![Page 1: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/1.jpg)
Diagonal subalgebras of bigraded algebras
Neeraj Kumar
Indian Institute of Technology Bombay
Joint work with H. Ananthnarayan and Vivek Mukundan
@ Diamond Jubilee Symposium of IIT Bombay,January 4-6, 2019
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Theme of this talk
Given a graded k-algebra, (k is a field), e.g.
I k[x1, . . . , xn]
I k[x1, . . . , xn]/(f1, . . . , fm)
I One study their (commutative) algebraic properties.
Some algebraic objects have natually bigraded structure,
I Associate to it graded algebras (diagonal subalgebras) via adiagonal functor.
I Study their (commutative) algebraic properties.
![Page 3: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/3.jpg)
Theme of this talk
Given a graded k-algebra, (k is a field), e.g.
I k[x1, . . . , xn]
I k[x1, . . . , xn]/(f1, . . . , fm)
I One study their (commutative) algebraic properties.
Some algebraic objects have natually bigraded structure,
I Associate to it graded algebras (diagonal subalgebras) via adiagonal functor.
I Study their (commutative) algebraic properties.
![Page 4: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/4.jpg)
Theme of this talk
Given a graded k-algebra, (k is a field), e.g.
I k[x1, . . . , xn]
I k[x1, . . . , xn]/(f1, . . . , fm)
I One study their (commutative) algebraic properties.
Some algebraic objects have natually bigraded structure,
I Associate to it graded algebras (diagonal subalgebras) via adiagonal functor.
I Study their (commutative) algebraic properties.
![Page 5: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/5.jpg)
Theme of this talk
Given a graded k-algebra, (k is a field), e.g.
I k[x1, . . . , xn]
I k[x1, . . . , xn]/(f1, . . . , fm)
I One study their (commutative) algebraic properties.
Some algebraic objects have natually bigraded structure,
I Associate to it graded algebras (diagonal subalgebras) via adiagonal functor.
I Study their (commutative) algebraic properties.
![Page 6: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/6.jpg)
Part I - Diagonal subalgebras
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Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 8: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/8.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 9: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/9.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 10: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/10.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1).
The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 11: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/11.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 12: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/12.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 13: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/13.jpg)
Diagonal subalgebras
i) c and e denote positive integers,
ii) ∆ = {(ci, ei) | i ∈ Z} is the (c , e)-diagonal of Z2.
iii) Let R = ⊕(i,j)∈Z2R(i,j) be a standard bigraded k-algebra,
iv) The (c , e)-diagonal subalgebra of R is R∆ =⊕
i≥0 R(ci,ei).
Example
v) S = k[x1, . . . , xn, y1, . . . , ym] is standard bigraded k-algebra withdeg xi = (1, 0) and deg yj = (0, 1). The (1, 1)-diagonal subalgebra of S is
S∆ = k[xiyj | 1 ≤ i ≤ n, 1 ≤ j ≤ m].
Main results of this talk:.
I Koszul property of the (c , e)-diagonal subalgebra R∆.
I Cohen-Macaulay property of the (c , e)-diagonal subalgebra R∆.
![Page 14: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/14.jpg)
Diagonal subalgebras of Rees algebra
i) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
ii) The Rees algebra of I is the subalgebra of k[x1, . . . , xn, t] defined as
R(I ) = k[x1, . . . , xn, f1t, . . . , fr t].
iii) R(I ) is standard bigraded, deg xi = (1, 0) and deg fj t = (0, 1)
Example
i) I = (x31 , x
32 ) ⊂ S = k[x1, x2].
ii) The Rees algebra of I is R(I ) = k[x1, x2, x31 t, x
32 t].
iii) Let 4 = (1, 1), then R(I )∆ is ?
Remark. R(I )∆ is the homogeneous co-ordinate ring of the twisted quarticcurve in the projective space P3.
![Page 15: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/15.jpg)
Diagonal subalgebras of Rees algebra
i) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
ii) The Rees algebra of I is the subalgebra of k[x1, . . . , xn, t] defined as
R(I ) = k[x1, . . . , xn, f1t, . . . , fr t].
iii) R(I ) is standard bigraded, deg xi = (1, 0) and deg fj t = (0, 1)
Example
i) I = (x31 , x
32 ) ⊂ S = k[x1, x2].
ii) The Rees algebra of I is R(I ) = k[x1, x2, x31 t, x
32 t].
iii) Let 4 = (1, 1), then R(I )∆ is ?
Remark. R(I )∆ is the homogeneous co-ordinate ring of the twisted quarticcurve in the projective space P3.
![Page 16: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/16.jpg)
Diagonal subalgebras of Rees algebra
i) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
ii) The Rees algebra of I is the subalgebra of k[x1, . . . , xn, t] defined as
R(I ) = k[x1, . . . , xn, f1t, . . . , fr t].
iii) R(I ) is standard bigraded, deg xi = (1, 0) and deg fj t = (0, 1)
Example
i) I = (x31 , x
32 ) ⊂ S = k[x1, x2].
ii) The Rees algebra of I is R(I ) = k[x1, x2, x31 t, x
32 t].
iii) Let 4 = (1, 1), then R(I )∆ is ?
Remark. R(I )∆ is the homogeneous co-ordinate ring of the twisted quarticcurve in the projective space P3.
![Page 17: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/17.jpg)
Diagonal subalgebras of Rees algebra
i) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
ii) The Rees algebra of I is the subalgebra of k[x1, . . . , xn, t] defined as
R(I ) = k[x1, . . . , xn, f1t, . . . , fr t].
iii) R(I ) is standard bigraded, deg xi = (1, 0) and deg fj t = (0, 1)
Example
i) I = (x31 , x
32 ) ⊂ S = k[x1, x2].
ii) The Rees algebra of I is R(I ) = k[x1, x2, x31 t, x
32 t].
iii) Let 4 = (1, 1), then R(I )∆ is ?
