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Guntram
Guntram
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combinatorial Combinatorial formula for bettiBetti numbers of k[xa $k[x,y] moduley]$-module

Suppose I have a (finitely-presented, say) graded module M$M$ over k[x,y]$k[x,y]$, and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with a≤c$a\leq c$ and b≤d$b \leq d$, as well as the Hilbert function $dim_k(M_{a,b})$$\dim_k(M_{a,b})$.

Recall that the bettiBetti numbers of M$M$ are defined to be the grades and multiplicities of the generators of a minimal free resolution.

Is there an explicit formula which expresses these bettiBetti numbers directly in terms of the ranks R_{(a,b),(c,d)}$R_{(a,b),(c,d)}$ and the hilbertHilbert function?

As motivation, there is a well-known expression for the alternating sums of the bettiBetti numbers in terms of the hilbertHilbert function, e.g. Corollay 1.10 in the"The Geometry of SyzygiesSyzygies" by Eisenbud.

combinatorial formula for betti numbers of k[x,y] module

Suppose I have a (finitely-presented, say) graded module M over k[x,y], and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with a≤c and b≤d, as well as the Hilbert function $dim_k(M_{a,b})$.

Recall that the betti numbers of M are defined to be the grades and multiplicities of the generators of a minimal free resolution.

Is there an explicit formula which expresses these betti numbers directly in terms of the ranks R_{(a,b),(c,d)} and the hilbert function?

As motivation, there is a well-known expression for the alternating sums of the betti numbers in terms of the hilbert function, e.g. Corollay 1.10 in the Geometry of Syzygies by Eisenbud.

Combinatorial formula for Betti numbers of a $k[x,y]$-module

Suppose I have a (finitely-presented, say) graded module $M$ over $k[x,y]$, and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with $a\leq c$ and $b \leq d$, as well as the Hilbert function $\dim_k(M_{a,b})$.

Recall that the Betti numbers of $M$ are defined to be the grades and multiplicities of the generators of a minimal free resolution.

Is there an explicit formula which expresses these Betti numbers directly in terms of the ranks $R_{(a,b),(c,d)}$ and the Hilbert function?

As motivation, there is a well-known expression for the alternating sums of the Betti numbers in terms of the Hilbert function, e.g. Corollay 1.10 in "The Geometry of Syzygies" by Eisenbud.

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Simon Segert
Simon Segert
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combinatorial formula for betti numbers of k[x,y] module

Suppose I have a (finitely-presented, say) graded module M over k[x,y], and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with a≤c and b≤d, as well as the Hilbert function $dim_k(M_{a,b})$.

Recall that the betti numbers of M are defined to be the grades and multiplicities of the generators of a minimal free resolution.

Is there an explicit formula which expresses these betti numbers directly in terms of the ranks R_{(a,b),(c,d)} and the hilbert function?

As motivation, there is a well-known expression for the alternating sums of the betti numbers in terms of the hilbert function, e.g. Corollay 1.10 in the Geometry of Syzygies by Eisenbud.