-modules. Write Proj-R for the subcategory
of projective modules, and proj-R for the subcategory of finitely generated
projective modules.

Two rings R, S are defined to be Morita equivalent if the categories Mod-R

and Mod-S are equivalent. In fact, any such equivalence will be additive: this
is a general fact about equivalences between abelian categories, since the sum
of two morphisms f, g : X → Y can be recovered as the composition
∆ f ⊕g ∇
f + g : X −→ X ⊕ X −−−→ Y ⊕ Y −→ Y

of f ⊕ g with the diagonal and fold maps. Note that any property defined
categorically will be preserved under equivalence: for example if
F : Mod-R −→ Mod-S is an equivalence, then F will take projective modules
to projective modules and hence induce an equivalence Proj-R −→ Proj-S.

1
 Moreover, an equivalence F as above wil induce an equivalence between
proj-R and proj-S, since the finitely generated projective modules are precisely
those projective modules P for which the functor HomR (P, −) distributes over
direct sums.

Example If R is a division ring, then R is Morita equivalent to all of its

matrix rings Mn (R). We’ll see a proof of a special case of this later on, using
quivers.

Remark One can show that Z(R) ∼ = End[Mod-R,Mod-R] (id), the endomorphism
ring of the identity functor of Mod-R (sometimes called the centre of Mod-R):
first note that id is just HomA (A, −) where A is regarded as an A − A-bimodule.
So by an appropriate version of the Yoneda lemma (we need to be careful about
A-linearity), we get that End[Mod-R,Mod-R] (id) ∼ = EndAe (A, A) ∼= Z(A), where
e op
A := A ⊗ A is the enveloping algebra. So if R and S are Morita equivalent,
then Z(R) ∼ = Z(S). In particular two commutative rings are Morita equivalent
if and only if they are isomorphic. So Morita equivalence is only interesting for
noncommutative rings!

A generator for a category C is an object G such that for any two parallel
morphisms f, g : X → Y with f 6= g, then there is some morphism h : G → X
such that f h 6= gh. Note that generators are preserved under equivalence. If C
is Mod-R, then a generator for C is the same thing as a module G such that
every R-module is a quotient of a (possibly infinite) direct sum of copies of G.

A progenerator in Mod-R is a finitely generated projective generator.

Progenerators always exist: as an easy example, R is a progenerator for Mod-R.
We care about progenerators because of the following result:
Theorem (Morita). Two rings R and S are Morita equivalent if and only if
there exists a progenerator P of Mod-R such that S ∼
= EndR (P ).

Note that we’re considering R as a right R-module. In particular, R ∼

= EndR (R).

We can easily prove one direction of this theorem now: If R and S are
Morita equivalent, then take an equivalence F : Mod-R −→ Mod-S. Since
R is a progenerator for Mod-R, F (R) is a progenerator for Mod-S. We have
isomorphisms R ∼= EndR (R) ∼= EndS (F (R)).

The converse is harder to prove: we need to consider an equivalent characterisation

of Morita equivalence in terms of tensor products of bimodules.

2
3 Bimodules
An R − S bimodule is an abelian group M which is both a left R-module and
a right S-module, such that the actions are compatible: (rm)s = r(ms). If M
is an R − S bimodule and N is an S − T bimodule then it makes sense to form
the tensor product and get an R − T bimodule M ⊗S N .

Theorem.
Two rings R and S are Morita equivalent if and only if there exists an S − R
bimodule P and an R − S bimodule Q such that P ⊗R Q ∼ = S and Q ⊗S P ∼ = R.
Note that these are isomorphisms of S − S bimodules and R − R bimodules
respectively.

Proof. To prove the ‘if’ direction, suppose we have P and Q such that P ⊗R Q ∼
=S
and Q ⊗S P ∼ = R. Then setting F := − ⊗R Q and G := − ⊗S P , we have
GF = (− ⊗R Q) ⊗S P ∼ = − ⊗R (Q ⊗S P ) ∼ = − ⊗R R ∼ = idR and similarly
FG ∼= Sid .

To prove the ‘only if’ direction we need a version of the Eilenberg-Watts

Theorem, which tells us that if F : Mod-R ←→ Mod-S : G is an equivalence,
then there exists an R − S bimodule Q such that F ∼= − ⊗R Q (and furthermore,
Q is a progenerator for Mod-S). So if we have an equivalence as above, applying
Eilenberg-Watts twice we get bimodules P and Q such that F ∼ = − ⊗R Q and
G ∼= − ⊗S P . Using idR ∼ = GF and idS ∼ = F G it’s not hard to check that
P ⊗R Q ∼ = S and Q ⊗S P ∼ = R.

4 Morita’s Theorem
Armed with this new characterisation of Morita equivalence we can prove the
other half of Morita’s Theorem. So suppose we have a progenerator P of Mod-R
with S ∼
= EndR (P ). The left action of EndR (P ) on P gives us a left action of
S on P making P into an S − R bimodule.

