C 0 N T E N T S

0. Introduction

1. Cohomology of Monoids and Monoidalgebras

2. Shuffling

3. Harrison Cohomology

4. The Relation between Harrison Cohomology and Algebra Cohomology

5. Harrison Cohomology of Monoid-like ordered Sets

6. Graded Harrison Cohomology

7. Harrison Cohomology of two-dimensional Torus Embeddings

8. An Example

9. References

p A (k,A;M)

Ap{A;M) p

Harr (A ;M)

p Ha (L; Z)

p HA (L;Z)

N 0 T A T I 0 N

algebra cohomology of A with values in M

algebra cohomology of a monoid A with values in M

Harrison cohomology of a monoid A with values in M

(differential ; d)

Homogenous Harrison cohomology of an ordered subset

L of A (differential ; o)

Inhomogenous Harrison cohomology of an orderen

subset L of A

Relative Harrison cohomology of an ordered set L

with respect to the ordered subset L-L 0

0. Introduction

Many authors (among others L&S, Ri, Sch 1, Ch) have tried to

calculate the algebra cohomology groups of singularities of various

special kind. In this paper we study monoid-algebras and give a

method for calculating the algebra cohomology by using the

combinatorial properties of the monoid. We introduce the Harrison

cohomology of commutative monoids as an important tool in the study

of commutative cohomology of commutative algebras. For the two­

dimensional torus embeddings over a field of characteristic zero, we

give a formula for computing the cohomology by looking carefully at

some finite subsets of the monoid.

Starting with a monoid-algebra we have different cohomology theories

of which two are very natural, one in the category of commutative

algebras and one in the category of commutative monoids. In the first

chapter a proof of Laudal is simplified to show that the two

cohomology theories coincide. In chapter 2 a number of usefull lemmas

concerning shuffling-theory are stated and proved, mainly as a tool

for working with Harrison cohomology. This cohomology is defined

using the Hochschild complex, dividing out by all shuffle-products.

In (Sch & S) Schlessinger and Stasheff shows that this theory is the

same as the algebra cohomology defined in chapter 1. We give another

proof for this fact, see chapter 3 and 4, using a quite different

method, more specific for monoids. The main point is to show that the

free commutative monoids are models for the theory, i.e. have

vanishing cohomology.

Looking at the graded cohomology groups we are led to the definition

of Harrison cohomology of monoid-like ordered sets. In chapter 5 and

6 we show the close relation between graded cohomology of the monoid

A and Harrison cohomology of monoid-like ordered subsets of A.

In the special case of two-dimensional torus embeddings we have

enough coherence in the monoid and the spectral sequence degenerates.

vve are able to state the results which give the algebra cohomology

groups as a direct sum of Harrison cohomology groups of finite

monoid-like subsets of the monoid. The well-known formulas

dimkA1 (A,k[A])=2(r-1) and dimkA2 (A,k[A])=r(r-2) are easily

verified. In addition we can compute the dimensions in the higher

dimensional cases.

- 1 -

1. Cohomology of monoids and monoidalgebras.

Let k-alg be the category of commutative k-algebras, and let k-free

be the full subcategory of free commutative k-algebras, i.e. polyno­

mial rings over k. Let A be an object of k-alg and denote by

& k-free/A the category where the objects are morphisms r + A of the

polynomial ring r into A, and the morphisms are co~tative dia-

grams

(jl r + r

1 . 2

6\ /&2 A

If M is any A-module we define the functor

r by the equality Derk(~6,M) = Derk(r,M) where M is considered as a

A

r-module via 6.

Definition 1 .1. With the notation as above we define the algebra

cohomology groups of A with values in the A-module M by

p A (k,A: M) =

(P) lim Derk(-,M) +--

(k-free/A)0

6 Let F be a free k-algebra and F + A a surjection. Consider the

semi-simplicial k-algebra

where

F . 0

F : A+-----•

•••• x F0 , the fibered product of A

p+1 copies of

- 2 -

Proposition 1.2. There exist a Leray spectral sequence with

• converging to the cohomology A (k,A:M) where F. is as above and

H is considered as a F -module via 6 • •

Proof. Follows immediately from the Leray spectral sequence of

[ La 1 ( 2 . 1. 3 ) ] .

Let ~ be the category of commutative monoids and let free ~ be

the full subcategory of free monoids, i.e monoids isomorphic to

for some n. Let A be an object of ~ and consider the category

6 free ~/A whre the objects are morphisms r + A, with r free.

Define the functor

Der(-,M): free ~/A + Ab

where M is a k[A]-module (and therefore a A-module) and

r Der(~6,M) = Der(r,M)

A

M is considered as a r-module via 6.

Definition 1.3. We define the cohomology groups of the monoid A

with values in M by

1 . ( p) ~m Der(-,M) p)Q.

~

( free mon/ A ) O

6 Let r + A be surjective, with r a free abelian monoid. Consider

the semi-simplicial monoid

0

- 3 -

r : A • r = r x r 0 1 0 .A

Proposition 1.4. There exists a Leray spectral sequence with

• converging to the cohomology A (A: M).

Proof. Follows from [La 1 (2.1 .3)].

D

Now observe that for an abelian monoid A "' Z + and the associated

monoid k-algebra k[A] we have the equality

Der(A: M) = Derk(k(A]: M)

Consider the semi-simplicial monoid •

defined above. It is easily r

seen that the associated k-algebra k[r.] is a semi-simplicial

object of the category k-alg. Moreover, there is a natural morphism

of semi-simplicial k-algebras

where

q,: k[r ] ~ k[r]

4> : n

.

k[r xr X•••Xr ] ~ k[r ] X k[r ] X ••• X k[r ] 0A 0A A O O k(A] O k[A] k[A] O

is defined by <r 1 ,···,y ) r 1 Yn

t n ~ (t ,•••,t ).

Suppose this map is surjective. Then it follows from the proof of

(2.1.3) in [La 1] that we may replace k[r]. by k[r.] in

proposition 1 .2. The reason is that if we replace the complex

c = { ( z, k[ r] ) , ~ } , 0 of lemma 2. 1 . 1 in [La 1 ] by • p p p ...

C' = {C(Z,k[r ]),o } , 0 it is still a resolution of C(Z,k[A)) in • p p P'

(k-free) 0 the category Ab ----

- 4 -

So we must show that ~n is surjective.

Let us assume A is finitely generated. k[r 0 ] has a natural

A-grading so we may work on the A-homogeneous parts. Pick AEA. Since

A is finitely generated there is a finite number of monomials

Y 1 Y m t ,•••,t such that

yi A o(t ) = t .

Let w be a homogeneous element of k[r0 ]p of degree A. He can

write

m y, w = ( l ai t ~

i=1

m y. m y. l b.t ~, ••• , l c.t ~

i=1 ~ i=l ~

m y, l d.t ~)

i=l ~

where a,, b,, c. and d. are elements of the ground field k. We ~ ~ ~ ~

shall prove the surjectivity of ~ by constructing an element

w E k[r ] such that ~ (W) = w. p p

It is easily seen that the element

m-1 (yi,y 1 , ... ,y 1 ) m-1 (yn,y 1 , ... ,y 1 ) m (y ,y,,y 1 , .. ,y1 ) H= L a.t +(b1- L a.)t + l b.t n ~

i=l ~ i=1 ~ i=l ~

m-1 ( Y I • • • 1 Y I Y 1 ) m-1 ( Y I • • • 1 Y I Y • ) + ••• + (d- l c )t n n + l d.t n n ~

1 i=1 1 i=1 ~

has this property.