Remark. R(I )∆ is the homogeneous co-ordinate ring of the twisted quarticcurve in the projective space P3.
![Page 18: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/18.jpg)
Diagonal subalgebras of Rees algebra
i) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
ii) The Rees algebra of I is the subalgebra of k[x1, . . . , xn, t] defined as
R(I ) = k[x1, . . . , xn, f1t, . . . , fr t].
iii) R(I ) is standard bigraded, deg xi = (1, 0) and deg fj t = (0, 1)
Example
i) I = (x31 , x
32 ) ⊂ S = k[x1, x2].
ii) The Rees algebra of I is R(I ) = k[x1, x2, x31 t, x
32 t].
iii) Let 4 = (1, 1), then R(I )∆ is ?
Remark. R(I )∆ is the homogeneous co-ordinate ring of the twisted quarticcurve in the projective space P3.
![Page 19: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/19.jpg)
An example of Diagonal subalgebra of Rees algebra
Proof of Remark.
i) R(I ) = k[x1, x2, x31 t, x
32 t] ∼= k[x1, x2, y1, y2]/(x3
1 y2 − x32 y1) = B/K .
ii) R(I )∆ = B4/K4, where B4 = k[x1y1, x1y2, x2y1, x2y2] and K4 is((x1y1)
2(x1y2)− (x2y1)3, (x2y1)(x2y2)
2 − (x1y2)3, (x1y1)(x1y2)
2 − (x2y1)2(x2y2)
).
iii) Define z0 7→ x1y1, z1 7→ x1y2, z2 7→ x2y1, z3 7→ x2y2, then
k[z0, . . . , z3]
(z1z2 − z0z3)∼= k[x1y1, x1y2, x2y1, x2y2] = B4
iv) We can describe
R(I )∆ =k[z0, . . . , z3]
(z1z2 − z0z3, z31 − z2z2
3 , z32 − z2
0 z1, z0z21 − z2
2 z3).
v) The vanishing locus of a twisted quartic curve defined from P −→ P3 bythe map [x0, x1] 7→ [x4
0 , x30 x1, x
20 x
21 , x0x
31 , x
41 ] are the polynomials
z1z2 − z0z3, z31 − z2z
23 , z
32 − z2
0 z1, z0z21 − z2
2 z3.
![Page 20: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/20.jpg)
An example of Diagonal subalgebra of Rees algebra
Proof of Remark.
i) R(I ) = k[x1, x2, x31 t, x
32 t] ∼= k[x1, x2, y1, y2]/(x3
1 y2 − x32 y1) = B/K .
ii) R(I )∆ = B4/K4, where B4 = k[x1y1, x1y2, x2y1, x2y2] and K4 is((x1y1)
2(x1y2)− (x2y1)3, (x2y1)(x2y2)
2 − (x1y2)3, (x1y1)(x1y2)
2 − (x2y1)2(x2y2)
).
iii) Define z0 7→ x1y1, z1 7→ x1y2, z2 7→ x2y1, z3 7→ x2y2, then
k[z0, . . . , z3]
(z1z2 − z0z3)∼= k[x1y1, x1y2, x2y1, x2y2] = B4
iv) We can describe
R(I )∆ =k[z0, . . . , z3]
(z1z2 − z0z3, z31 − z2z2
3 , z32 − z2
0 z1, z0z21 − z2
2 z3).
v) The vanishing locus of a twisted quartic curve defined from P −→ P3 bythe map [x0, x1] 7→ [x4
0 , x30 x1, x
20 x
21 , x0x
31 , x
41 ] are the polynomials
z1z2 − z0z3, z31 − z2z
23 , z
32 − z2
0 z1, z0z21 − z2
2 z3.
![Page 21: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/21.jpg)
An example of Diagonal subalgebra of Rees algebra
Proof of Remark.
i) R(I ) = k[x1, x2, x31 t, x
32 t] ∼= k[x1, x2, y1, y2]/(x3
1 y2 − x32 y1) = B/K .
ii) R(I )∆ = B4/K4, where B4 = k[x1y1, x1y2, x2y1, x2y2] and K4 is((x1y1)
2(x1y2)− (x2y1)3, (x2y1)(x2y2)
2 − (x1y2)3, (x1y1)(x1y2)
2 − (x2y1)2(x2y2)
).
iii) Define z0 7→ x1y1, z1 7→ x1y2, z2 7→ x2y1, z3 7→ x2y2, then
k[z0, . . . , z3]
(z1z2 − z0z3)∼= k[x1y1, x1y2, x2y1, x2y2] = B4
iv) We can describe
R(I )∆ =k[z0, . . . , z3]
(z1z2 − z0z3, z31 − z2z2
3 , z32 − z2
0 z1, z0z21 − z2
2 z3).
v) The vanishing locus of a twisted quartic curve defined from P −→ P3 bythe map [x0, x1] 7→ [x4
0 , x30 x1, x
20 x
21 , x0x
31 , x
41 ] are the polynomials
z1z2 − z0z3, z31 − z2z
23 , z
32 − z2
0 z1, z0z21 − z2
2 z3.
![Page 22: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/22.jpg)
More on diagonal subalgebras
i) R is a (standard) graded commutative ring with 1.
ii) The Symmetric algebra of Rm is Sym(Rm) = R[t1, . . . , tm].
iii) Sym(Rm) is standard bigraded.
iv) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
v) The Symmetric algebra of I is
Sym(I ) =S [t1, . . . , tr ]
Jφ, where Jφ =
( [t1 t2 . . . tr
]· φ)
and φ is the presentation matrix of I , i.e., Sp φ−→ S r −→ I −→ 0.
vi) Sym(Rm)∆ and Sym(I )∆ are diagonal subalgebras.