Set Q := HomR (P, R). Q has a right action by S ∼

= EndR (P ) where we
precompose a morphism P → R with an endomorphism of P , and a left action
by R ∼
= EndR (R) where we compose with an endomorphism of R. This turns
Q into an R − S bimodule.

If we can prove that P ⊗R Q ∼= EndR (P ) and Q ⊗S P ∼ = R then we’re done.

Define maps φ : Q ⊗S P → R and ψ : P ⊗R Q → EndR (P ) by φ(f ⊗ p) = f (p)
and ψ(p ⊗ f ) = pf . The map φ is onto since P is a generator. The map ψ
is onto since P is a direct summand of Rn and so any endomorphism of P
is a sum of endomorphisms factoring through R. Note that we have identities
ψ(x ⊗ f )(y) = xφ(f ⊗ y) and g ◦ ψ(x ⊗ f ) = φ(g ⊗ x)f , for x, y ∈ P and f, g ∈ Q.

3
 We just need to prove that φ and ψ are injective maps. We’ll only show that
ψ is injectivePsince the argument
P for φ is similar. Since
P ψ is surjective, first find
an element xi ⊗ fi with ψ( xi ⊗ fi ) = 1. Let yi ⊗ gi be any element of
i i i
PP⊗R Q. Then:
yi ⊗ gi
Pi
= (yi ⊗ gi )ψ(xj ⊗ fj ) (multiplying by 1 and using linearity of ψ)
i,j
P
= yi ⊗ φ(gi ⊗ xj )fj (using the second identity)
i,j
P
= yi φ(gi ⊗ xj ) ⊗ fj (pulling an element of R through the tensor product)
i,j
P
= ψ(yi ⊗ gi )(xj ⊗ fj ) (using the first identity)
i,j

P P
So if ψ( yi ⊗ gi ) = 0 then yi ⊗ gi = 0, and so ψ is injective.
i i

5 Induced equivalences between subcategories

We know that an equivalence F : Mod-R −→ Mod-S induces equivalences
between the subcategories Proj-R and Proj-S, as well as an equivalence between
proj-R and proj-S. What about mod-R and mod-S? It turns out that we
have the following very nice theorem:
Theorem (Morita). Mod-R is equivalent to Mod-S if and only if mod-R is
equivalent to mod-S.
Proof. If F : Mod-R −→ Mod-S is an equivalence, then by the Eilenberg-Watts
theorem F ∼ = − ⊗R P for some R − S bimodule P that’s a progenerator for
Mod-S. Since P is a finitely generated S−module, F restricts to a functor
mod-R −→ mod-S, and furthermore this is an equivalence since the inverse of
F restricts in the same way.

Conversely if mod-R and mod-S are equivalent, then we can run the proof
at the end of Section 2 again to obtain a progenerator P of mod-R such that
S ∼
= EndR (P ). But a progenerator for mod-R is a progenerator for Mod-R,
and hence, applying Morita’s Theorem, R and S are Morita equivalent.

4
6 Basic algebras
We are interested in finite dimensional algebras over a field. Therefore for any
such algebra A we want to find a ‘nice’ algebra A0 Morita equivalent to A. Then
we can prove results about Mod-A0 and get results about Mod-A. Here we
describe how to build one such A0 , the basic algebra of A.

Let A be a finite dimensional algebra over a field. Then A admits a decomposition

n
∼ L ei A as right A-modules, where the ei are a complete set of primitive
A=
i=0
orthogonal idempotents. We call A basic if for every i 6= j, the two A-modules
ei A and ej A are not isomorphic.

In general, A need not be basic (see the example below). But there is an easy
way to construct a basic algebra from A. Suppose we have a decomposition of A
as above. Choose a subset {ea1 , . . . , eam } of {e1 , . . . , en } maximal with respect
to the condition that eai A  eaj A whenever i 6= j. In particular every ei A is
isomorphic to some eaj A.

Set e := ea1 + · · · + eam , and put Ab := eAe. Then Ab is a basic algebra. Up

to isomorphism, Ab will not depend on the choice of {ea1 , . . . , eam }. Call Ab the
basic algebra associated to A. Then A and Ab are Morita equivalent: one
can show that eA is a progenerator for Mod-A, and its endomorphism ring is
precisely Ab .

Example Set A = Mn (k), and let ei be the matrix with a 1 in position (i, i)
and zeroes elsewhere. Then the ei are a complete set of primitive orthogonal
idempotents, and for all i and j, ei A ∼
= kn ∼
= ej A. So Ab ∼
= e1 Ae1 is a copy of
k. This provides an alternate proof that Mn (k) is Morita equivalent to k.