We have thus proved the following

Theorem 1 . 5. With the notation as above we have an isomorphism of

cohomology groups

p p A (A; M) =A (k,k(A]; M) p>O.

0

- 5 -

2. Shuffling.

Let I be a totally ordered set with n elements. Devide I into

two blocks I = (I <I ) 1 2

where =I =p. 1

Definition 2.1. A (p,n-p)-shuffling a of I into a totally

ordered set J is a bijective map a: I ~ J such that

a ( i ) < a ( j ) if i < j E I 1 or i < j E I 2 .

This definition is slightly more general than the usual definition,

which is obtained by setting I=J={1,2, ... n}.

Lemma 2.2. Let a be a (p,n-p)-shuffling of {1, •.• ,n} into it-

self. Then we have

i) a-1 (1) = 1 or a- 1 (1) = p+l

ii) -1 -1 a (n) = p or a (n) = n

Proof. Suppose a- 1 (1) = j. Two cases are possible. 1 ( j<p+1 implies

a ( 1 ) -1 if p+1<:1<:j<:n a(p+1) -1

( a a ( 1 ) = and we have ( a a ( 1) = 1 .

ii) is proved in a similar way. 0

Remark 2.3. Using lemma 2.2 we can describe the shufflings recursi-

vely. A (p,n-p)-suffling a: {1, •.. ,n} ~ {1, ... ,n} is determined as

follows:

Either 1) a(l )=1 and a: {2, ••• ,n} ~ {2, ••. ,n} a (p-1 ,n-p)-shuffling

1\ 2) a(p+1)=1 and a:{1, •.. p+1, •.. ,n} ~ {2, ... ,n}

a (p,n-p-1)-shuffling.

- 6 -

Starting in the other end we get an alterntive description:

. " 1} a(p}=n and a:{1, ..• ,p, •.. ,n} + {1, ... ,n-1} a (p-l,n-p}-shuffling.

2} a(n}=n and a:{1, ..• ,n-1} + {1, ... ,n-1} a (p,n-p-1}-shuffling.

The third description is as follows. If i E {1, ... ,n} is different

from the maximal element of {1, .•. ,n}, there is a 1-1 correspondance

between (p,n-p}-shufflings a:{1, ... ,n} + {1, .•. ,n} such that

a(i}+1=a(i+1) and (p-l,n-p}-shufflings (resp.(p,n-p-1)-shufflings)

a:{1, ... ,i+1, ... ,n}+{1, ... ,a(i+1), ... ,n} if i E lower block (resp.

upper block.) Moreover we have the mod 2-equality

I cr I = I 0" I + d ( 0" ( j ) I j ) + l ( mod 2 )

where d(a(j),j) is the number of elements in {1, ... ,n} strictly

between a(j) and j. If equality a(j)=j holds, put d(j,j) = -1.

He also need a lemma to produce new shufflings from given ones .

Lemma 2.4. Let ..... I + {1 n} "" I • 0 • I be a (p,n-p)-shuffling, and let

1(i<n. Suppose a-1 (i} and a-1 (i+1) are not in the same block. Let

~ be the transposition which permutes a-1 (i) and a-1 (i+1). Then

ao~ is a (p,n-p)-shuffling of I into {1, ..• ,n}.

Proof. Let s<t be in the same block of I. If {s,t}n{a-1 (i),

a - 1 ( i + 1 ) } = ¢ we have a o 't ( s) = a ( s ) < a ( t) = a ( t) = a o.,; ( t) . Sup-

pose s = a-l (i). Then a(s)= i < a(t) and a o ,;(s)= aat(i+1)= i+l. -1

Assume a(t) = i+l. then t =a (i+l). But s and t are in the

same block which contradicts the fact tht a- 1 (i) and a-1 (i+1) are

not in the same block. So i+1 < a(t) = ao,;(t) and ao,;(s) < ao,;(t).

If s = a- 1 (i+1) we have ao,;(s)= aa-l (i)= i<i+1 = a(s} < a(t}= ao~(t}.

The two other cases are treated similarly.

0

- 7 -

If I is a totally ordered set, with =I=n., it is possible to de-

fine a bijective, order-preserving map

o:I:{l, ..• ,n} +I

For a shuffling cr:I + J we have a diagram

0: I

{l, ... ,n} I

o:J lcr {l, ... ,n} I

Since all the maps are bijective we obtain a permutation -1

o: J ocr o o: I = p of {l, ... ,n}. This construction is unique and we may

define the sign I cr I of the suffling as the sign of the corresponding

permutation.

3. Harrison cohomology

Let r be a commutative semi-group with unit, i.e. a monoid, and M

a r-module. \Je use the notation

Mor(~r,M) = {~:~r+Mj~(y 1 , ... ,y )=0 if 3i such that y,=O} n n n 1

Let ~EMor(llr,M). We say that ~ vanishes on all shuffle-products if n

for every (y 1 , ... ,y )E~r, and every l<p<n the sum n n

where cr runs through all (p,n-p)-shufflings of {y 1 , ... ,yn}, ordered

by the indicies.

The subset of Mor(~r,M) of morphisms vanishing on all shuffle­n

products is denoted by Mor8 ~r,M). Notice that this generalizes the

definition of algebra cohomology in [Ha].

- 8 -

There is a differential map d:Mor(.u.r,M)+Mor( ..ll1r,M), n)l n n+

defined

by

n .

d<j> ( y 1' • · • 'Y n+l) =y 14> ( Y 2' · · · 'y n+ 1) + i~l ( -l) 1 4> ( y 1' · • • 'y i +y i +1' · ' · 'y n+ 1 )

n+l + (-l) ell(yl, ... ,yn)·yn+l

Lemma 3.1. The differential satisfies the following two conditions:

i) d 2 = 0

ii) If <j>EMor8 (*r,M), then d4>EMor8 (n~1 r,M).

Proof i) See for instance [c&E].

ii) If we n~ke no distinction between a shuffling of {r 1 , .•. ,yn+1 }

and a shuffling of the index set {1, ..• ,n} we can write

-1 d<j>(a (y1' · · .,yn+l)) = Y -1 <j>(y -1 '· •• ,y -1 )

a (1) a (2) a (n+l) n .

+ 2 ( -1 ) ~ell ( y -1 ' • . . ' y -1 +y -1 ' . . • ' y -1 ) i=l a (1) a (i) a (i+l) a (n+1)

n+l + (-1 ) <j>(y -1 , ... ,y -1 )y -1

a (1) a (n) a (n+l)

where a is a (p,n+l-p)-shuffling of {l, ••. ,n+l}.

