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Denis T
Denis T
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There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of epimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.

UPD: I need to do a due diligence and add reference. I can wholeheartedly recommend to anyone interested in quadratic algebras and combinatorial algebra in general to read small book "Quadratic algebras" by A. Polishchuk and L. Positselski. https://bookstore.ams.org/ulect-37

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of epimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of epimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite quadratic algebra to have something in degrees $\geq 3$ if you want.

UPD: I need to do a due diligence and add reference. I can wholeheartedly recommend to anyone interested in quadratic algebras and combinatorial algebra in general to read small book "Quadratic algebras" by A. Polishchuk and L. Positselski. https://bookstore.ams.org/ulect-37

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Denis T
Denis T
  • 4.6k
  • 2
  • 21
  • 32

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j)$.$$\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of eipmorphismepimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of $R$ being as low as $m^2/2$. One of those algebras is $\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j)$. Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of eipmorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of epimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.

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Denis T
Denis T
  • 4.6k
  • 2
  • 21
  • 32

There are plentiful examples of $m$-generated quadratic algebras $A$ with $A_3 = 0$, and dimension of $R$ being as low as $m^2/2$. One of those algebras is $\Bbb k\langle x_1, y_1, \dots, x_n, y_n \rangle / (x_i y_j, x_i x_j - y_i y_j)$. Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A_3 = 0$ and be Koszul.

Obviously, any proper quadratic quotient of those will give you an example of eipmorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite finite-dimensional quadratic algebra to have something in degrees $\geq 3$ if you want.