5
Another random document with
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the bed. “Uncle Dave!” he cried.
In a moment David Hollis had clasped his nephew’s hands in his
own. “You’ve had a hard time, Donald, my boy,” he said. “How do
you feel?”
“All right, except for my ankle; I gave it a bad twist when I fell.”
“Yes; Glen has told me. I hope you’ll be able to walk soon.” David
Hollis looked at his nephew anxiously.
“In two or three days maybe,” said the trapper.
Don groaned. “Not until then?” he asked. “Meanwhile Aunt Martha is
all alone.”
“Yes, and she needs you, Donald.” David Hollis was plainly worried.
“The worst of it is,” he continued, “that the King’s soldiers have
fortified the Neck and are mighty watchful. There’s no way of getting
in or out.”
“You’re wrong there,” said Glen. “The back harbor’s dry at low water,
you know.”
David Hollis looked doubtful. “It’d be too great a risk to try and cross
that way,” he said. “If anything should happen, I’d never forgive
myself.”
“Now, listen here,” said Glen; “I promised the boy I’d get him back,
and, by thunder, I’m a man of my word. A dark night, a little fog, and
nothing’s easier.”
Don’s uncle said nothing for several minutes. At last he grasped the
old trapper’s hand. “Glen,” he said, “I’ve never yet known you to fail
in an undertaking. May you succeed in this. I see no other way.”
The next day was Friday, and thanks to the trapper’s ointment Don
was able to walk a very little. In the evening his uncle came to talk
with him again. “I probably shan’t see you again for some time,” he
said. “My company is leaving Cambridge. When you see your Aunt
Martha I want you to say this to her: tell her first of all that I’m safe
and well and that she needn’t worry. Second, tell her that at the first
opportunity I want her to leave the town; it’s the height of folly to
remain. And, Donald”—David Hollis spoke in a low voice,—“tell her I
love her. And now, my boy, good-bye, and God bless you!”
That was the last that Don saw of his uncle for many, many weeks.
The next day he and the trapper went for a short walk among the
narrow, twisting streets of the town. Soldiers were quartered in many
of the houses, and people were talking of others that were soon to
arrive. One man remarked that as a result of the British attack on
Concord and Lexington an army of twenty thousand Provincials had
arisen almost overnight. There was much brave talk of attacking the
King’s troops in Boston and of driving them headlong into the sea.
By Tuesday, Don’s ankle was strong again, but he had to walk with
great care. Then early one foggy morning Glen Drake announced
that the time had come to cross the flats.
The two had a hot supper together down in the kitchen, and an hour
or so later they started toward the river.
Glen led the way and in spite of the heavy fog and the darkness
stepped boldly yet as silently as a cat. They had gone beyond the
last fringe of dwelling houses when the trapper put the end of a
buckskin thong into Don’s hand. “Keep tight hold,” he whispered,
“and don’t talk.”
Don thought he had never seen a blacker night—blackness and fog
overhead, blackness and fog all round them, with here and there a
dim yellow light. Several times, at the sound of footsteps, Glen
paused to let a Provincial sentry pass unseen ahead of them. Once
they turned sharply to the left and walked for almost half an hour
over uneven grassy land. Then they turned to the right, and soon
Don felt his feet sink into cool mud. Glen put his mouth close to the
boy’s ear and whispered, “How’s the ankle?”
“All right, Glen,” Don replied softly.
They pressed forward slowly. Sometimes reeds and cattails swept
against their hands; sometimes they seemed to be walking on firm
sand. The fog, cold and oppressive, was blowing in from the east
and seemed to deaden all sounds, even the quash, quash of their
feet. Don’s fingers were like ice as he clung to the thong. He had no
idea in what direction he was going, but he had confidence in his
sturdy guide. Then a bell tolled somewhere ahead, and a few
minutes later he heard a horse neigh loudly.
A quarter of an hour passed, then half an hour. Finally they were
among more cattails. Glen led the way cautiously among them and
at last climbed a gentle slope. They had reached the Boston side.
They were making their way upward, when a stick cracked close at
hand, and a sharp voice rang out: “Halt! Who’s there?”
Don felt Glen’s arm go around his shoulders, and in a twinkling the
two were flat on their faces.
“Who’s there?” came the voice of the sentry again.
Don felt his heart pounding at his ribs and the trapper’s great arm
pressing downward on him like a heavy weight. He heard the sentry
advance and knew that Glen had reached into his belt for something.
Crunch, crunch sounded the footsteps, each louder than the last
one. Glen had drawn back his arm and was gathering himself for a
spring, when the footsteps ceased. A moment later the two heard
them begin again, but now they were growing fainter and fainter.
Glen got softly to his feet and pulled Don upward. Together they
hurried forward and did not stop till they reached a clump of trees by
the side of what appeared to be a path.
“Do you know where you are?” whispered Glen.
“No,” replied Don.
“Well, this is Cambridge Street. You’ll have to follow it alone. Go
carefully, and if you meet anyone—well, don’t let ’em see you; that’ll
be best. And now, good-bye, Don. Take good care of your Aunt
Martha.”
They shook hands in the darkness, and a moment later Don was
alone.
 CHAPTER VII
JUD APPLETON