Using remark 2.3 we get for p*l and p*n

y -1 <j>(y -1 , ... ,y -1 ) = a (1) a (2) a (n+l)

y14>(y 1 , ... ,y 1 ) a- (2) a- (n+l)

y p+ 1 ell ( y -1 I 0 0 0 I y -1 ) a (2) a (n+1)

where in the first case a:{2, .•. ,n+l}+{2, ... ,n+l} is a (p-l,n+l-p)-1\

shuffling and in the second case a:{l, ••• ,p+l, •.• ,n+l}+{2, ... ,n+l}

is a (p,n-p)-suffling. Taking the alternating sum of all (p,n+l-p)-

shufflings and using the fact that 4> vanishes on all shuffle-

products we get

IC-1)1aly -1 <j>(y -1 , ••. ,y -1 ) = 0 a a (1) a (2) a (n+l)

- 9 -

when o runs through all (p,n+1-p)-shufflings of {1, ... ,n+1}. In a

similar way we obtain

"(-l)lol 41 (y ) L ,..-1( 1 )•••••Y -1 y -1 o u o ( n) o ( n+ 1 )

= 0

If p=1 we get, using the vanishing of 41,

I ( -1 ) I o I Y -1 41 ( Y -1 ' ... ' Y -1 ) = Y 1 41 ( Y 2 ' ••• ' Y n+ 1 ) o o ( 1 ) o ( 2) o ( n+ 1 )

But we also get

, I o l+n+1 I ( -1 ) 41 ( Y _1 , ... , Y _ 1 ) Y _ 1 = -41 ( Y 2 , · · • , Y n+ 1 ) • Y 1 o o (1) o (n) o (n+1)

And the two sums cancel each other. Similarily for p=n. So it

remains to show

~· ~(- 1 )1ol+i~(y ) = o L L 't' 1 I ••• I y 1 +y 1 I • • • I y 1 o i=l o- (1) o- (i) o- (i+l) o- (n+l)

If o-1 (i) and o- 1 (i+1) belong to different blocks, and if ' is

the transposition which changes o-1 (i) and o-1 (i+1), o and oo'

are shufflings with opposite sign (Lemma 2.4) and they will be

cancelled.

Assume and -1

0 (i+1) are in the same block. Then we have

the equality -1 -1

o (i)+l=o (i+1). Let jE{1, .•. ,n+1} satisfy

and j*n+1 and let X.={x 1 , ... ,x.+x. 1 , •.. ,x 1 } be ordered by the J J J+ n+

indicies such that xj+1 <xj+xj+l<xj+ 2 if xj_ 1 exists. We have an

order-preserving bijection

b A {x1 , ... ,xj+xj+1 , ... ,xn+1 } ~ {1, ... ,j+1, ... ,n+l}

Let j be in the lower block (resp. upper block). Consider all

(p-1, n+1-p)-sufflings (resp. (p,n-p)-shufflings)

A 0 {1, ... ,j+1, ... ,n+1} ~ {l, ... ,n}

- 10 -

F'or each a we have a bijective, order-preserving map

{1, .•. ,n} a A -~ {1, •.. ,a(j)+1, ... ,n+1} = Ia(j)+1

1\ so we can consider the shuffling as a map {1, ... ,j+l, ... ,n+l}~

1\ {l, ... ,o(j)+l, ... ,n+l}. Since ~EMor5 (f,ir,M) we have

\' \' I a I -1 ~ 4 L (-1} Ha (l, ... ,a(J}+l, ..• ,n+l)) = 0 J a:I, ~r

J+l n+l

It is easily seen that this is the sum which remains to show

vanishes.

0

Having proved this lemma we can make the following definition

Definition 3.2. We define the n-th Harrison cohomology of r with

values in M by

n n Harr (r,M} = H (Mor 5 (~r,M),d) n)l

The motivation for this definition is the results of the two next

propositions, which are stated here, and proved in the rest of this

section.

Proposition 3.3. Let M be a r 1- and a r 2-module, and therefore a

r 1xr2-module. Then there is an isomorphism of cohomology groups

for all n> 1 .

Proposition 3.4.

- 11 -

Let M be a Z -module. Then we have +

The main result of this section follows as an immediate corollary.

r Theorem 3.5. If r=z is a free abelian monoid, and M is a

+ r-module, it follows that

n Harr (r,M) = 0 n)l

\Je start by proving proposition 3. 3 •

where

and y.=(y. 1 y:)Er 1 ~r2 . Define maps -~ ~ ~

Mor5 (~r 1 ,M) Mor5 (~r 2 ,M) i Mor5 (~(r 1 xr 2 ),M) ex

13 ( <P 1 I <P 2 ) (!) = y 1 ° 0 0 y n <P 2 ( y i I 0 0 0 I y ~) +y i 0 0 0 y ~ <P 1 ( y 1 I 0 0 0 I y n )

a (<P)l(yl, ... ,yn) = q,((y1 10), ••• 1 (yn10 ))

a(q,) 2 (y~~···~Y~) = <P((O,yl), ••• 1 (0 1 y~))

Lemma 3.6. With the notation as above, we have

i) doj3 = j3od

ii) doa = aod

Proof: Follows easily from the definitions.

We have the equality aoj3=id, but j3oa*id. We shall construct a

homotopy j.J.:Mor5 (-ft(r1xr2 ),M)+ Mor5 (*(r 1 )(r2 )~M) such that

j3oa=id+doiJ.+iJ.od. Thus a and 13 will induce isomorphisms in

cohomology.

0

- 12 -

( 1.1 I o o o I l.n-1 )

<r.2 1 ' ' 0 1 I.n)

( y 1 I o • • I Y n I Y i + o • • +y ~ )

<r.,~···~r.n~A) for some

~ is defined recursively by

Let cr (~)(r.)=do~ 1 (~)(y)+~ 2 od(~)(r.)-y 1' ... y'~(y 1 ~···~Y) n - n- - n- - n n

- 13 -

n n n +~n-lod(cpoSY )(dn(~))-(-1) Ln~n-l(cp)(dn(~))

-n n-1 n-1 n n n-1 n

-(-1) ~n-2(cpoSLn-l+Ln)(dn-ldn(~))-(-l) Ln-l~n-2($oSLn)(dn-ldn(!)) 0 n 0 n

- Yl~n-2(cpoSy 1 +y )(dn-ldn(t)) + Ll~n-2($oSY )(dn-ldn(t)) 1 -n -n

+ $ ( y 1 I o o o I y 1 I y 11 + o o • y I 1 +y ) - y 11 o o o y I 1$ ( y 1 I o o o I y 1 I y ) n- n- -n n- n- -n n 0 0

+ (-1) (~n-lod){cpoSYi)(dn(~))- Ll~n-l<P(dn(X:))

0 n n-1 0 + Yl~n-2($oSY +y 1 )(dn-ldn(~))- YlLn~n-2($oSY 1 )(dn-ldn(!))

~ 1 1 n 0 0

- (-l) YlY2~n-2(cpoSyi+Y2)(dn-ldn(~))

+(-l)nYll2~n-2(cpoSY{)(d~-ld~(t))+(-l)nylcp(y21'''1Ynlyi+ ... +y~)

- ( -1) ny 1 Y 2 ... Y ~ <P ( Y 2 I ••• I Y n' Y i) + ( -1) n-1 d<P( ~~ ( ~) ) -y i ... Y ~ $ ( Y 1, ... I Y n)

n n 0 =a 1 (cpoS )(d (x_)) + (-1) y1o 1 (cpoS 1)(d (x_)) n- Ln n - n- y 1 n -

This recursion can be used to give a closed formula for (J (cp). n

Lemma 3.8. There is a closed formula for a (cp) given by n

0

where A= S( )(~ 1 , ... ,~ ) n-q,q n

is the set of all (n-q,q)-shufflings

CJ : { ~ 1 1 • • • 1 ~ n} + { 1 1 • • • 1 n} and

[Li+q

A. = I

~ Y n+l-i

l<i<n-q

n-q+l<i<n

are ordered by the indices. E(n~q) is given by the formula

E(n 1q) = (-l)nE(n-l,q-1) 1 E(n 10)=-l ~n)O,

Proof. By induction on the index n

Assume the formula is proved for k<n-1.

- 14 -

Using Remark 2.3 we may rewrite the sum

I a I -1 e:(n,q)}:(-1) 4>(a (l, •.• ,n))y 1 ••• y

. a q -1 .