![Page 23: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/23.jpg)
More on diagonal subalgebras
i) R is a (standard) graded commutative ring with 1.
ii) The Symmetric algebra of Rm is Sym(Rm) = R[t1, . . . , tm].
iii) Sym(Rm) is standard bigraded.
iv) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
v) The Symmetric algebra of I is
Sym(I ) =S [t1, . . . , tr ]
Jφ, where Jφ =
( [t1 t2 . . . tr
]· φ)
and φ is the presentation matrix of I , i.e., Sp φ−→ S r −→ I −→ 0.
vi) Sym(Rm)∆ and Sym(I )∆ are diagonal subalgebras.
![Page 24: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/24.jpg)
More on diagonal subalgebras
i) R is a (standard) graded commutative ring with 1.
ii) The Symmetric algebra of Rm is Sym(Rm) = R[t1, . . . , tm].
iii) Sym(Rm) is standard bigraded.
iv) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
v) The Symmetric algebra of I is
Sym(I ) =S [t1, . . . , tr ]
Jφ, where Jφ =
( [t1 t2 . . . tr
]· φ)
and φ is the presentation matrix of I , i.e., Sp φ−→ S r −→ I −→ 0.
vi) Sym(Rm)∆ and Sym(I )∆ are diagonal subalgebras.
![Page 25: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/25.jpg)
More on diagonal subalgebras
i) R is a (standard) graded commutative ring with 1.
ii) The Symmetric algebra of Rm is Sym(Rm) = R[t1, . . . , tm].
iii) Sym(Rm) is standard bigraded.
iv) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
v) The Symmetric algebra of I is
Sym(I ) =S [t1, . . . , tr ]
Jφ, where Jφ =
( [t1 t2 . . . tr
]· φ)
and φ is the presentation matrix of I , i.e., Sp φ−→ S r −→ I −→ 0.
vi) Sym(Rm)∆ and Sym(I )∆ are diagonal subalgebras.
![Page 26: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/26.jpg)
More on diagonal subalgebras
i) R is a (standard) graded commutative ring with 1.
ii) The Symmetric algebra of Rm is Sym(Rm) = R[t1, . . . , tm].
iii) Sym(Rm) is standard bigraded.
iv) S = k[x1, . . . , xn], and I = 〈f1, . . . , fr 〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree.
v) The Symmetric algebra of I is
Sym(I ) =S [t1, . . . , tr ]
Jφ, where Jφ =
( [t1 t2 . . . tr
]· φ)
and φ is the presentation matrix of I , i.e., Sp φ−→ S r −→ I −→ 0.
vi) Sym(Rm)∆ and Sym(I )∆ are diagonal subalgebras.
![Page 27: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/27.jpg)
Part II - Cohen-Macaulay and Koszul properties
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Linear Algebra and Commutative Algebra
Finitely generated (f.g.) module over a ring ←→ f.g. vector space over a ring.
Usually f.g. modules have generators and relations among them instead ofhaving basis as in Linear algebra.
Let R be any commutative ring with 1. Then Rn is a vector space of rank nover R with basis e1, e2, . . . , en. We say Rn is a free R-module. Moduleshaving basis are rare.
Example. Let R = k[x , y ]/(x2, y3). Then R is a module over k[x , y ] withgenerators x , y and relations x2 = 0, y3 = 0 and fx2 + gy3 = 0 for some f , g in
k[x , y ]. (For instance, take f = y3 and g = −x2).
Free modules are most fundamental (in other words, most elementary) objectsin the theory of modules in commutative algebra.
![Page 29: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/29.jpg)
Linear Algebra and Commutative Algebra
Finitely generated (f.g.) module over a ring ←→ f.g. vector space over a ring.
Usually f.g. modules have generators and relations among them instead ofhaving basis as in Linear algebra.
Let R be any commutative ring with 1. Then Rn is a vector space of rank nover R with basis e1, e2, . . . , en. We say Rn is a free R-module. Moduleshaving basis are rare.
Example. Let R = k[x , y ]/(x2, y3). Then R is a module over k[x , y ] withgenerators x , y and relations x2 = 0, y3 = 0 and fx2 + gy3 = 0 for some f , g in
k[x , y ]. (For instance, take f = y3 and g = −x2).
Free modules are most fundamental (in other words, most elementary) objectsin the theory of modules in commutative algebra.
![Page 30: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/30.jpg)
Linear Algebra and Commutative Algebra
Finitely generated (f.g.) module over a ring ←→ f.g. vector space over a ring.
Usually f.g. modules have generators and relations among them instead ofhaving basis as in Linear algebra.
Let R be any commutative ring with 1. Then Rn is a vector space of rank nover R with basis e1, e2, . . . , en. We say Rn is a free R-module. Moduleshaving basis are rare.
Example. Let R = k[x , y ]/(x2, y3). Then R is a module over k[x , y ] withgenerators x , y and relations x2 = 0, y3 = 0 and fx2 + gy3 = 0 for some f , g in
k[x , y ]. (For instance, take f = y3 and g = −x2).
Free modules are most fundamental (in other words, most elementary) objectsin the theory of modules in commutative algebra.
![Page 31: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/31.jpg)
Free resolution
I Let R = k[x1, . . . , xn] be a polynomial ring and M a f.g. R-module.
D. Hilbert (1890): Used generators of module M and relations among them todescribe the properties of module M, by approximating it with free R-modules.This process of approximation by free module is known as free resolution.