Luck seemed to walk hand in hand with Don after Glen Drake had
vanished into the darkness. The boy set out at once along
Cambridge Street, walking slowly, pausing frequently, and keeping
well at the side of the road, where the shadows were thickest. When
he came within sight of the first house he stopped to consider, but
the sudden barking of a dog caused him to turn abruptly into the field
at the right. He crossed George Street and skirted Beacon Hill. Near
Valley Acre he came unexpectedly on a large overhanging rock with
two scrub pines growing in front of it; the spot was so sheltered and
so fragrant and dry with pine needles that he decided to remain there
till dawn.
Aunt Martha was an early riser, and it was well that she was, for
shortly before six o’clock the knocker rose and fell heavily three
times on the door. She left her stove and hastened to answer the
knocks. The next moment she was perhaps the most astonished
woman in Boston. “Why—why, Donald!” she cried, and then caught
her nephew in her arms.
Don had the breath almost crushed from his body, and the little
prepared speech of greeting that he had had all ready seemed to
have fled from his memory. “Aunt Martha,” he gasped. “I didn’t know
—you were so—so strong!”
“Now,” said his aunt, releasing him at last, “tell me everything,
Donald,—everything!”
Hungry as Don was, he made no mention of food but sat down in the
low white rocker beside the window and began with the thing that
was most vivid in his mind—the skirmish at Concord.
And all the while that he talked, Aunt Martha sat pale and rigid in the
chair beside him. Only once were her eyes moist, and that was when
Don gave her the last of his uncle’s three messages; but she said
nothing and merely nodded for him to continue.
“Well, I guess that’s all,” said Don at last. “You know, Aunt Martha, I’d
have been home long ago except for my ankle.”
“I know, Donald; and I’m thankful, I hope. It might have been worse.
And now let me get you something to eat. Oh, Donald, you’ll never
know how glad I am to have you with me again.”
It was a long while before Aunt Martha ceased to ask questions; and
then it was Don’s turn. A great change, he learned, had come over
the town even in the few days that he had been away from it. It was
in a state of siege, cut off from the outside world, and food was
scarce among the poor. There were suffering and distress; many
persons, like Aunt Martha, had relatives and friends in the
Continental army and thought with dread and apprehension of what
might happen if the besiegers should attack.
“I don’t know what’s to become of us, truly I don’t,” said Aunt Martha.
“With your uncle and Glen with the army, it’s most too much to bear.
Fortunately, though, we shall not lack for food; the store’s well
stocked.”
“And that stuff in the cellar, is it still there?” asked Don.
“Yes, and it’s likely to remain.”
“We might be able to sell it,” Don suggested hopefully. Then he
added, “If we could only get it to the army in Cambridge!”
But Aunt Martha only smiled and shook her head. “Don,” she said,
“would you rather be in Cambridge, or perhaps with your cousin in
Concord, than here?”
“I want to be with you,” Don replied firmly and then wondered at the
look of quick relief that came over his aunt’s face.
The next day he learned the reason for it. General Gage had agreed
to allow those families who wished to leave the town to go in safety.
But Aunt Martha had not changed her mind. In spite of the
supplications of her husband, whom she loved dearly, and in spite of
the risks that she ran in remaining, she would not leave the little
house in which she had been born and had lived most of her life. If
she was stubborn, it was stubbornness of a defiant, heroic sort, and
those who knew her respected her for it, though some called her a
“foolish woman.”
As a result of General Gage’s permission hundreds of families did
leave the town—a circumstance that greatly alarmed the Tories, who
believed that as long as there were women and children in the town
the Continentals would not attack. So at last the general withdrew his
permission, and the town settled down to wait and to watch.
Though there was no longer any school for Don to attend he found
plenty of things to keep him busy. He helped his aunt about the store
in the daytime and sat and talked with her at night. And the
conversation always was of his uncle and of Glen Drake and the
army, of which they knew little enough. Then always before they
went to bed Aunt Martha would spread the old thumb-worn Bible on
her knees and read a chapter aloud.
Frequently of an afternoon Don would take Sailor and go for a long
walk as he used to do. One bright warm day early in May the two
were on their way home from a long jaunt, and were walking along
between the elms on Common Street, when Don observed a group
of Redcoats some distance in front of him. “Here, Sailor,” he called,
but the terrier paid no heed and ran on ahead.
When Don was within a few yards of the group he recognized two
familiar figures—Tom Bullard, who as aide to General Ruggles of the
Tories, now wore a white sash round his left sleeve, and Harry
Hawkins, the Redcoat, whose life Don had saved. The two were
laughing and talking together.
“Here’s one of the young rebels,” cried Tom as Don drew near. “And
here’s his rebel dog. Get out of here, you pup.”
Don made no answer but spoke sharply to Sailor, and the dog trotted
to his side.
“Good day to you, young sire,” said Hawkins pleasantly.
“Good day,” replied Don, and then colored as he observed a boy of
perhaps his own age who happened to be passing with a fishing pole
over his shoulder.
“Do you know him, Hawkins?” inquired Tom in astonishment and
then as Sailor left Don’s side and started back toward the group he
added angrily: “Git, you pup, git!”
But Sailor was all friendliness as he trotted toward the soldiers.
“Come here, Sailor!” ordered Don, stopping and snapping his
fingers.
But at that instant Tom’s foot shot out and, striking the terrier in the
chest, lifted him into the air. With a loud yelp the dog landed on his
back and then, scrambling to his feet, ran to Don and stood beside
him, trembling.
“I’ll learn a rebel dog a trick or two,” cried Tom. “And before long——”
But Tom never finished the sentence. Before Don could take more