= e:(n,q) L Ha (l, ... ,n-l),Xnyl .•• yq aEA

~ lal+q -1 + e:(n,q) L.. (-1) q,(a (l, ..• ,n-l),X )y 1 ... y aEB n-q q

A where B = S( l )(X 1 , •.. ,X , ... ,X) n- -q,q n-q n

The sign function satisfies the equality

e:(n,q)•(-l)q = (-l)n+(n-l)+ •.. +(n-(q-l))e:(n-q,O)(-l)q

= (-l)n+(n-l)+ •.. +(n-(q-l))e:(n-q-1,0)(-l)q

= (-l)(n+l)+ ••• +(n-q)e:(n-q-1,0)

= e: (n-1, q)

Putting this together, using the induction hypothesis and the

recursion for a (<t>) we obtain the postulated formula. n

0

If 4>EMor (~r,M) <f> vanishes on most of the sums in the expression S n

of Lemma 3.8. The remaining part, for q=O or q=n,

is treated in the next lemma.

Lemma 3.9. Let <f> vanish on all (p,n-p)-shuffle products. Then we

have the equality

- 15 -

Proof. The vanishing hypothesis implies

n-1 I I L (-1)1+2+ .•. +(n-p)};(-1) a l!l(a-1(1, ... ,n)) = 0 p=1 0'

£y. 1 <:i<p where for given p; A,- ~ 1 .

~ Yn+p+1-i p+ <~<:n and a runs through all

(p,n-p)-shufflings of {1, ... ,n}.

Every bit of the (p,n-p)-product must be of one out of two types;

ljl( ••• ,y , ••• ,y 1) p p+

or ljl( ••• ,y , ••• ,y) p+1 p

Since p=O and p=n do not occure in the sum, every n-tuple of the

two given types, except for Ill (y 1 , .•• ,yn) (.:):p=n-1) and ljl{yn' •.. ,y 1 )

(~:p=1), occures twice. The first case also in the (p+1,n-p-1)-pro­

duct, and the second case also in the (p-1,n-p+1)-product. An

important remark is that these are the only places they can occure.

Checking the signs it is easily seen that the two bits annihilate

each other. What is left then is the sum

where 6 is the cyclic permutation which brings the first element

into the last position. lol=n-1 and we have proved the lemma.

Thus we have proved

do~n- 1 {ljl)('~) + ~nod(ljl)(~) - ~oa:(ljl){x,) + ljl(~)

=a {l!l)(x.)- r 1 ···Y Hr 1~ •..• ,y~) + lll<r 1 •••• ,l. > n - n n - n

0

( ) ( 1 ) ~n ( n-1 ) + 1 ( I I ) ( 1 I ) { ) 0 = -Ill ~ + - Ill Y n ' • • · ' Y 1 Y 1 · · • Y n -y 1 · • • Y n Ill Y 1 ' · • • ' Y n + Ill ~ . =

- 16 -

and ~ is a homotopy between ~oa and the identity. The conclusion

of the proposition follows immediately.

0

The next step is to prove Proposition 3.4, i.e. calculate Harrison

cohomology of the monoid z+. We shall do this by showing that

where

f ( ) t i or the complex Mor8 ~Z+,M •

z+-module M. The calculation is

=

n-1 . + l: <-1>1.

i=2

+

is a homotopy

is the multiplication of i on the

He have proved the equality doh+hod+id=O and the vanishing of the

cohomology groups follows as an immediate consequence.

0

- 17 -

4. The relation between Harrison cohomology and algebra cohomology

The purpose of this section is to prove the coincidence of Harrison

cohomology and algebra cohomology of a monoid.

\'ve can consider 1 for M a A-module 1

Mar (ll-~M):free man/A+ complex of ab.gr. s • ---- --- - -

as a contravariant functor from the category of free monoids over

A into the category of complexes of abelian groups. If we let • 0

c ((free .!!!2!2/A) ~-) be the resolving complex of the functor

lim -+- 0

(free ~/A)

we get the double complex

• 0 K"" = c ((free man/A) 1Mor8 (~-~M))

and the two spectral sequences

'Eplq = 2

lim (p) 0 Hq(Mor8 ~-I M)) (free ~/A)

= Hq li:m(p) 0 (Mor8 (~-~M)) (free !!EE./A)

both converging to the cohomology of the double complex K"". We have

shown that for a free monoid r

For q=1 we get

1 Harr (r~M) = ker{Mor8 (r~M)+Mor8 (r~r~M)}

= {~EMor(fiM)j~(y1+y2)=y1~(y2)+y2~(y1)}

= Der(f 1M)

- 18 -

and the first spectral sequence

To calculate the other sequence \te need a lemma.

Lemma 4. 1 • With the above assumptions and if char k = 0 we have

i) l.!m 0Mor8 (J!.-, M) = Mor S (J!.A, M) (free ~/A)

ii) (p)

lj;m 0Mor 8 (.Jl-, M). = 0 for q)l and p)l (~~/A) q

Proof, Consider a semi-simple monoid r.tA, and the induced cochain­

complex

Mor(A 1M) + Mor(r0 1M) + Mor(r 1 ,M) + ••• (*)

The semi-simplicial monoid is A-graded and A is finitly generated

so it is enough to consider the homogeneous case

d() d M + TI M -+ TI M -l

A}.. A~

where ')..EA -1

and A}..=~ (}..)I a finite set.

Let s:TI M + TI 1M be the map given by A n n-

A ~

A}.. • s { ~ ) { a 1 1 • • • 1 an_ 1 ) =

It is easily seen that this forms a homotopy between zero and the

identity of the complex (TI M1d ). i.e. s6-os=id. So for finitely An n

A

generated A the sequence (*) is exact. Now observe that

ftr =(Hr 0 ) . The sequence q p q p

- 19 -

+ •••

may thus be considered as a subcomplex of the acyclic complex

Mor{ {~r 0 )., M). The homotopy s

S {!; ) ( y 1 I • • • I y n-1 )

extends to Mor{r ,M) n

through

-1 where A=~ {~{yi)) and IAI= A. For !;EMors{(~r0 )n,M) an easy

computation shows that s{!;)EMors{(~r0 )n_ 1 ,M). So the homotopy

restricts to the subcomplex, which is acyclic as well. By [La 1] we

have lim 0Mor{Jl-,M)"'Mor{.J1A,M) {free+~/A) q q

and as a consequence

l!m 0Mors{.ll.-,M)"'Mors(.llA,M) for {free mon/A) q q

q)1. A repeated use of Leray

spectral sequence [La 1] to compute lim{p) Mor{ll- M) + 0 I

(free ~/A) q now

proves the lemma.

Thus we have proved the following

Theorem 4.2. With notation as above, we have an isomorphism

0

Combining this theorem with Definition 3.2 we obtain

Corollary 4.3.

0

5. Harrison cohomology of Monoid-like ordered Sets.

Let A be a cancellative monoid with no non-trivial subgroups. Then

A has the structure of an ordered set given by ~ 1 ~ ~ 2 E A if there

exists ~EA such that ~ 1 + ~ = ~ 2 . Now let L c A be some

sub-ordered set.

- 20 -

Definition 5.1. L c A is said to be a monoid-like ordered set if

for all relations A1< A2E L there exists ~EL such that A1+ ~ = A2 as elements of the monoid.

Now define the set

S (L) = {(A 11 .•. 1A) E Lnjw(A) E L} n n -

n where the weight w(~) of is the element w(~) = L A,E A. The

i=1 ].

permutation group ~ acts on s (L) Ln n by

or = 0' (.A 1 1 ••• 1 An) = (A I • • • I A_ ) 0'-1(1) -1 0' ( n)

Let c (L) be the free abelian group on s (L) . The action of n n

on s (L) induces an action on c (L) by permuting the basis n n

elements.

