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Let me discuss the graded case, that is the ring is the polynomial ring and the matrices have general homogeneous entries of degree 1$1$. The local version should follows by taking "lowest order part" as you do in example 1$1$.

Consider fistfirst of the case of the polynomial ring S$S$ in variables x_{ij} 's$x_{ij}$ and y_{ij} 's$y_{ij}$ and matrices X=(x_{ij}) of$X=(x_{ij})$ of size axb$a\times b$ and Y=(y_{ij})$Y=(y_{ij})$ of size cxd$c\times d$ with a\leq b$a\leq b$ and c\leq d$c\leq d$. Then take the ideal I$I$ of minors of size a$a$ of X$X$ and the ideal J$J$ of minors of size c$c$ of Y$Y$.

Set A=S/I+J$A=S/I+J$.

The minimal free resolution of A$A$ is obtained by taking the tensor product of the resolution of S/I$S/I$ with that of S/J$S/J$ because the ideals are are generated by polynmialspolynomials in different varaiblesvariables. We may use this fact, in combination with the fact that I$I$ and J$J$ are resolved by the Eagon-Northcot complex, toto compute the Castelnuovo-Mumford reg(A)regularity $\operatorname{reg}(A)$ of A$A$, its projective dimension and its dimension. WeWe have:

reg(A)=reg(S/I)+reg(S/J)=(a-1)+(b-1).$\operatorname{reg}(A)=\operatorname{reg}(S/I)+\operatorname{reg}(S/J)=(a-1)+(b-1)$

dim(A) = ab+cd-(b-a+1)-(d-c+1)$\dim(A) = ab+cd-(b-a+1)-(d-c+1)$

and A$A$ is Cohen-Macaulay.

Now we specialize generically the x_{ij}$x_{ij}$'s and the y_{ij}$y_{ij}$'s generically to linearlinear forms inin variables z_1,..,z_n$z_1,..,z_n$. The ring you want to udenrstandunderstand gets identified with A/L $A/L$ where L$L$ is generated by ab+cd-n$ab+cd-n$ general linear forms in the x_{ij}$x_{ij}$ and y_{ij}$y_{ij}$.

Now if n\leq (b-a+1)+(d-c+1)$n\leq (b-a+1)+(d-c+1)$ then ab+cd-n\geq ab+cd-(b-a+1)-(d-c+1)$ab+cd-n\geq ab+cd-(b-a+1)-(d-c+1)$ and ab+cd-(b-a+1)-(d-c+1)$ab+cd-(b-a+1)-(d-c+1)$ of the general linear forms generating L$L$ form a maximal regular sequence in A$A$. Let U$U$ be the ideal generated by ab+cd-(b-a+1)-(d-c+1) of$ab+cd-(b-a+1)-(d-c+1)$ of the general linear forms generating L$L$.

Then reg(A/U)=reg(A)=(a-1)+(b-1)$\operatorname{reg}(A/U)=\operatorname{reg}(A)=(a-1)+(b-1)$ and dim(A/U)=0$\dim(A/U)=0$. The regularity for a 0$0$-dimensional module M is$M$ is the largest index i$i$ such that M_i\neq 0$M_i\neq 0$. It follows that the (A/U)_i=0$(A/U)_i=0$ for i>(a-1)+(b-1)$i>(a-1)+(b-1)$. And the same is true for A/L $A/L$ (becasuebecause it is a quotinetquotient of A/U$A/U$). Hence we have (A/L)_i=0$(A/L)_i=0$ for i>(a-1)+(b-1)$i>(a-1)+(b-1)$. On the size of the z's$z$'s it says that the ideal (z_1,..,z_n)$(z_1,..,z_n)$ to the power (a-1)+(b-1)+1$(a-1)+(b-1)+1$ is contained in ideal of definitondefinition, which is exactly what you wanted to prove.

Let me discuss the graded case, that is the ring is the polynomial ring and the matrices have general homogeneous entries of degree 1. The local version should follows by taking "lowest order part" as you do in example 1.

Consider fist of the case of the polynomial ring S in variables x_{ij} 's and y_{ij} 's and matrices X=(x_{ij}) of size axb and Y=(y_{ij}) of size cxd with a\leq b and c\leq d. Then take the ideal I of minors of size a of X and the ideal J of minors of size c of Y.

Set A=S/I+J.

The minimal free resolution of A is obtained by taking the tensor product of the resolution of S/I with that of S/J because the ideals are are generated by polynmials in different varaibles. We may use this fact, in combination with the fact that I and J are resolved by the Eagon-Northcot complex, to compute the Castelnuovo-Mumford reg(A) of A, its projective dimension and its dimension. We have:

reg(A)=reg(S/I)+reg(S/J)=(a-1)+(b-1).

dim(A) = ab+cd-(b-a+1)-(d-c+1)

and A is Cohen-Macaulay.