than two steps forward, and before any of the soldiers could
interfere, the boy whom Don had just passed dropped his fishing
pole, and, lowering his head, rushed at Tom. One of his fists struck
the Tory squarely in the mouth and sent him reeling; the other struck
him on the ear and sent him crashing to the ground.
Tom was a big boy and very active. In a moment he was on his feet
and had closed with his opponent, who was easily twenty pounds the
lighter.
“Fight!” cried a Redcoat. “Clear the way there!”
But there was no fight; at least it lasted only until Harry Hawkins
could spring forward and pull the two apart. “Stop it!” he cried and
pushed Tom’s assailant away. “And you,” he said sharply to Tom,
“get along and be quick about it! I thought better than that of you!”
“Why, Hawkins——”
“Never mind that; you deserve a licking, and if the boy hadn’t been
smaller than you, I’d have stood and watched you take it. Kick a dog!
You ought to be kicked, yourself!”
Tom Bullard’s mouth opened and closed. He gulped several times
and then turned for sympathy to the other soldiers; but they were
laughing at him. With low mutterings he picked up his hat and strode
abruptly off across the Common. The soldiers, still laughing, started
toward the tented area.
Don gathered Sailor in his arms and walked to where the boy was
standing; he had shouldered his fishing pole and was blowing on the
knuckles of his right hand.
He was a boy very much like Don in general appearance—sturdy,
active and alert-looking. His hair was of a reddish brown, and his
eyes, dark and sparkling, seemed to flash with little points of fire. As
Don approached him, a smile played about the corners of his rather
large mouth.
Don extended his hand, and the boy grasped it. “I want to thank
you,” said Don, “for thrashing Tom Bullard. My name is Donald
Alden; I live in Pudding Lane.”
The boy grinned. “Mine’s Jud Appleton.” He patted the head of the
terrier. “Nice looking dog you have. When that big Tory kicked him I
couldn’t help sailing into him. He’d have licked me, though, if it hadn’t
been for the Redcoat. My, but didn’t he talk hot to him afterward!”
The two boys laughed heartily. “You surely hit him hard,” said Don.
“Did I?” said Jud. “Well, not hard enough, I reckon. Anybody who’d
kick a dog—my, how I hate ’em! I hate Redcoats too, and Tories
worse—and when a Tory kicks a dog I just boil over.”
The boy’s eyes were flashing again, and his fists were tightly
clenched. Don felt an instant liking for him.
“Say,” said Jud quickly, “do you know that Redcoat? I saw him speak
to you.”
“Well, yes,” Don replied and colored again. “You see, I—I saved him
from drowning once.”
“From drowning!”
“Yes; that is—it was before Concord.”
“Oh, I see.” Jud seemed somewhat relieved. “Do you know the
Tory?”
“We used to be good friends once. His name is Tom Bullard.”
“Oh, yes; so you said. Say, come on along home with me, won’t
you? I live just down here in Hog Alley. I’ve got the finest bunch of
kittens you ever saw.”
“You like kittens?”
“I like all kinds of animals,” Jud replied gravely.
That was enough for Don, and he accompanied his new friend past
West Street and along toward the alley.
“It’s no fun, living so close to the Common these days,” said Jud. “All
you see is Redcoats. And how I hate ’em! My father and my two
brothers are in the army, and I only wish I could be there too. A
drummer boy is what I’d like to be.”
“So would I,” replied Don. “I was up at Concord and saw the fight
——”
“Did you!” cried Jud. “Tell me about it. And how did you ever get
back?”
By the time Don had told him something of the skirmish and of Glen
Drake and his Uncle David the two boys were at Jud’s house. A
poor, miserable-looking, one-story little place it was, with a cracked
weather-worn door and a window on either side that looked out
across the road on a large triangular field covered with clover and
dandelions.
“That’s our cow over there,” said Jud, “and those are our chickens.
We had twenty-six, but we lost four the other night. Ma thinks a
skunk got ’em, but I think it was Redcoats.”
He led the way to a shed behind the house, and a moment later Don
was looking at six fluffy black and white kittens nestled in the folds of
a burlap bag. As he bent over them the mother cat came running
from a corner of the shed, and he started backward. Sailor backed
away and sat down; he had suffered enough for one day.
“She won’t hurt you,” said Jud, laughing. “Will you, puss?” He played
with the kittens for several minutes, stroking and calling each by
name while the mother cat sat by and watched contentedly. “They’re
pretty well grown now and about ready to shift for themselves. That’s
a good dog of yours to sit there like that. I had a hard time to keep
my dog away from them at first. Say, wouldn’t you like to have one?
Ma says I can’t keep ’em all.”
“Yes, I would,” replied Don. “We haven’t any, and a cat might be
good company for my aunt.”
“Well, here’s a nice one,” and Jud lifted one of the kittens that was all
black except for one white foot. “See, she has one white shoe on;
she lost the others.”
“I’ll call it Whitefoot, then,” said Don and laughed.
“Judson, are you home?” came a woman’s voice from the house.
“Just got home, Ma.”
“Well, come here; we lost two more chickens last night.”
“Thunder!” exclaimed Jud in a low voice.
“Yes, two more,” repeated Mrs. Appleton, appearing at the door of
the shed. “I counted them just now, and there’s only twenty. Oh!” she
exclaimed at sight of Don.
“This is Don Alden,” said Jud; “he lives up in Puddin’ Lane.”
Don found Jud’s mother a pleasant, talkative little woman who in
some ways reminded him of his aunt, though she was not so old.
When Jud had explained to her about their adventure with Tom
Bullard and about Don’s trip to Concord she insisted that he stay and
have something to eat with them.
Later as Don was about to set out with his new pet, Jud whispered to
him: “I’m going to stay up to-night and catch that chicken-thief. I wish
you could be here with me. Can’t you come back?”
“I don’t know,” Don replied doubtfully. “I’d like to, but there’s my aunt,
you know; I don’t like to leave her alone. Have you got a gun or
anything?”
“No; but I’ve got a hickory club, and I can throw a stone pretty
straight.”
“I’d like to sit up with you,” said Don.
 CHAPTER VIII
THE BOYS SET A TRAP