We also define the dual groups: Cn(L) = Homz(Cn(L)~z>~ with the

action of In given by

and rJ E L • n

ln

There is a shuffle-product • C (L) p

® C (L) + C (L) defined by q p+q

~ lol -1 = L(-1) 0' (A 11 ... 1A +) 0' p q

where w(A 11 ... 1A ) E L and where 0' runs through all p+q

(p~q)-shufflings of {1121'''1p+q}. Denote by Sh (L) the n

z-submodule of C (L) generated by all shuffle-products. n

- 21 -

Next we define some differential maps

6 , a : C (L) + C ( L) n n n n-1

(resp. 6n,an: Cn-1 (L) + Cn(L))

by the action on basis elements of

n-1 .

C (L) (resp. n

6 (A1, ... ,A) =I (-l)~(A1''''tA.+A, 1'''''A) n n i=1 ~ ~+ n

n an(A1''"'An) = (A2''"'An) + (-1) (Al'"''An-1)

n-1 . (resp. 6nE(A 1 , ••• ,An) =I (-1)~E(A 1 , ... ,A.+A, 1 , ..• ,A )

. 1 ~ ~+ . n ~= .

n n a E(Al'''''An) = ~0.2, ... ,A.n) + (-1) E(A.1'"''"'n-1))

For the case n=1 we put

It is easily seen that the differentials

we have

6~ (~) = E6 (A) n-and

a nE (~) = Ea (A) n-

Lemma 5.2. We have the equalities

i) 0 0 = 0 n-1 n

are dual,

ii) a & + 6 a + a a = 0 n-1 n n-1 n n-1 n

Proof. A simple computation.

i.e. for

Let D = 6 + a • As a consequence of the lemma D D = 0. The n n n n-1 n

relations between the differentials and the shuffle-products are

stated in the next lemma.

0

- 22 -

Lemma 5.3. With the notation as above the following equations hold

for xEC (L), yEC (L) : p q

i}

ii}

iii}

and as a consequence

iv)

x•y = (-l)p•qy•x

&p+q(x•y) = &P(x)•y +

= a (x} •y + (-1 )px•a (y} p q

D (x•y) = D (x)•y + (-l)Px•D (y) p+q p q

Proof. Another simple calculation.

0

Using these lemmas we may define the Harrison (co-}homology of the

set L.

Definition 5.4.

The homogenous Harrison homology Ha (L) (resp. cohomology n

a} n

Ha (L,Z)) of the ordered set L is the (co-}homology of the complex

c:(L) =c. (L)/Sh.{L) {resp.

the homogenous differential

c;(L) ={Ill E C8 (L)!4l(Sh.(L)) = 0}

& (resp. on). n

with

b) The inhomogenous Harrison (co-)homology HA (L) (resp.HAn(L,Z)) n

8 of the ordered set L is the (co-)homology of the complex c.(L)

• n (resp. Ds(L)) with the inhomogenous differential Dn (resp. D).

Remark 5.5 There is also a relative version of Harrison

(co-)homology. Let L0 ~ L c A and suppose L0 is full in L, i.e.

if yEL, y 0EL0 and y">y 0 , then yEL0 • The relative Harrison complex

is given by

s Cn(L-L0 ,L) = Cn(L-Lo~L)/Shn(L)

(resp. c:(L-L0 ,L) = {4>ECn(L-L0 ,L!4l(Shn(L)) = 0}

where c n ( L-L 0 , L) = c n ( L) I { ( ~ 1 1 ••• 1 ~ n) I w ( ~) E 1.-r. 0 }

(resp. Cn(L-L0 ,L) = {4>ECn(L)!4>(~ 11 ••• ,~n)=O for w(~)E L-L0 }

- 23 -

Proposition 5.6. With the notation as above there is a long-exact

sequence

1 1 1 0 ~ HA (L-L0 ,L:k) + HA (L:k) + HA (L-L0 :k)

+ HA2 (L-L0 ,L:k) +

relating Harrrison cohomology of the ordered sets L and L-L . 0

Proof. The relative complex gives rise to a short-exact sequence of

complexes

where L-L 0

is an ordered set since L0 is full in L.

0

The next theorem is the main result of this section. It relates the

"local" cohomology HAP(~:k) for elements A.EL to the "global"

cohomology HA• (L:k).

Theorem 5.7. There exist a spectral sequence given by

= lim(p)HAq(~:k) m

• converging to HA (L:k).

Proof. Using the definition of the complex it is easily seen that

for some ordered set L

• C (L:k) s

• 1\ = lim C (-·k) s I

tE"L 1\

where A. = {A.'Ej~u·<A.}. The shuffle-products are homogenous and the 1\

inhomogenous differential is well-defined on the sets A..

• Now let D (L,-) be the resolving complex for lim. Denote by K -L

the double complex

• •

- 24 -

• • • • 1\

K = D (L,Cs(-~k))

We have the two associated spectral sequences

and

'Ep,q = HpHq(D• (L,C·(~;k}}} 2 s

= Hp(D• (L,HAq(~;k}}}

= lim(p}HAq(~;k} + L

,.--, q p • • 1\

= H H (D (L,Cs(-;k}}}

is surjective and by [La 2] we have

p > 0

and for p = 0

The double complex is situated in the first quadrant and the two

spectral sequences have the same abutment. The second sequence

d€generates to HAq(L;k) and the theorem follows.

0

- 25 -

6. Graded Harrison Cohomology

Suppose A+(-A)=Zr. We shall equipe the complex MorS(qA,k[A]) with

r d' E r . a Z -gra ~ng. For AO Z we def~ne

Ao Mors (~A,k[A]) = {~EMors(~A,k[A])I$:homogenous of degree A0 }

Homogenous means that $(~) is homogenous and that the element

is independent of choice of ~. This element is called the degree of

~- It is easily seen that the differential respects the grading, and

that the degree of the differential is 0.

Definition 6.1. The graded Harrison Cohomology of A with values in

k[Aj is defined by

for n)Q, AEZr.

Put as an abbreviation M~= Mors(*A,k[A]) and M~'A= Mor~(ftA,k[A]).

Proposition 6.2. 0

a) The inclusion Jl M•, A s

AEZr

-+ M s of complexes induces an inclusion

at the cohomology level;

n, A [ ] n [ ] l1. Harr (A,k A ) + Harr (A,k A ) n ) 0.

AEZr

b) The inclusion is an isomorphism whenever Harrn(A,k[A]) is a

r Z -graded group.

Proof.

a) Let ~EMn,A be homogenous and suppose •EMn-l satisfies d•=~. s s Let •A be the A-graded homogenous part of •· Since deg d=O we

must have $=d•A·

- 26 -

b) Suppose Harrn(A,k[A]) r

is Z -graded and let ~EM ,d~ = 0. Then s we may replace (mod im(d)) ~ by some ~O which is sum of

homogenous components.

The graded Harrison cohomology groups are closely related to the

Harrison cohomology of ordered sets, as defined in the previous

chapter.

0

Theorem 6.3. With the notations as above and in chapter 5 there is

an isomorphism in cohomology

where = (-~+A)nA , and +

A = A-{0} +

Proof. Put where ~ 0 (~) E k. The map

is easily seen to induce an isomorphism of vector spaces

--+

It also takes the graded version of the differential d into the

inhomogenous differential D. Recall that in the definition of

Mor~{llA,k[A]) we agreed that ~(~) = 0 if 3i such that s •

~i = 0. This is the reason why we use the positive part. A+ in stead

of A.