Free resolution of M over R:
· · · → Rβ2
∂2=
(relations onthe relations
in ∂1
)−−−−−−−−−−−−→ Rβ1
∂1=
(relations on
the generatorsof M
)−−−−−−−−−−−−−→ R#M
(generators
of M
)−−−−−−−→ M → 0,
where β1 is the minimal number of relations on the generators of M, and so on.
I Poincare-Betti series of M is PRM(t) =
∑i≥0 βi · ti
(Assuming that free resolution is minimal).
I Hilbert series of M is HRM(t) =
∑i≥0 dimR Mi · ti.
![Page 32: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/32.jpg)
Free resolution
I Let R = k[x1, . . . , xn] be a polynomial ring and M a f.g. R-module.
D. Hilbert (1890): Used generators of module M and relations among them todescribe the properties of module M, by approximating it with free R-modules.This process of approximation by free module is known as free resolution.
Free resolution of M over R:
· · · → Rβ2
∂2=
(relations onthe relations
in ∂1
)−−−−−−−−−−−−→ Rβ1
∂1=
(relations on
the generatorsof M
)−−−−−−−−−−−−−→ R#M
(generators
of M
)−−−−−−−→ M → 0,
where β1 is the minimal number of relations on the generators of M, and so on.
I Poincare-Betti series of M is PRM(t) =
∑i≥0 βi · ti
(Assuming that free resolution is minimal).
I Hilbert series of M is HRM(t) =
∑i≥0 dimR Mi · ti.
![Page 33: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/33.jpg)
Free resolution
I Let R = k[x1, . . . , xn] be a polynomial ring and M a f.g. R-module.
D. Hilbert (1890): Used generators of module M and relations among them todescribe the properties of module M, by approximating it with free R-modules.This process of approximation by free module is known as free resolution.
Free resolution of M over R:
· · · → Rβ2
∂2=
(relations onthe relations
in ∂1
)−−−−−−−−−−−−→ Rβ1
∂1=
(relations on
the generatorsof M
)−−−−−−−−−−−−−→ R#M
(generators
of M
)−−−−−−−→ M → 0,
where β1 is the minimal number of relations on the generators of M, and so on.
I Poincare-Betti series of M is PRM(t) =
∑i≥0 βi · ti
(Assuming that free resolution is minimal).
I Hilbert series of M is HRM(t) =
∑i≥0 dimR Mi · ti.
![Page 34: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/34.jpg)
Koszul algebras
I R = ⊕i≥0Ri is a graded ring with R0 = k (e.g. R = S/I ).
I Let mR = ⊕i≥1Ri be the unique homogeneous maximal ideal of R.
I R is standard graded if mR is generated by R1.
Let F be the minimal graded free resolution of k = R/mR over R:
F · · · → Rβi∂i−→ Rβi−1 → · · · ∂2−→ Rβ1 ∂1−→ Rβ0 → 0,
Let Xi be the matrix corresponding to the map ∂i.
Priddy (1970): R is Koszul if all the non-zero entries of Xi are in degree one,
Xi =
? 0 · · · ?? ? · · · 0...
... · · ·...
0 ? · · · ?
.
Using Mumford (1966) definition of regularity, R is Koszul if
regR(k) := supi∈N{j − i : dimk Tor
Ri (k, k)j 6= 0} = 0.
Lofwall (1983): R is Koszul if and only if HR(t) · PRk (−t) = 1.
![Page 35: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/35.jpg)
Koszul algebras
I R = ⊕i≥0Ri is a graded ring with R0 = k (e.g. R = S/I ).
I Let mR = ⊕i≥1Ri be the unique homogeneous maximal ideal of R.
I R is standard graded if mR is generated by R1.
Let F be the minimal graded free resolution of k = R/mR over R:
F · · · → Rβi∂i−→ Rβi−1 → · · · ∂2−→ Rβ1 ∂1−→ Rβ0 → 0,
Let Xi be the matrix corresponding to the map ∂i.
Priddy (1970): R is Koszul if all the non-zero entries of Xi are in degree one,
Xi =
? 0 · · · ?? ? · · · 0...
... · · ·...
0 ? · · · ?
.
Using Mumford (1966) definition of regularity, R is Koszul if
regR(k) := supi∈N{j − i : dimk Tor
Ri (k, k)j 6= 0} = 0.
Lofwall (1983): R is Koszul if and only if HR(t) · PRk (−t) = 1.
![Page 36: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/36.jpg)
Koszul algebras
I R = ⊕i≥0Ri is a graded ring with R0 = k (e.g. R = S/I ).
I Let mR = ⊕i≥1Ri be the unique homogeneous maximal ideal of R.
I R is standard graded if mR is generated by R1.
Let F be the minimal graded free resolution of k = R/mR over R:
F · · · → Rβi∂i−→ Rβi−1 → · · · ∂2−→ Rβ1 ∂1−→ Rβ0 → 0,
Let Xi be the matrix corresponding to the map ∂i.
Priddy (1970): R is Koszul if all the non-zero entries of Xi are in degree one,
Xi =
? 0 · · · ?? ? · · · 0...
... · · ·...
0 ? · · · ?
.
Using Mumford (1966) definition of regularity, R is Koszul if
regR(k) := supi∈N{j − i : dimk Tor
Ri (k, k)j 6= 0} = 0.
Lofwall (1983): R is Koszul if and only if HR(t) · PRk (−t) = 1.
![Page 37: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/37.jpg)
Koszul algebras
I R = ⊕i≥0Ri is a graded ring with R0 = k (e.g. R = S/I ).
I Let mR = ⊕i≥1Ri be the unique homogeneous maximal ideal of R.
I R is standard graded if mR is generated by R1.