The next day was fair and warm, but on the following day the wind
changed, and the drab, suffering town of Boston was shrouded in a
thick blanket of fog. Don rolled over in bed and stretched and
yawned.
“Donald,” came the voice of his aunt, “it’s high time you were down
here to breakfast. You’re awake, ’cause I hear the bed a-creaking.
Come on now; Mrs. Lancaster is coming to-day.”
Don lay and blinked for a moment; then he sprang out of bed. If Mrs.
Lancaster were coming, probably she would stay all night—she
usually did. Don had almost given up hope of going to Jud’s and of
sitting up with him to catch the skunk or whatever was stealing his
chickens; but now, if Mrs. Lancaster were coming, he would not mind
leaving his aunt for a while in the evening.
At breakfast Aunt Martha said that her visitor would remain
overnight; and when Don had told her what he wanted to do she
objected at first, as he knew she would, and then consented after he
had promised her to keep far away from any skunk that might come
after Jud’s chickens.
At evening when Don set out for Hog Alley the fog was still heavy.
The houses on the opposite side of Pudding Lane, which was one of
the narrowest streets in town, could hardly be seen. And on the
Common even the scarlet-coated soldiers were almost invisible at a
distance of twenty yards.
“I don’t know but what Ma was right,” said Jud when Don reached
the shabby little house in Hog Alley. “There was a skunk round here
last night—a big fellow too, from the smell of him. But I had the hen-
house locked tight and all the chickens inside; so he didn’t get a one.
I was wishing you’d been here though—are you going to stay to-
night?”
“For a while, if you want me.”
“I surely do!” Jud was very positive about it. There was no doubt that,
even on such short acquaintance, he liked Don quite as well as Don
liked him. “Well, I’ve got a plan,” he said eagerly. “I want you to tell
me what you think of it.”
“Let’s hear it,” said Don.
“Well, come around to the chicken yard and I’ll explain,” said Jud.
“Now here,” he said a few moments later, “you see our chicken yard
has a high fence and a small gate at the far end.”
“I see,” said Don; “the gate opens out and latches on the outside.”
“Yes, and it’s a strong latch too. Well, I thought we could leave the
gate open and attach a long rope to it and run it through the fence on
this side and back to the wagon shed here, where you and I could
wait. Then if Mr. Skunk comes along and enters the yard, all we’ll
have to do is to pull the gate shut and we’ll have him. Of course he
won’t be able to hurt the chicks ’cause they’ll be locked tight in the
hen-house. What do you think of the idea, Don?”
“Mighty good; but what’ll we do with the skunk when we catch it?”
“Oh, Fred Ferguson next door will kill it for us in the morning.”
“And what if it shouldn’t be a skunk? What if it should be a Redcoat?”
Jud laughed. “I guess we shan’t catch a Redcoat,” he replied. “I hate
’em so much I guess I was unfair the other day. It’s a skunk all right;
you’ll see.”
“I hope so,” said Don. “We’d be in a nice fix if we caught a Redcoat.”
“Let’s set our trap,” said Jud. “The first thing is to find enough rope.”
The boys at once began to search the wagon shed, and by the time
they had found enough lengths, had fastened them to one another
and had tied one end of the improvised rope to the gate of the
chicken yard, darkness had set in in earnest. Carrying the other end
of the rope across the yard and passing it between the wires of the
fence, they retired with it to the door of the wagon shed to wait.
“Just a moment,” said Jud and crossed the yard to the house.
When he returned he carried with him a pan of cornbread and two
large apples. “This is going to be fun,” he said. “It’s like being out in
the woods, trapping.”
“It is a little,” Don agreed; and then he told Jud more about Glen
Drake and about the trips that the old trapper and he had made
together. “You’ll have to come to the house sometime when he’s
there,” he said.
“I’d like to,” said Jud, “but if he’s with the army, it’ll be a long time
before he can come to Boston again.”
“Oh, I don’t know,” replied Don. “If Glen wanted to come very much
he’d come, and the King’s men would never catch him either!”
For a while the boys sat silent, munching cornbread and apples in
the doorway of the old shed. All round them was darkness, damp
and chill. Up on Common Street a wagon creaked past; the driver,
whoever he was, was singing a boisterous song. After a while he
passed out of hearing; and only the occasional challenge of a sentry
far across the Common broke the stillness.
Don’s head was beginning to nod; but Jud, rope in hand, was wide
awake. “Not asleep, are you, Don?” he whispered.
“What? Oh, yes.” Don shook his head from side to side several
times. “Guess I was asleep. Wonder what time it is?”
“Don’t know; I’ve been listening for a bell.”
“It won’t do to fall asleep,” muttered Don.
But in a few minutes his head was on his chest, and his shoulder
was resting comfortably against the side of the doorway.
Half an hour passed, and at the end of it Jud was nodding between
sleep and wakefulness. Suddenly he felt a slight tug on the rope in
his hands. With a start he sat bolt upright, and the next instant the