0

We end this chapter with a couple of results about the graded •

Harrison cohomology. A close study of the complexes Cs(A+-A0 ,A+~k)

for various ~Ezr gives the next proposition.

Proposition 6.3. Fix some n)l. The cohomology groups

Harrn'~(A,k[A]) are equal for all ~EA.

- 27 -

Proof. If A.EA we have A =(-A.+A )nA =A 0 + + + and

means that for every A.EA we study the same complex.

0

Corollary 6.4. Suppose the cohomology group An(A,k[A]) is of

finite dimension over k. Then

n+l,A. Harr (A,k[A]) = 0

for all A.EA.

Proof. k[A] has infinite dimension over k.

0

7. Harrison Cohomology of two-dimensional Torus Embeddin~s.

The simplest, but still maybe the most important family of

monoid-algebras are the two-dimensional torus embeddings k[A] over

a field of caracteristic zero.

2 Let A c z+ be a commutative saturated monoid and let

positive part, i.e. A= A-{o}. For A. E A we define + +

A+ be the

A(A.) = A.+A.

He want to study the "local" and "global" Harrison cohomology of

monoids, that is, the cohomology of monoidlike subsets of the monoid

as well as the monoid itself. Also the submonoids A+-A(A.) for

various A. E A are of great interest, and we start with a closer +

look at these objects.

Let and be the generators for the one-dimensional faces of

A. (See for instance [k] for details), and define

r. = r.(A.) ={A.' E A I ~ t E z, A.'+t•y, t A(A.)} i = 1,2 1 1 + 1

- 28 -

r. ].

is an ordered set with the same ordering as A. Furthermore it is

easily seen that

and

where L'A. is the "strong" link defined as follows: There is a

unique description of A. given by

A. = a • l

where the a.'s are non-negative rational numbers. We make the ].

definition

L'A. ={A.'= by+ byE A lo < b < a,,i = 1,2} 1 1 2 2 + i ].

(Note: For the normal link the definition is 0 < b ~a,, but A.'*A.) i l.

Proposition 7.1. With the notation as above'there is a

Mayer-Vietoris sequence

for all

Proof. Using the functor HAq(~) on the system of inclusions

- 29 -

of ordered sets we get

and for the higher derivatives, a spectral sequence

Since for p:t:O, 1 the spectral sequence degenerates to the

two exact sequences

and

0 -+ El,p -+ 2

E O,p+l -+ 0 2

0 -+ EO,p 2

-+ !im(p)HAq(~) x }im(p)HAq(~) -+

r 1 r 2

-+ El,p -+ 0 2

Putting this together we obtain the long-exact sequence of the

proposition.

0

To state and prove the next proposition we need some notation and

definitions.

For ~ 0 E A we let C(Lq(~ 0 )) be the vector space on the set

and consider the complex (C(L·(~0 )),&) where the differential is

the homogenous differential of definition 5.3.

Denote by

- 30 -

the subset of A where the q-th homology of the given complex

vanish.

Proposition 7.2. The map qA ql\

HA (:>.. 2 ;k) + HA {:>-. 1 ;k), q;>l, induced by 1\ 1\

the inclusion :>-. 1 ~ :>-. 2 is an isomorphism whenever x2-x 1c Uq, where

u is defined as above. q

Proof. It is enough to show the proposition for L c L'c U where q

L'-L = {u} is a one-element set and u is minimally greater than

L, that is if u;>u', then u'E L. This is because we have a

filtration

where = {u.}, u. ~

is minimally greater than L. ~

and u.

belongs to u . q

~

Consider the exact sequences of complexes

0--+ D~(u;k)--+ C~(L';k) ~ C~(L;k)--+ 0

where

~

and the differential is the dual of the homogenous differential given

above.

• We must show that the complex (D (u;k) ,d) s is acyclic. Dualizing the

problem we are led to the study of the short-exact sequence of

complexes

• 0 --+ Sh(L (u);k)

• s • ~ C(L (u);k) --+ C (L' (u);k ~ 0 (*)

where Sh(Lq(u);k) is the subspace of C(Lq(u);k) consisting of all

shuffle-products x•y with x E C(LP(u);k) and y E C(Lq-p(u);k),

and

- 31 -

The differentials are the homogenous ones.

• The question is whether a homotopy for C(L (u):k) will induce a

homotopy for the subcomplex Sh(L.(u):k). We are working over a field

of caracteristic zero and the following lemma gives an answer .

• Lemma 7.3. Let g be a homotopy for C(L (u);k). The map

• • h: Sh(L (u);k) ~ Sh(L (u);k)

defined by

h(x•y) = 1/2(g(x)•y + (-l)px•g(y))

for x E C(Lp(u) ;k), y E C(Lq-p(u) ;k) is a homotopy for the • subcomplex Sh(L (u);k).

Proof.

(dh + hd)(x•y) = d(~ (g(x)•y)) + d(~(-l)P(x•g(y)))

+ h(d(x)•y) + (-1)Ph(x•d(y))

= ~ (d(g(x))•y + (-1 )p-1g(x)•d(y)

+ (-1)Pd(x)•g(y) + (-1) 2Px•d(g(y)))

+ ~ (g(d(x) )•y + (-1 )p-ld(x)•g(y)

+ (-1)Pg(x)•d(y)+(-1) 2Px•g(d(y))

=~((dog+ god)(x)•y + ((-1)p-l+ (-1)P)g(x)•d(y)

+ ((-1)p+ (-1)p-1 )d(x)•g(y) + x•(dog + god)(y))

= ~ (x•y + x•y)

= x•y

The assumption in the proposition ensures that the complex

0

C(L.(u);k) is acyclic and since working over a field, dualizing of

the complex (*) will give an inclusion of acyclic cochain complexes

0

- 32 -

The assumption in Prop 7.2. is that the set sits inside •

So we must study the set Uq, or better, the complex C(L (u)) with

differential &.

A basis for C(Lq(u)) consists of all tuples (n 1 , .•• ,nq) with

In.= u. These elements may also be written as ordered tuples; J

+ n 1 < n1+ ..• + n = u q- q

Observ that all the tuples has u as their maximal element. Removing

this top element we obtain an ordered tuple of the ordered set L(u).

It is easy to see that this sets up a bijection between U Lq(u) and q

the simplicial set associated to the set L(u).

The homogenous differential coincide through the bijection with the

usual differential of the ordered set, the alternating sum of the

face maps.

The homology of the simplicial sets L(u) are studied in [La & Sl]

and we state, without proof, one result from this paper.

Let a be the right-most minimal element of A excepting the +

generator of the face y 1 , and be the left-most minimal element

excepting the generator of the face y 2 . Denote by U the subset of

A+ given by

Lemma 7.4. For all we have the inclusion

Proof. See Lemma 2.5 of [La & Sl].

ucu . q·

0

Corollary 7.5. The morphism HAq(A 1 < A2 ) is an isomorphism for all 1\ 1\

q)l whenever A2 -A 1c u.

- 33 -

Proof. Combine Proposition 7.2 and Lemma 7.4

D

To calculate the graded algebra cohomology groups we have seen that

we need information about invers limits of the pre-sheaves HAq(~) over various ordered subsets of the monoid A. In [La 2] it is shown

that these calculations can be made over even smaller subsets under

the assumption of cofinality.

