presentation-IMS12345678990123456778.pdf

Betti Numbers talk given at Kolkata Conference IMS

1. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Betti Numbers of Quotients of Rings and Hilbert
Polynomials Associated to Derived Functors
By
Ganesh S. Kadu
Assistant Professor
Department of Mathematics
Savitribai Phule Pune University
Pune-411 007
Email:[email protected]
22
nd
November 2019
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

2. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Motivation
Let (A;m) be a Cohen-Macaulay local ring of Krull dimension
d1:LetMbe a nitely generatedAmodule with andIan
m-primary ideal ofA:It is well known that the numerical function
n7!ℓ(M=I
n+1
M)
is given by a polynomial for sufficiently large values ofn:
This
follows from the classical work of Hilbert, Samuel and Serre. This
polynomial is known as the Hilbert-Samuel polynomial and is
known to contain important geometric, combinatorial and
homological information. For example normalized leading
coefficient of this polynomial can be interpreted in algebraic
geometry as the intersection multiplicity of algebraic varieties.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

3. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Motivation
Let (A;m) be a Cohen-Macaulay local ring of Krull dimension
d1:LetMbe a nitely generatedAmodule with andIan
m-primary ideal ofA:It is well known that the numerical function
n7!ℓ(M=I
n+1
M)
is given by a polynomial for sufficiently large values ofn:
This
follows from the classical work of Hilbert, Samuel and Serre. This
polynomial is known as the Hilbert-Samuel polynomial and is
known to contain important geometric, combinatorial and
homological information. For example normalized leading
coefficient of this polynomial can be interpreted in algebraic
geometry as the intersection multiplicity of algebraic varieties.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

4. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Introduction
One notices that
M=I
n+1
M

=Tor
A
0(M;A=I
n+1
)
Hence it is natural to look at the the modules Tor
A
i
(M;A=I
n+1
) for
i1:We investigate the following numerical function fori1;
n7!ℓ(Tor
A
i(M;A=I
n+1
))
It is interesting to see how the lengths of the modules above grow
as a function ofn:It follows from a result of V. Kodiyalam that
this function is given by a polynomialt
A
i
(M;z) of degree at most
l(I)1 forn≫0:We mainly look at the following cases
(a)M=k(k=A=mis the residue eld ofA:)
(b)Mis maximal Cohen-MacaulayAmodule.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

5. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Introduction
One notices that
M=I
n+1
M

=Tor
A
0(M;A=I
n+1
)
Hence it is natural to look at the the modules Tor
A
i
(M;A=I
n+1
) for
i1:We investigate the following numerical function fori1;
n7!ℓ(Tor
A
i(M;A=I
n+1
))
It is interesting to see how the lengths of the modules above grow
as a function ofn:It follows from a result of V. Kodiyalam that
this function is given by a polynomialt
A
i
(M;z) of degree at most
l(I)1 forn≫0:We mainly look at the following cases
(a)M=k(k=A=mis the residue eld ofA:)
(b)Mis maximal Cohen-MacaulayAmodule.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

6. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Introduction
One notices that
M=I
n+1
M

=Tor
A
0(M;A=I
n+1
)
Hence it is natural to look at the the modules Tor
A
i
(M;A=I
n+1
) for
i1:We investigate the following numerical function fori1;
n7!ℓ(Tor
A
i(M;A=I
n+1
))
It is interesting to see how the lengths of the modules above grow
as a function ofn:It follows from a result of V. Kodiyalam that
this function is given by a polynomialt
A
i
(M;z) of degree at most
l(I)1 forn≫0:
We mainly look at the following cases
(a)M=k(k=A=mis the residue eld ofA:)
(b)Mis maximal Cohen-MacaulayAmodule.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

7. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Introduction
One notices that
M=I
n+1
M

=Tor
A
0(M;A=I
n+1
)
Hence it is natural to look at the the modules Tor
A
i
(M;A=I
n+1
) for
i1:We investigate the following numerical function fori1;
n7!ℓ(Tor
A
i(M;A=I
n+1
))
It is interesting to see how the lengths of the modules above grow
as a function ofn:It follows from a result of V. Kodiyalam that
this function is given by a polynomialt
A
i
(M;z) of degree at most
l(I)1 forn≫0:We mainly look at the following cases
(a)M=k(k=A=mis the residue eld ofA:)
(b)Mis maximal Cohen-MacaulayAmodule.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

8. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Main Theorem
It is well known for instance that whenM=kthen
dimk(Tor
A
i
(k;A=I
n
)) is thei
th
Betti number ofA=I
n
denoted by
i(A=I
n
):
Thei
th
Betti number ofA=I
n
is the rank ofi
th
free
A-module in the minimal free resolution ofA=I
n
:Thus asngrows
the Betti numbersi(A=I
n
) as a function ofnfollow a polynomial
growth. The degree of this polynomial is however not known or is
known only in very few cases.We give a large class of ideals for
which the degree of this polynomial giving Betti numbers is known.
We state one of the main theorems of this article.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

9. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Main Theorem
It is well known for instance that whenM=kthen
dimk(Tor
A
i
(k;A=I
n
)) is thei
th
Betti number ofA=I
n
denoted by
i(A=I
n
):
Thei
th
Betti number ofA=I
n
is the rank ofi
th
free
A-module in the minimal free resolution ofA=I
n
:Thus asngrows
the Betti numbersi(A=I
n
) as a function ofnfollow a polynomial
growth. The degree of this polynomial is however not known or is
known only in very few cases.
We give a large class of ideals for
which the degree of this polynomial giving Betti numbers is known.
We state one of the main theorems of this article.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

10. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Main Theorem
It is well known for instance that whenM=kthen
dimk(Tor
A
i
(k;A=I
n
)) is thei
th
Betti number ofA=I
n
denoted by
i(A=I
n
):
Thei
th
Betti number ofA=I
n
is the rank ofi
th
free
A-module in the minimal free resolution ofA=I
n
:Thus asngrows
the Betti numbersi(A=I
n
) as a function ofnfollow a polynomial
growth. The degree of this polynomial is however not known or is
known only in very few cases.We give a large class of ideals for
which the degree of this polynomial giving Betti numbers is known.
We state one of the main theorems of this article.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

11. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Main Theorem
Theorem
Let(A;m)be a Cohen-Macaulay ring of dimensiond1:LetIbe
anm-primary ideal withcurv(I
n
)>1for alln1:Then for
id1;
n7!ℓ(Tor
A
i(k;A=I
n
))
is given by a polynomialt
A
i
(k;z)of degreed1forn>>0:
The curvature ofA-moduleMis dened as
curvM= lim sup
n!1
n
√

A
n(M)
We treat idealIas anA-module and so we have curv(I)2N:We
give many examples of ideals for which curv(I
n
)>1 for alln1:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

12. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Main Theorem
Theorem
Let(A;m)be a Cohen-Macaulay ring of dimensiond1:LetIbe
anm-primary ideal withcurv(I
n
)>1for alln1:Then for
id1;
n7!ℓ(Tor
A
i(k;A=I
n
))
is given by a polynomialt
A
i
(k;z)of degreed1forn>>0:
The curvature ofA-moduleMis dened as
curvM= lim sup
n!1
n
√

A
n(M)
We treat idealIas anA-module and so we have curv(I)2N:We
give many examples of ideals for which curv(I
n
)>1 for alln1:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

13. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Examples
Let (A;m;k) be a Noetherian local ring of dimensiond1 that is
not a complete intersection. LetJbe any ideal with ht(J)>0 and
fori1;letI=m
i
J:SinceAis not complete intersection so by a
theorem of Avramov curvk>1:So we obtain curv(I
n
)>1 for all
n1:Thus there are many examples of ideals satisfying the
condition curv(I
n
)>1 for alln1:
To prove the main theorem we argue by induction on the
dimensiondofA:We rst prove the result in the case ringAhas
dimensiond= 1:We then go modulo a supercial sequence of the
idealIto reduce the dimension and prove the result by induction.
For anI-supercial sequencex=x1;x2; :::;xd1in idealIofAand
for 1ldwe dene,Al(x) =A=(x1; :::;xdl):
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

14. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Examples
Let (A;m;k) be a Noetherian local ring of dimensiond1 that is
not a complete intersection. LetJbe any ideal with ht(J)>0 and
fori1;letI=m
i
J:SinceAis not complete intersection so by a
theorem of Avramov curvk>1:So we obtain curv(I
n
)>1 for all
n1:Thus there are many examples of ideals satisfying the
condition curv(I
n
)>1 for alln1:
To prove the main theorem we argue by induction on the
dimensiondofA:We rst prove the result in the case ringAhas
dimensiond= 1:We then go modulo a supercial sequence of the
idealIto reduce the dimension and prove the result by induction.
For anI-supercial sequencex=x1;x2; :::;xd1in idealIofAand
for 1ldwe dene,Al(x) =A=(x1; :::;xdl):
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

15. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Examples
Let (A;m;k) be a Noetherian local ring of dimensiond1 that is
not a complete intersection. LetJbe any ideal with ht(J)>0 and
fori1;letI=m
i
J:SinceAis not complete intersection so by a
theorem of Avramov curvk>1:So we obtain curv(I
n
)>1 for all
n1:Thus there are many examples of ideals satisfying the
condition curv(I
n
)>1 for alln1:
To prove the main theorem we argue by induction on the
dimensiondofA:We rst prove the result in the case ringAhas
dimensiond= 1:We then go modulo a supercial sequence of the
idealIto reduce the dimension and prove the result by induction.
For anI-supercial sequencex=x1;x2; :::;xd1in idealIofAand
for 1ldwe dene,Al(x) =A=(x1; :::;xdl):
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

16. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Techniques
Following is an important ingredient in the proof of our main result:
Lemma
Let(A;m)be a Cohen-Macaulay local ring withdimA=d2:
SetA1=A1(x):LetIbe anm-primary ideal withcurv(I
n
)>1for
somen1:Then
(i)projdim
A
A1
I
n
A1
=1and
(ii)Tor
A
i
(k;
A1
I
n
A1
)̸= 0for alli1:
We now give an outline of the proof of main theorem. The proof is
by induction on the dimension of ringA:We rst prove the result
in dimension one and then use supercial sequence to reduce the
dimension.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

17. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Techniques
Following is an important ingredient in the proof of our main result:
Lemma
Let(A;m)be a Cohen-Macaulay local ring withdimA=d2:
SetA1=A1(x):LetIbe anm-primary ideal withcurv(I
n
)>1for
somen1:Then
(i)projdim
A
A1
I
n
A1
=1and
(ii)Tor
A
i
(k;
A1
I
n
A1
)̸= 0for alli1:
We now give an outline of the proof of main theorem. The proof is
by induction on the dimension of ringA:We rst prove the result
in dimension one and then use supercial sequence to reduce the
dimension.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

18. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Proof of the Main Theorem
Proof.
We already have degt
A
I;i
(k;z)d1 (V. Kodiyalam). Now let
d= 1:Since curvI
n
>1 for alln1;we getℓ(Tor
A
i
(k;A=I
n
))̸= 0
for alli1 andn1:So degt
A
i
(k;z) =d1 = 0:This proves
the result in dimensiond= 1:
Recall thatAl(x) =A=(x1; :::;xdl):
Ford2;consider the following exact sequence
0!
Al+1
I
n
Al+1
!
Al+1
I
n+1
Al+1
!
Al(y)
I
n+1
Al(y)
!0:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

19. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Proof of the Main Theorem
Proof.
We already have degt
A
I;i
(k;z)d1 (V. Kodiyalam). Now let
d= 1:Since curvI
n
>1 for alln1;we getℓ(Tor
A
i
(k;A=I
n
))̸= 0
for alli1 andn1:So degt
A
i
(k;z) =d1 = 0:This proves
the result in dimensiond= 1:Recall thatAl(x) =A=(x1; :::;xdl):
Ford2;consider the following exact sequence
0!
Al+1
I
n
Al+1
!
Al+1
I
n+1
Al+1
!
Al(y)
I
n+1
Al(y)
!0:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

20. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Proof of the Main Theorem
Proof.
We already have degt
A
I;i
(k;z)d1 (V. Kodiyalam). Now let
d= 1:Since curvI
n
>1 for alln1;we getℓ(Tor
A
i
(k;A=I
n
))̸= 0
for alli1 andn1:So degt
A
i
(k;z) =d1 = 0:This proves
the result in dimensiond= 1:Recall thatAl(x) =A=(x1; :::;xdl):
Ford2;consider the following exact sequence
0!
Al+1
I
n
Al+1
!
Al+1
I
n+1
Al+1
!
Al(y)
I
n+1
Al(y)
!0:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

21. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Proof of the Main Theorem
Proof.
So forn>>0 it follows that for allidl+ 1;
t
Al(y)
i
(k;n+ 1) =t
Al+1
i
(k;n+ 1)t
Al+1
i
(k;n):
One then observes that degt
Al
i
(k;z) =l1 for allid1:
Hence degt
Al+1
i
(k;z) =lforid1:This proves that
degt
Al
i
(k;z) =l1 for allid1:Notice that whenl=dwe
haveAd=A:So degt
A
i
(k;z) =d1 for allid1:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

22. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Proof of the Main Theorem
Proof.
So forn>>0 it follows that for allidl+ 1;
t
Al(y)
i
(k;n+ 1) =t
Al+1
i
(k;n+ 1)t
Al+1
i
(k;n):
One then observes that degt
Al
i
(k;z) =l1 for allid1:
Hence degt
Al+1
i
(k;z) =lforid1:This proves that
degt
Al
i
(k;z) =l1 for allid1:Notice that whenl=dwe
haveAd=A:So degt
A
i
(k;z) =d1 for allid1:
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

23. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Interesting Question
For any non-principalm-primary idealIin a one dimensional local
ring we have
Lemma
Let(A;m)be a Cohen-Macaulay local ring of dimensiond= 1:
SupposeIis a non-principalm-primary ideal ofA:Then
Tor
A
i
(k;A=I
n
)̸= 0for anyn1andi1:
Thus in this case
A
i
(A=I
n
) is a polynomial of degree zero i.e.

A
i
(A=I
n
) =cwherecis a positive integer.
Following is then a natural question:
Question :IfIis a non-parameter ideal ind-dimensional ringA
then is
A
i
(A=I
n
) a polynomial of degreed1?
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

24. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Interesting Question
For any non-principalm-primary idealIin a one dimensional local
ring we have
Lemma
Let(A;m)be a Cohen-Macaulay local ring of dimensiond= 1:
SupposeIis a non-principalm-primary ideal ofA:Then
Tor
A
i
(k;A=I
n
)̸= 0for anyn1andi1:
Thus in this case
A
i
(A=I
n
) is a polynomial of degree zero i.e.

A
i
(A=I
n
) =cwherecis a positive integer.
Following is then a natural question:
Question :IfIis a non-parameter ideal ind-dimensional ringA
then is
A
i
(A=I
n
) a polynomial of degreed1?
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

25. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
References
Avramov, L,Innite free resolutions. Six lectures on commutative
algebra, Birkhuser, Basel, 1998.
Kodiyalam V,Homological invariants of powers of an ideal.Proc.
Amer. Math. Soc. 118 (1993), no. 3, 757764
Tony Puthenpurakal,Hilbert-coefficients of a Cohen-Macaulay
moduleJ. Algebra 264 (2003), no. 1, 8297.
Kadu, Ganesh S.; Puthenpurakal, Tony J.Analytic deviation one
ideals and test modules, 8999, Ramanujan Math. Soc. Lect. Notes
Ser., 17,
Joseph Rotman,An introduction to Homological Algebra, Pure and
Applied Mathematics, vol. 85, Academic Press Inc., New York, 1979.
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors

26. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Betti Numbers of Quotients of Rings and Hilbert Polynomials
Thank You!
Ganesh S. Kadu Betti Numbers of Quotients of Rings and Hilbert Polynomials Associated to Derived Functors