chickens in the hen-house began to cackle furiously.
“Don! Don!” whispered Jud and seized his friend by the shoulder.
“What!” Don was wide awake in a flash.
But before Jud could reply something struck the fence. Jud gave a
mighty heave on the rope, and as the gate came shut with a harsh
bang both boys heard someone exclaim aloud.
“A Redcoat!” gasped Jud. “What shall we do?”
“Quick, call somebody!” cried Don, springing to his feet.
Both boys raised their voices and then rushed toward the house. The
chickens were making a terrible noise now; and Jud’s dog, which he
had tied at the back of the wagon shed, was barking at the top of his
voice.
Whoever was in the chicken yard was having a hard time getting out.
Don, standing at the corner of the house, could hear the fellow
pounding furiously at the gate and shaking it with all his might.
In the midst of the commotion a window opened in the house next
door, and then a light gleamed within. “There’s Fred Ferguson,” said
Jud. “O Fred, O Fred!” he shouted. “Come quick!”
“Judson, Judson, what on earth is the matter?” It was the voice of
Mrs. Appleton.
Jud did not reply, for at that moment Fred Ferguson, partly dressed
and carrying a lighted candle, which he was shading with his hand,
appeared on the back doorstep of the Ferguson house. He was a big
raw-boned young fellow, and both boys noticed that he was carrying
a heavy stick under one arm. “What’s wrong?” he shouted and
advanced toward the fence.
“Somebody’s in our chicken yard,” replied Jud. “Come on, Don,” he
added, and the boys hurried toward Fred.
“Open the gate and let me out of this!” came a voice out of the fog,
and Don started.
The fence shook violently, and the dog and the chickens increased
their clamor.
“Open the gate, I say!”
“Leave off shaking that fence,” cried Fred. “Who are you, and what
are you doing in there? Leave off shaking that gate, I tell you—if you
break it, I’ll whale ye!”
“Open up, then!”
“Come here, you boys, and tell me who it is,” said Fred and held the
candle above his head.
Both boys got a brief glimpse of the person within the yard, and Jud
said quickly, “’Tain’t a Redcoat.”
“No; ’tain’t a Redcoat,” said Fred. “Now come here,” he said in a loud
voice. “Come here and let me see ye, and tell me what you’re a-
doing in there.”
“Open that gate and stand aside—or—or, by thunder, I’ll shoot!”
“Judson! Come here!” cried his mother from the doorway. “Donald,
you too!”
There was a moment of silence, and then Fred said evenly: “I’ll risk a
shot from a chicken-thief.”
With those words he unlatched the gate and threw it open. “Now
come here and let’s see what kind of a person ye are,” he said and
waited with club poised in one hand.
“Let me hold the candle,” said Don.
He was advancing to take it when the fellow in the yard made a
sudden rush. Don saw Fred’s club descend and heard it strike
something hard. Then Fred went over backward, but just before the
candle went out Don had a glimpse of the intruder’s face as the
fellow rushed past and vanished into the darkness. It was Tom
Bullard!
“Tarnation!” exclaimed Fred, getting to his feet. “Can’t see a thing.
He’s gone, blast him! What a tormented fool I was to let him rush me
like that!”
The quick footsteps of the thief were becoming fainter and fainter in
the distance. Then they ceased abruptly.
“Who was it, Fred?” asked Jud.
“Don’t know.” Fred was angry with himself and spoke sharply. “Didn’t
get much of a look at him and wouldn’t know him again if I saw him.
Well, he won’t come back; that’s certain.”
“Judson, didn’t I call you?”
“Yes’m. Don, where are you? Come into the house for a minute.”
“No; I’d best be going,” replied Don quickly.
But before he went he whispered to Jud: “Do you know who the
fellow was? It was Tom Bullard!”
“Tom Bullard! The fellow who kicked your dog?”
“Yes; I’m sure of it; I saw his face just before the candle went out.”
Jud whistled softly.
“Judson Greenleaf Appleton, if you don’t come into the house right
straight this minute——”
“Good night, Jud,” said Don and hurried out into the alley.
A bell was striking the hour of ten o’clock as Don reached
Marlborough Street. Almost no one was abroad at that late hour, and
only here and there a light gleamed soft and yellow through the
heavy fog. He passed the Old South Meeting-House and a few
minutes later was in Pudding Lane.
Mrs. Lancaster and Aunt Martha were just preparing to go to bed,
when Don entered, out of breath and red of face.
“Well, Donald,” said his aunt, “I was thinking it was high time you
returned.”
“Did you catch your skunk?” inquired Mrs. Lancaster.
Don could not help grinning. “Well, yes; I guess we did.”
“You guess!” Aunt Martha was mildly astonished. “Just what do you
mean, Donald?”
“It wasn’t a real skunk that was after Jud Appleton’s chickens,” Don
replied. “It was Tom Bullard.”
“Goodness!” exclaimed both ladies.
And Don hastened to explain what had happened while he was
gone.
“Wasn’t I just a-saying,” said Mrs. Lancaster when Don had finished,
“wasn’t I just a-saying, Martha, that you can’t trust a Tory out of your
sight? Wasn’t I, now?”
“You were, Hannah.”
“And Tom Bullard—well, I always said he was a bad one.”
And Don was thinking the same thing as he climbed the stairs to bed
a few minutes later.
 CHAPTER IX
THE REGULARS EMBARK