Let we put B (r) = {y•e r h'> y}. y 0 0

Definition 7.6. A subset r c r 0

is called cofinal if the following

two conditions are satisfied:

i) For every yEr, we have BY(r0 ) * !i1 ii) For every finite family y ,y , .•. ,y

1 2 s of elements of a < r > y 0

there exists a y OE BY(r0 ) such that for every i=l,2, ... ,s we

have either or

Using the theory for cofinal subsets it is rather easy to prove the

next proposition.

Proposition 7.7.

a) For a two-dimensional torus embedding A=k[A] we have

b) For the subsets r 1 and r 2 , defined above we have the same

equation

}_.im(p)HAq(~) = 0 for i=1, 2 , p)1, q)l. r. ~

Proof. In all the sets A n U, r n U and r n u there are cofinal + 1 2

subsets isomorphic to z+. Equipe these with the constant presheaf

HAq(~). It is well-known that the higher derivatives vanish. Using

the cofinality of the subset and Th 1 .2.4 of [La 2] the proposition

follows.

D

- 34 -

The following is also true:

Proposition 7.8. For a two-dimensional torus embedding A=k[A] we

have

Proof. An irnrnideate consequence of corollary 6.4 and the facrt that

A is an isolated singularity (see for instance [Pi]) and therefore

has finite dimensional cohomology groups.

0

Combining these results we obtain the next theorem, concerning the

vanishing of the cohomology groups HAq(~;k).

Proposition 7.9.

let A. E U c A,

2 Let A c Z+ be a commutative saturated monoid and

A. >> 0. U is as defined above. Then we have

Proof. The set u has two components, one for each face of the

monoid and we put u = v u 1 v2. Using definition 7.6 it is easily seen

that v', i=l , 2, are cofinal ·~

in A+. Thus we have an isomorphism of

derived functors

A consequence of this last equality is that the spectral sequence

· degenerates and we have an isomorphism

The right side of the equation vanishes (Prop. 7.8.) and so does the

invers limit. But HAq(~) is constant on U as seen in Cor. 7.5. and

the result of the theorem follows.

a

- 35 -

We may use Prop. 7.8 in another context, namely together with the

result of Prop. 5.5 concerning the relation between Harrison

Cohomology and relative Harrison Cohomol09Y·

Remark 7 . 1 0 • If we put L=A +

and L = A -A(A.) 0 + into Proposition

5.5. and use Proposition 7.8. a) we get an exact sequence

0 ~ HA1 (A+-A(A.),A+;k)--+ HA1 (A+;k) ~ HA1 (A+-A(A.);k)

2 --+ HA (A+-A(A.),A+;k) --+ 0

and isomorphisms

\~e will come back to the use of this in the next chapter.

As a consequence of Proposition 7.7.b) the Mayer-Vietoris sequence of

Proposition 7.1. splits up into the following exact sequences

(2)

The isomorphism of (2) proves the next theorem which states that the

algebra Cohomology Groups of a monoid-algebra may be calculated as

the Harrison Cohomology of finite monoid-like subsets of the monoid.

Theorem 7.11. Let A be as above and let A.EA. There is a spectral

sequence

•

.li!!l (p)HAq(~;k) A+=i\TA.)

}-im(p- 1 )HAq(~;k) L'A.

converging to HA (A+-A(A.);k).

p = 0,1

p ) 2

- 36 -

Proof. The theorem is just a reformulation of theorem 5.6. using the

isomorphism (2) above.

8. An Example

• ~ie want to end this paper with the computation of A for the

two-dimensional torus embedding with all multiples e.= 2. ].

0

For this purpose we need a description of the monoid-algebra A and

we give it as the invariant set of the group action of Z/(r+l) on

the free algebra k[x1 ,x2 ] given by

x1 --+ l;•x 1

x2 --+ l;•x 2

where I; is a primitive (r+1)-th root of unity (see [K]). This

gives A as the monoidalgebra k[A] where A consists of all pairs

(i,j) E z! such that i+j = 0 (mod r+1) (see [La & SL]). It has a

natural z+-grading given by

deg(i, j) = (i+j) / (r+l)

He want to use the information from chapter 7, but we need some

specific calculations. The results are listed below, each'With a

short argument.

The values of I; E ker o1 is determined by the values on the

elements (O,r+l) and (r+l,O). An explicit formula is easily

given.

b)

There are at least 2 minimal elements in A -A(A) and for these +

e'lements the value of I; E ker D 1 is zero. But the values on two

minimal elements generates all the values of 1;.

- 37 -

t2 if A is non-minimal or A=(O,r+1) or A=(r+l,O)

c) dimkHA1(A+-A(A):k) = 3 if A= (1,r) or A= (r,1) 4 if A= (a,b) with 1<a<r, b = r+1-a

Just counting of the number of linear independent vectors in 1 ker D .

From chapter 7 (Remark 7.10.) we have the long-exact sequence

0 ~ 1

HA (A+-A(A),A+:k) --+ 2 --+ HA (A+-A(A),A+:k) -4- 0

We divide into different cases depending on A and put the values of

a), b) and c) above into the sequence.

Case 1: A is non-minimal

Counting the dimensions gives

Case 2: A = (O,r+1) or A = ( r+1 , 0)

di~HA2 (A+ -A(A) ,A+:k) = 0

Case 3: A = ( 1 , r) or A = ( r, 1 )

dim HA2 (A -A(A),A :k) k + +

= 1

Case 4: A is minimal, but none of case 2 or 3

dimkHA2 (A+-A(A),A+:k) = 2

Summing up for the four cases we get

Computation 8.1.

0

For the higher cohomology groups the main tool is the spectral

sequence

- 38 -

q A But as we have seen in Proposition 7.9. HA (A;k)=O for A>>O and

AEU. Now for A>>O AEA+-A(A) implies AEU and therefore

for A> >0 and AEA+- A (A). The spectral sequence

degenerates and what is left are the groups with q=1 . Ttn~s we have

the equation

Ep-1,1 = 2

and we can use theorem 7.11. which

strong links L'A and the,groups

suggest a closer look

lim(p-l)HA 1 (~;k). -L'A

at the

To do this we introduce some new subsets of the monoid. Let

2 (a,b) E Z+, and define

1\ (a,b) = {A= (i,j) E Ala<i, b<j}

Lemma 8.2. Let A be as above and let (a,b) E z2 . Suppose +

a+b) 2(r+1) + 1 and ab * 0. Then we have

1 A di~HA ((a, b) ;k) = 2

If a=O or b=O the equality is

1 A di~HA ((a, b) ;k) =

Proof. The second case is the linear ordered set and the cohomology

is easily calculated. In the first case suppose temporarily that

a+b = 2(r+1) + 1. Then (a-1,b) and (a,b-1) are elements of the

monoid A. Let s and t be the number of minimal elements beneath A

the two maximal elements in the set (a,b) with s>t. A carefull

look at the map o 1 and the group c0 ((a,b)",o) shows that ker o1

consist of s+2 variables and (s+t+l)/2 = s (since s=t+l)

independent relations.

- 39 -

1 A 2 It follows that HA ((a,b) :k) = k. A consequence of this is tll9r ~}'e

1 - i i!

value of ~E ker D is determined by the value on two arbitra~X

minimal elements. If we concider a more general subset of the monoid, A 1

say ~. elements ~ E ker D are still determined by the values on

the minimal elements. Looking at subsets of the type described above,

it is easily seen that the value on two minimal elements will

determine everything.