Let F be the minimal graded free resolution of k = R/mR over R:
F · · · → Rβi∂i−→ Rβi−1 → · · · ∂2−→ Rβ1 ∂1−→ Rβ0 → 0,
Let Xi be the matrix corresponding to the map ∂i.
Priddy (1970): R is Koszul if all the non-zero entries of Xi are in degree one,
Xi =
? 0 · · · ?? ? · · · 0...
... · · ·...
0 ? · · · ?
.
Using Mumford (1966) definition of regularity, R is Koszul if
regR(k) := supi∈N{j − i : dimk Tor
Ri (k, k)j 6= 0} = 0.
Lofwall (1983): R is Koszul if and only if HR(t) · PRk (−t) = 1.
![Page 38: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/38.jpg)
Few definitions and facts
I S = k[x1, . . . , xn] be a polynomial ring over a field k.
I I = (f1, f2, . . . , fr ) ⊂ S , where fi ’s are homogeneous polynomials.
I R = S/I is Quadratic if deg(fi ) = 2 for all i .
I R = S/I is defined by Grobner basis of quadrics, if w.r.t some coordinatesystem of S1 and some term order τ on S , I can be deformed to an idealgenerated by monomials of degree 2 without changing the Hilbert seriesof R.
I I = (f1, f2, . . . , fr ) ⊂ S is generated by regular sequence if for each i , theimage of fi+1 is a nonzero divisor in S/(f1, . . . , fi ).
I R = S/I is complete interection if I is generated by regular sequence.
I We say R is Cohen-Macaulay if the length of maximal regular sequence inR is same as the dimension of R, i.e., dimR = depthR.
![Page 39: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/39.jpg)
Few definitions and facts
I S = k[x1, . . . , xn] be a polynomial ring over a field k.
I I = (f1, f2, . . . , fr ) ⊂ S , where fi ’s are homogeneous polynomials.
I R = S/I is Quadratic if deg(fi ) = 2 for all i .
I R = S/I is defined by Grobner basis of quadrics, if w.r.t some coordinatesystem of S1 and some term order τ on S , I can be deformed to an idealgenerated by monomials of degree 2 without changing the Hilbert seriesof R.
I I = (f1, f2, . . . , fr ) ⊂ S is generated by regular sequence if for each i , theimage of fi+1 is a nonzero divisor in S/(f1, . . . , fi ).
I R = S/I is complete interection if I is generated by regular sequence.
I We say R is Cohen-Macaulay if the length of maximal regular sequence inR is same as the dimension of R, i.e., dimR = depthR.
![Page 40: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/40.jpg)
Few definitions and facts
I S = k[x1, . . . , xn] be a polynomial ring over a field k.
I I = (f1, f2, . . . , fr ) ⊂ S , where fi ’s are homogeneous polynomials.
I R = S/I is Quadratic if deg(fi ) = 2 for all i .
I R = S/I is defined by Grobner basis of quadrics, if w.r.t some coordinatesystem of S1 and some term order τ on S , I can be deformed to an idealgenerated by monomials of degree 2 without changing the Hilbert seriesof R.
I I = (f1, f2, . . . , fr ) ⊂ S is generated by regular sequence if for each i , theimage of fi+1 is a nonzero divisor in S/(f1, . . . , fi ).
I R = S/I is complete interection if I is generated by regular sequence.
I We say R is Cohen-Macaulay if the length of maximal regular sequence inR is same as the dimension of R, i.e., dimR = depthR.
![Page 41: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/41.jpg)
Few definitions and facts
I S = k[x1, . . . , xn] be a polynomial ring over a field k.
I I = (f1, f2, . . . , fr ) ⊂ S , where fi ’s are homogeneous polynomials.
I R = S/I is Quadratic if deg(fi ) = 2 for all i .
I R = S/I is defined by Grobner basis of quadrics, if w.r.t some coordinatesystem of S1 and some term order τ on S , I can be deformed to an idealgenerated by monomials of degree 2 without changing the Hilbert seriesof R.
I I = (f1, f2, . . . , fr ) ⊂ S is generated by regular sequence if for each i , theimage of fi+1 is a nonzero divisor in S/(f1, . . . , fi ).
I R = S/I is complete interection if I is generated by regular sequence.
I We say R is Cohen-Macaulay if the length of maximal regular sequence inR is same as the dimension of R, i.e., dimR = depthR.
![Page 42: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/42.jpg)
Motivation for studying Koszul algebras
i) Mumford (1969) - Any projective variety X ⊂ Pn, its Veronese embeddingvd0 (X ) ⊂ PN0 is cut out by quadrics, for d0 � 0.
ii) Backlin (1986) - If R is any standard graded k-algebra, then Veronese ringR(d1) is even Koszul, for d1 ≥ d0 � 0.
iii) Eisenbud, Reeves and Totaro (1994) - Sufficiently high Veronese ring R(d2)
is defined by Grobner basis of quadrics, for d2 ≥ d1 � 0.
Grobner basis of quadrics ⇒ Koszul ⇒ Quadratic
⇑c.i. quadratic
(a) Kaplansky (1957), Serre (1965) - Whether Poincare-Betti series Pk(R) isrational? (Motivated by Tate’s (1957) construction of free resolution).
(b) Froberg (1975) - Poincare-Betti series is rational for Koszul algebras.
(c) Anick (1980), Roos and Sturmfels (1998) - Poincare series can be irrational.
![Page 43: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/43.jpg)
Motivation for studying Koszul algebras
i) Mumford (1969) - Any projective variety X ⊂ Pn, its Veronese embeddingvd0 (X ) ⊂ PN0 is cut out by quadrics, for d0 � 0.
ii) Backlin (1986) - If R is any standard graded k-algebra, then Veronese ringR(d1) is even Koszul, for d1 ≥ d0 � 0.
iii) Eisenbud, Reeves and Totaro (1994) - Sufficiently high Veronese ring R(d2)
is defined by Grobner basis of quadrics, for d2 ≥ d1 � 0.