Early the next morning Don was hard at work washing the windows
at the front of the store. He had cleaned them on the inside and was
about to start on the outside, when Jud crossed the square and
hailed him. Over his shoulder he was carrying two fishing poles.
“Where are you going?” asked Don.
“Up to the mill-pond. I thought maybe you’d come along, so I brought
an extra pole.”
“Sure,” said Don; “but I’ll have to finish these windows first.”
“I’ll help you,” Jud replied promptly and, setting down the poles,
rolled up his sleeves.
While the two boys were cleaning and polishing the glass Tom
Bullard happened to turn into the lane from King Street. It was clear
that he had not expected to meet the boys and did not want them to
see him; for he had no sooner spied them than he stopped and
made as if to turn back; but Jud’s sharp eyes had already caught
sight of him. “Here’s the chicken-thief, Don,” he whispered.
Don stopped work to look. It is to Tom’s credit perhaps that he did
not turn on his heel then and there. What he did was to lift his chin a
trifle and, choosing the opposite side of the street, march past
without looking either to the right or to the left. It was really a hard
thing to do, for Don and Jud were staring at him and grinning frankly.
“He’s got his head pretty high, hasn’t he?” said Jud in a loud whisper.
“But not high enough to hide that bump above his left eye,” replied
Don.
“That’s where Fred’s stick landed,” said Jud. “Just look how high he
holds his head—just like a chicken!”
Both boys chuckled, and a moment later they laughed outright when
Tom’s foot struck an upraised brick, and he stumbled. At the corner
of Water Street, Tom turned and shook his fist.
Jud’s eyes flashed, but Don was silent. “And to think,” he said at last,
“that he used to be my best friend!”
“He’s not worth thinking about,” said Jud shortly. “Come on, Don,
let’s finish these windows in a hurry. I wonder how the fish are
biting?”
But there were other things beside fish to wonder about on that day
in early May. The people of Boston knew little enough of what was
going on round them. Every other person was wondering how soon
the American army would attack the British, and whether the
Redcoats would risk going out and fighting in the open. Already there
had been skirmishes and they continued to occur off and on
throughout the rest of the month; but although the Americans were
generally successful, the skirmishes really did not amount to much.
Word had somehow seeped into the beleaguered town that the
Continental force consisted of sixteen thousand men and that
fortifications were being prepared in Cambridge and along the Mystic
River; and it was whispered that men from all the other Colonies as
far south as Virginia were flocking to join the army. But Gage’s men
scoffed at such reports; and although none of them dared venture
outside the town they also scoffed at the idea that they were in a
state of siege. A body of undisciplined farmers oppose them, the
King’s soldiers? Preposterous!
What the King’s men did not realize was that many of them,
especially the officers, had fought in the French wars. Oddly enough
the terrible experience of the nineteenth of April was lost upon the
over-confident British; they supposed that the men who had fought
so valiantly at Concord and Lexington would run like frightened
sheep in an encounter in the open.
Numerous things had occurred to exasperate the good people of
Boston, but one of the worst was a proclamation that Gage issued; it
declared martial law and referred to all who were bearing arms