From chapter 7 we have the equality

1 A 1 A ~im HA (-:k) = HA (~;k)

A ~

0

In addition the cofinality of the one-element top set of implies the

vanishing

of the higher derivatives. A repeated use of a degenerated version of

the Mayer-Vietori& sequence of Prop. 7. 1 • (In fact the excision

theorem) together with the fact that lim(p)F = 0 whenever the L

of the ordered set (the length of the longest ordered chain)

than the exponent of derivation of the functor, gives us the

following

Proposition 8.3. Let ~EA+ be of degree p. Then we have

whenever q * p-2.

is

depth

less

0

- 40 -

Corollary 8.4. Let AEA+ be of degree p. The spectral sequence of

Th. 7.11 degenerates completely and we have

Er,q- 0 2 -

whenever r * p-2 or q:l:1

Proof. A direct consequence of prop. 8.3.

We shall use this to compute the dimension of the second algebra

cohomology group of the monoid algebra k[A].

The corollary above tells us that

and from Prop. 7.1. we have the long-exact sequence

0 ~ 1A 1A 1A

~ HA (-:k) --+ tim HA (-:k) x tim HA (-rk) A+:.A\A) r 1 r 2

. 1" . (1) 1" --? };~m HA (-rk) ~ t~( HA (-:k) --+ 0 L'A A+-A A)

Putting this into c) above we can divide into different cases:

(1) 1" (G = J:i{ HA (-rk))

A+ -A A)

A = (2r+2,0) or A = (0,2r+2) 0 + k2 + k2 + 0 + G + 0

f... (2r+1,1) A ( 1 , 2r+1 ) 0 + k2 2 + 0 = or = + k X k + k + G

A (2r,2) f... (2,2r) 0 2 k 2 X k2 + k2 + G + 0 = or = + k +

f... = (a, b) a+b = 2r+2 J(a(r 0 + k2 + k2 X k 2 + ka + G + 0

a=r+1 0 + k2 + k 2 X k2 + kr + G + 0

3<b(r 0 + k2 + k 2 X k2 + kb + G + 0

Summing up this diagram, we get

0

- 41 -

1+2+ ... +(r-1)+(r-2)+(r-2)+(r-2)+(r-1) + •.. +2+1

for the second cohomology group, which makes up to the formula.

Computation B.S.

0

Before we can make further computations we need some more specific

calculations for the monoid A= {(i,j) E z;li+j = 0 (mod r+1)}. We

list them below, each with a short argument.

d) There are r+2 minimal elements of A+' and for A minimal we

have , 1 (A ) du~HA A ;k • 1.

Obvious.

e) There are 2r+3 elements of degree 2 in A+: we denote them by

(a,b) where a+b = 2(r+1) and Q(a,b(2(r+1). Furthermore:

, 1 A d~~HA ((a, b) :k) = min

ra+21 ,b+21 ( -2- I -2- ).

Drawing a picture will be a good idea.

f) 2r~ 2 . ( fa+21 12r+4-a1)

£ m1n ~ , 2 = a=O

An easy computation based on induction on r, together with the

formula

t Q tt t(t+2) ~ a -L "2" -

a=l i (t+1)2

when t is even

when t is odd

g) There are d•r + (d+1) elements of degree d in A+, and

1 A l di~HA (A:k) = 2d•(r+1) deg=d

- 42 -

There are d•r+(d-1) elements with 1 1\

di~HA (A;k)=2, two elements

where the dimension equals one and alltogether it adds up to

2d • ( r+1 ) .

The methode we are going to use for the rest of the computations can

be described as follows:

We consider the resolving complex • 1 1\

C (L'A,HA (-;k)) for

. 1 1\ t J.m HA (- ; k ) ; L'A

with differential as described in section 1 .3. of [La 1 ].

He have seen (Prop. 8.4.) that for A of degree p the complex

is acyclic.

vle have as an abbrivation have put Cq(L'A,HA1 (~;k) = Cq(A)). We

are interested in the total dimension of the algebra cohomology and

we have the equalities .

To calculate this sum we use the additivity property of an exact

sequence and obtain the formula

for p)3.

We just outline how this alternating sum is c.alculated in the case

p=3 0

- 43 -

1. There are 3r+4 elements of degree 3 (see g) above).

If A. = (0, 3r+3) or A. = (3r+3, 0), then di~ ttr HAl (~;k) = 0

If A. = ( 1 , 3r+2) or A. = (3r+2,1), then di~ t~l HA1 (~;k) = 1

1 1\ If none of the above cases, then dimk !im HA (-;k) = 2

L'A. Which adds up the formula

I dimk lim 1 1\

6r+2 HA (-;k) = deg A.=3 L'A.

2. There are r+2 minimal elements in A+. Each minimal element is

contained in 2r+1 ordered sets of type L'A., deg A. = 3. Every

degree-2-elements is contained in r ordered sets of this type. On

the other hand the s~ of the dimensions of HA1 (~:k) with 1 2 deg A.= 2 are in e) and f) above shown to be 2(r +7r+8). Thus we

have

1 2 = (r+2)(2r+l) + r 2<r +7r+8)

3. The number of ordering relations are calculated as follows: There

are r+2 minimal elements, each are related to r+2

degree-2-elements. Each of the degree-2-elements lies in r ordered

sets of type L' A. with deg A. = 3. The conclusion must be

I dimkc1 (A.) = (r+2)(r+2)r deg A.=3

Making an alternating sum out of these numbers we have made

Computation 8.6.

Similar calculations can be made for p)4. Knowing the method in

details we drop the tedious details and just list

0

- 44 -

Computation 8.7.

a)

b)

dlmkA4 (A,k[A]) = ~ r(r-1 ) 3

dimkA5 (A,lc.[A.J) = ~ r 2 (r-1 ) 3

0

- 45 -

REFERENCES

[C&E] CARTAN, H. and ElLENBERG, S., Homological Algebra Princeton Univ. Press (1956) '

[ch] CHRISTOPHERSEN, J., Monomial curves and obstructions on cyclic quotient singularities Proc of Symp, Lambrect 1985. Lecture Notes in Math. 1273 pp. 117-133 Springer-Verlag.

[Ha] HARRISON, D.K., Commutative Algebras and Cohomology. Trans. Amer. Math. Soc. 104 (1962) pp. 191-204

[K] Kempf. G., Knudsen F., Mumford D., Saint-Donat B., Toroidal Embeddings I ,Lecture Notes in Math. 339, Springer Verlag.

[L&S] LAUDAL, O.A. and SLETSJ¢E, A.B., Bettinumbers of Monoid Algebras. Application to 2-dimensional Torus E;mbeddings. Math. Scand. 56 (1985) pp. 145-162

[La 1] LAUDAL, O.A., Formal Moduli of Algebraic Structures

Lecture Notes in Math. 754 Springer Verlag.

[LA 2] LAUDAL, O.A., Sur la theorie des limites projectives et inductives. Theorie homologique des ensembles ordonnes. Annals Sci. de l'Ecole Normal Superreure. 3 serie t. 82 (196S),pp. 241-296.

[Pij PINKHAM , H.C., Deformation of Quotient Surface sigularities

Proc. of Symp. in Pure Math. AMS vol. 30 (1977)

[Ri] RIEMENSCHNEIDER, 0., Deformationen von Quotienten­sigularitaten. Mathematische Annalen 209 (1974) pp. 211-248

(Sch 1]SCHLESSINGER, M., Rigidity of Quotient Singularities Iventiones Math. 14 (1971) pp. 17-26

[Sch&S]SCHLESSINGER, M. and STASHEFF, J., The Lie algebra structure of tangent cohomology and deformation theory J. Pure Appl. Algebra 38 (1985) no. 2-3 pp. 313-322.