Grobner basis of quadrics ⇒ Koszul ⇒ Quadratic
⇑c.i. quadratic
(a) Kaplansky (1957), Serre (1965) - Whether Poincare-Betti series Pk(R) isrational? (Motivated by Tate’s (1957) construction of free resolution).
(b) Froberg (1975) - Poincare-Betti series is rational for Koszul algebras.
(c) Anick (1980), Roos and Sturmfels (1998) - Poincare series can be irrational.
![Page 44: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/44.jpg)
Motivation for studying Koszul algebras
i) Mumford (1969) - Any projective variety X ⊂ Pn, its Veronese embeddingvd0 (X ) ⊂ PN0 is cut out by quadrics, for d0 � 0.
ii) Backlin (1986) - If R is any standard graded k-algebra, then Veronese ringR(d1) is even Koszul, for d1 ≥ d0 � 0.
iii) Eisenbud, Reeves and Totaro (1994) - Sufficiently high Veronese ring R(d2)
is defined by Grobner basis of quadrics, for d2 ≥ d1 � 0.
Grobner basis of quadrics ⇒ Koszul ⇒ Quadratic
⇑c.i. quadratic
(a) Kaplansky (1957), Serre (1965) - Whether Poincare-Betti series Pk(R) isrational? (Motivated by Tate’s (1957) construction of free resolution).
(b) Froberg (1975) - Poincare-Betti series is rational for Koszul algebras.
(c) Anick (1980), Roos and Sturmfels (1998) - Poincare series can be irrational.
![Page 45: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/45.jpg)
Motivation for studying Koszul algebras
i) Mumford (1969) - Any projective variety X ⊂ Pn, its Veronese embeddingvd0 (X ) ⊂ PN0 is cut out by quadrics, for d0 � 0.
ii) Backlin (1986) - If R is any standard graded k-algebra, then Veronese ringR(d1) is even Koszul, for d1 ≥ d0 � 0.
iii) Eisenbud, Reeves and Totaro (1994) - Sufficiently high Veronese ring R(d2)
is defined by Grobner basis of quadrics, for d2 ≥ d1 � 0.
Grobner basis of quadrics ⇒ Koszul ⇒ Quadratic
⇑c.i. quadratic
(a) Kaplansky (1957), Serre (1965) - Whether Poincare-Betti series Pk(R) isrational? (Motivated by Tate’s (1957) construction of free resolution).
(b) Froberg (1975) - Poincare-Betti series is rational for Koszul algebras.
(c) Anick (1980), Roos and Sturmfels (1998) - Poincare series can be irrational.
![Page 46: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/46.jpg)
Motivation for studying Koszul algebras
i) Mumford (1969) - Any projective variety X ⊂ Pn, its Veronese embeddingvd0 (X ) ⊂ PN0 is cut out by quadrics, for d0 � 0.
ii) Backlin (1986) - If R is any standard graded k-algebra, then Veronese ringR(d1) is even Koszul, for d1 ≥ d0 � 0.
iii) Eisenbud, Reeves and Totaro (1994) - Sufficiently high Veronese ring R(d2)
is defined by Grobner basis of quadrics, for d2 ≥ d1 � 0.
Grobner basis of quadrics ⇒ Koszul ⇒ Quadratic
⇑c.i. quadratic
(a) Kaplansky (1957), Serre (1965) - Whether Poincare-Betti series Pk(R) isrational? (Motivated by Tate’s (1957) construction of free resolution).
(b) Froberg (1975) - Poincare-Betti series is rational for Koszul algebras.
(c) Anick (1980), Roos and Sturmfels (1998) - Poincare series can be irrational.
![Page 47: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/47.jpg)
Results
i) S = k[x1, . . . , xn], and I = 〈f1, f2, f3〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree d .
ii) The Rees algebra of I is R(I ) = k[x1, . . . , xn, f1t, f2t, f3t].
iii) Let ∆ = (c , e).
Theorem (Kumar (2014)). R(I )∆ is Koszul if c ≥ d2 and e > 0.
Known results
Theorem (Conca, Herzog, Trung, and Valla (1997)). R(I )∆ is Koszul if c ≥ 2d3
and e > 0.Theorem (Caviglia (2009)). Let ∆ = (1, 1) and I = 〈x2
1 , x22 , x
23 〉, then R(I )∆ is
Koszul.Theorem (Conca, Caviglia (2013)). Let ∆ = (1, 1) and I = 〈f1, f2, f3〉 withdeg fi = 2 , then R(I )∆ is Koszul.
![Page 48: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/48.jpg)
Results
i) S = k[x1, . . . , xn], and I = 〈f1, f2, f3〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree d .
ii) The Rees algebra of I is R(I ) = k[x1, . . . , xn, f1t, f2t, f3t].
iii) Let ∆ = (c , e).
Theorem (Kumar (2014)). R(I )∆ is Koszul if c ≥ d2 and e > 0.
Known results
Theorem (Conca, Herzog, Trung, and Valla (1997)). R(I )∆ is Koszul if c ≥ 2d3
and e > 0.Theorem (Caviglia (2009)). Let ∆ = (1, 1) and I = 〈x2
1 , x22 , x
23 〉, then R(I )∆ is
Koszul.Theorem (Conca, Caviglia (2013)). Let ∆ = (1, 1) and I = 〈f1, f2, f3〉 withdeg fi = 2 , then R(I )∆ is Koszul.
![Page 49: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/49.jpg)
Results
i) S = k[x1, . . . , xn], and I = 〈f1, f2, f3〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree d .
ii) The Rees algebra of I is R(I ) = k[x1, . . . , xn, f1t, f2t, f3t].
iii) Let ∆ = (c , e).
Theorem (Kumar (2014)). R(I )∆ is Koszul if c ≥ d2 and e > 0.
Known results
Theorem (Conca, Herzog, Trung, and Valla (1997)). R(I )∆ is Koszul if c ≥ 2d3
and e > 0.
Theorem (Caviglia (2009)). Let ∆ = (1, 1) and I = 〈x21 , x
22 , x
23 〉, then R(I )∆ is
Koszul.Theorem (Conca, Caviglia (2013)). Let ∆ = (1, 1) and I = 〈f1, f2, f3〉 withdeg fi = 2 , then R(I )∆ is Koszul.
![Page 50: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/50.jpg)
Results
i) S = k[x1, . . . , xn], and I = 〈f1, f2, f3〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree d .
ii) The Rees algebra of I is R(I ) = k[x1, . . . , xn, f1t, f2t, f3t].
iii) Let ∆ = (c , e).
Theorem (Kumar (2014)). R(I )∆ is Koszul if c ≥ d2 and e > 0.
Known results
Theorem (Conca, Herzog, Trung, and Valla (1997)). R(I )∆ is Koszul if c ≥ 2d3
and e > 0.Theorem (Caviglia (2009)). Let ∆ = (1, 1) and I = 〈x2
1 , x22 , x
23 〉, then R(I )∆ is
Koszul.
Theorem (Conca, Caviglia (2013)). Let ∆ = (1, 1) and I = 〈f1, f2, f3〉 withdeg fi = 2 , then R(I )∆ is Koszul.
![Page 51: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/51.jpg)
Results
i) S = k[x1, . . . , xn], and I = 〈f1, f2, f3〉 an ideal of S generated byhomogeneous (homog.) polynomials of same degree d .
ii) The Rees algebra of I is R(I ) = k[x1, . . . , xn, f1t, f2t, f3t].
iii) Let ∆ = (c , e).
Theorem (Kumar (2014)). R(I )∆ is Koszul if c ≥ d2 and e > 0.
Known results
Theorem (Conca, Herzog, Trung, and Valla (1997)). R(I )∆ is Koszul if c ≥ 2d3
and e > 0.Theorem (Caviglia (2009)). Let ∆ = (1, 1) and I = 〈x2
1 , x22 , x
23 〉, then R(I )∆ is
Koszul.Theorem (Conca, Caviglia (2013)). Let ∆ = (1, 1) and I = 〈f1, f2, f3〉 withdeg fi = 2 , then R(I )∆ is Koszul.
![Page 52: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/52.jpg)
Residual Intersections
1) Artin and Nagata (1972): An ideal J of Noetherian ring R is an s-residualintersection of I if there exists an ideal K = (g1, . . . , gs) ⊂ I such thatJ = K : I and ht J ≥ s.
2) Huneke and Ulrich (1982): An ideal J ⊂ R is a geometric s-residualintersection of I if J is an s-residual intersection of I and ht(I + J) ≥ s + 1.
Bruns, Kustin and Miller (1990):
I Let φ be an n ×m matrix (m ≥ n) with linear entries in y1, . . . , yp.
I Let R = S [y1, y2, . . . , yp] be a polynomial ring over S = k[x1, . . . , xn], andmS = (x1, . . . , xn).
I R is bigraded with deg(xi ) = (1, 0), deg(yj) = (0, 1).
I Consider an ideal J = (z1, . . . zm) + In(φ), where[z1 z2 . . . zm
]=[x1 x2 . . . xn
]· φ
and In(φ) is the ideal generated by maximal minors of φ.
Then J ⊂ R is geometric m-residual intersection of mS , if ht(J) ≥ m, and
ht(In(φ)) ≥ m − n + 1.
![Page 53: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/53.jpg)
Residual Intersections
1) Artin and Nagata (1972): An ideal J of Noetherian ring R is an s-residualintersection of I if there exists an ideal K = (g1, . . . , gs) ⊂ I such thatJ = K : I and ht J ≥ s.
2) Huneke and Ulrich (1982): An ideal J ⊂ R is a geometric s-residualintersection of I if J is an s-residual intersection of I and ht(I + J) ≥ s + 1.
Bruns, Kustin and Miller (1990):
I Let φ be an n ×m matrix (m ≥ n) with linear entries in y1, . . . , yp.
I Let R = S [y1, y2, . . . , yp] be a polynomial ring over S = k[x1, . . . , xn], andmS = (x1, . . . , xn).
I R is bigraded with deg(xi ) = (1, 0), deg(yj) = (0, 1).
I Consider an ideal J = (z1, . . . zm) + In(φ), where[z1 z2 . . . zm
]=[x1 x2 . . . xn
]· φ
and In(φ) is the ideal generated by maximal minors of φ.
Then J ⊂ R is geometric m-residual intersection of mS , if ht(J) ≥ m, and
ht(In(φ)) ≥ m − n + 1.
![Page 54: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/54.jpg)
Residual Intersections
1) Artin and Nagata (1972): An ideal J of Noetherian ring R is an s-residualintersection of I if there exists an ideal K = (g1, . . . , gs) ⊂ I such thatJ = K : I and ht J ≥ s.
2) Huneke and Ulrich (1982): An ideal J ⊂ R is a geometric s-residualintersection of I if J is an s-residual intersection of I and ht(I + J) ≥ s + 1.
Bruns, Kustin and Miller (1990):
I Let φ be an n ×m matrix (m ≥ n) with linear entries in y1, . . . , yp.
I Let R = S [y1, y2, . . . , yp] be a polynomial ring over S = k[x1, . . . , xn], andmS = (x1, . . . , xn).
I R is bigraded with deg(xi ) = (1, 0), deg(yj) = (0, 1).
I Consider an ideal J = (z1, . . . zm) + In(φ), where[z1 z2 . . . zm
]=[x1 x2 . . . xn
]· φ
and In(φ) is the ideal generated by maximal minors of φ.
Then J ⊂ R is geometric m-residual intersection of mS , if ht(J) ≥ m, and
ht(In(φ)) ≥ m − n + 1.
![Page 55: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/55.jpg)
Residual Intersections
1) Artin and Nagata (1972): An ideal J of Noetherian ring R is an s-residualintersection of I if there exists an ideal K = (g1, . . . , gs) ⊂ I such thatJ = K : I and ht J ≥ s.
2) Huneke and Ulrich (1982): An ideal J ⊂ R is a geometric s-residualintersection of I if J is an s-residual intersection of I and ht(I + J) ≥ s + 1.
Bruns, Kustin and Miller (1990):
I Let φ be an n ×m matrix (m ≥ n) with linear entries in y1, . . . , yp.
I Let R = S [y1, y2, . . . , yp] be a polynomial ring over S = k[x1, . . . , xn], andmS = (x1, . . . , xn).
I R is bigraded with deg(xi ) = (1, 0), deg(yj) = (0, 1).
I Consider an ideal J = (z1, . . . zm) + In(φ), where[z1 z2 . . . zm
]=[x1 x2 . . . xn
]· φ
and In(φ) is the ideal generated by maximal minors of φ.
Then J ⊂ R is geometric m-residual intersection of mS , if ht(J) ≥ m, and
ht(In(φ)) ≥ m − n + 1.
![Page 56: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/56.jpg)
Diagonal Subalgebras of Residual Intersections
Ananthnarayan, Kumar and Mukundan (2018):
We describe the bigraded resolution and compute the homological invariantsdepth and regularity of modules in the bigraded resolution of geometricm-residual intersection ring R/J:
i) (R/J)∆ is Koszul for all c ≥ 1 and e ≥ n2 .
ii) Suppose p > m, then depth(R/J)∆ ≥ p + n − (m + 1) for all ∆.
iii) If p = m + 1, then (R/J)∆ is Cohen-Macaulay for all ∆.
Conca, Herzog, Trung, and Valla (1997): If p = m + 1, then (R/J)∆ isCohen-Macaulay for large ∆.
![Page 57: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/57.jpg)
Diagonal Subalgebras of Residual Intersections
Ananthnarayan, Kumar and Mukundan (2018):
We describe the bigraded resolution and compute the homological invariantsdepth and regularity of modules in the bigraded resolution of geometricm-residual intersection ring R/J:
i) (R/J)∆ is Koszul for all c ≥ 1 and e ≥ n2 .
ii) Suppose p > m, then depth(R/J)∆ ≥ p + n − (m + 1) for all ∆.
iii) If p = m + 1, then (R/J)∆ is Cohen-Macaulay for all ∆.
Conca, Herzog, Trung, and Valla (1997): If p = m + 1, then (R/J)∆ isCohen-Macaulay for large ∆.
![Page 58: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/58.jpg)
Diagonal Subalgebras of Residual Intersections
Ananthnarayan, Kumar and Mukundan (2018):
We describe the bigraded resolution and compute the homological invariantsdepth and regularity of modules in the bigraded resolution of geometricm-residual intersection ring R/J:
i) (R/J)∆ is Koszul for all c ≥ 1 and e ≥ n2 .
ii) Suppose p > m, then depth(R/J)∆ ≥ p + n − (m + 1) for all ∆.
iii) If p = m + 1, then (R/J)∆ is Cohen-Macaulay for all ∆.
Conca, Herzog, Trung, and Valla (1997): If p = m + 1, then (R/J)∆ isCohen-Macaulay for large ∆.
![Page 59: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/59.jpg)
Diagonal Subalgebras of Residual Intersections
Ananthnarayan, Kumar and Mukundan (2018):
We describe the bigraded resolution and compute the homological invariantsdepth and regularity of modules in the bigraded resolution of geometricm-residual intersection ring R/J:
i) (R/J)∆ is Koszul for all c ≥ 1 and e ≥ n2 .
ii) Suppose p > m, then depth(R/J)∆ ≥ p + n − (m + 1) for all ∆.
iii) If p = m + 1, then (R/J)∆ is Cohen-Macaulay for all ∆.
Conca, Herzog, Trung, and Valla (1997): If p = m + 1, then (R/J)∆ isCohen-Macaulay for large ∆.
![Page 60: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/60.jpg)
Diagonal Subalgebras of Residual Intersections
Ananthnarayan, Kumar and Mukundan (2018):
We describe the bigraded resolution and compute the homological invariantsdepth and regularity of modules in the bigraded resolution of geometricm-residual intersection ring R/J:
i) (R/J)∆ is Koszul for all c ≥ 1 and e ≥ n2 .
ii) Suppose p > m, then depth(R/J)∆ ≥ p + n − (m + 1) for all ∆.
iii) If p = m + 1, then (R/J)∆ is Cohen-Macaulay for all ∆.
Conca, Herzog, Trung, and Valla (1997): If p = m + 1, then (R/J)∆ isCohen-Macaulay for large ∆.
![Page 61: Neeraj Kumar Indian Institute of Technology Bombay Joint ...djsym/talk/Day1/Neeraj.pdf · Diagonal subalgebras of bigraded algebras Neeraj Kumar Indian Institute of Technology Bombay](https://reader033.vdocuments.site/reader033/viewer/2022050403/5f80a81389f5e73b7b3f6366/html5/thumbnails/61.jpg)
Thank you for your attention!