Now we specialize generically the x_{ij}'s and the y_{ij}'s generically to linear forms in variables z_1,..,z_n. The ring you want to udenrstand gets identified with A/L where L is generated by ab+cd-n general linear forms in the x_{ij} and y_{ij}.

Now if n\leq (b-a+1)+(d-c+1) then ab+cd-n\geq ab+cd-(b-a+1)-(d-c+1) and ab+cd-(b-a+1)-(d-c+1) of the general linear forms generating L form a maximal regular sequence in A. Let U be the ideal generated by ab+cd-(b-a+1)-(d-c+1) of the general linear forms generating L.

Then reg(A/U)=reg(A)=(a-1)+(b-1) and dim(A/U)=0. The regularity for a 0-dimensional module M is the largest index i such that M_i\neq 0. It follows that the (A/U)_i=0 for i>(a-1)+(b-1). And the same is true for A/L (becasue it is a quotinet of A/U). Hence we have (A/L)_i=0 for i>(a-1)+(b-1). On the size of the z's it says that the ideal (z_1,..,z_n) to power (a-1)+(b-1)+1 is contained in ideal of definiton, which is exactly what you wanted to prove.

Let me discuss the graded case, that is the ring is the polynomial ring and the matrices have general homogeneous entries of degree $1$. The local version should follows by taking "lowest order part" as you do in example $1$.

Consider first of the case of the polynomial ring $S$ in variables $x_{ij}$ and $y_{ij}$ and matrices $X=(x_{ij})$ of size $a\times b$ and $Y=(y_{ij})$ of size $c\times d$ with $a\leq b$ and $c\leq d$. Then take the ideal $I$ of minors of size $a$ of $X$ and the ideal $J$ of minors of size $c$ of $Y$.

Set $A=S/I+J$.

The minimal free resolution of $A$ is obtained by taking the tensor product of the resolution of $S/I$ with that of $S/J$ because the ideals are generated by polynomials in different variables. We may use this fact, in combination with the fact that $I$ and $J$ are resolved by the Eagon-Northcot complex, to compute the Castelnuovo-Mumford regularity $\operatorname{reg}(A)$ of $A$, its projective dimension and its dimension. We have:

$\operatorname{reg}(A)=\operatorname{reg}(S/I)+\operatorname{reg}(S/J)=(a-1)+(b-1)$

$\dim(A) = ab+cd-(b-a+1)-(d-c+1)$

and $A$ is Cohen-Macaulay.

Now we specialize generically the $x_{ij}$'s and the $y_{ij}$'s generically to linear forms in variables $z_1,..,z_n$. The ring you want to understand gets identified with $A/L$ where $L$ is generated by $ab+cd-n$ general linear forms in the $x_{ij}$ and $y_{ij}$.

Now if $n\leq (b-a+1)+(d-c+1)$ then $ab+cd-n\geq ab+cd-(b-a+1)-(d-c+1)$ and $ab+cd-(b-a+1)-(d-c+1)$ of the general linear forms generating $L$ form a maximal regular sequence in $A$. Let $U$ be the ideal generated by $ab+cd-(b-a+1)-(d-c+1)$ of the general linear forms generating $L$.

Then $\operatorname{reg}(A/U)=\operatorname{reg}(A)=(a-1)+(b-1)$ and $\dim(A/U)=0$. The regularity for a $0$-dimensional module $M$ is the largest index $i$ such that $M_i\neq 0$. It follows that $(A/U)_i=0$ for $i>(a-1)+(b-1)$. And the same is true for $A/L$ (because it is a quotient of $A/U$). Hence we have $(A/L)_i=0$ for $i>(a-1)+(b-1)$. On the size of the $z$'s it says that the ideal $(z_1,..,z_n)$ to the power $(a-1)+(b-1)+1$ is contained in ideal of definition, which is exactly what you wanted to prove.

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Let me discuss the graded case, that is the ring is the polynomial ring and the matrices have general homogeneous entries of degree 1. The local version should follows by taking "lowest order part" as you do in example 1.

Consider fist of the case of the polynomial ring S in variables x_{ij} 's and y_{ij} 's and matrices X=(x_{ij}) of size axb and Y=(y_{ij}) of size cxd with a\leq b and c\leq d. Then take the ideal I of minors of size a of X and the ideal J of minors of size c of Y.

Set A=S/I+J.

The minimal free resolution of A is obtained by taking the tensor product of the resolution of S/I with that of S/J because the ideals are are generated by polynmials in different varaibles. We may use this fact, in combination with the fact that I and J are resolved by the Eagon-Northcot complex, to compute the Castelnuovo-Mumford reg(A) of A, its projective dimension and its dimension. We have:

reg(A)=reg(S/I)+reg(S/J)=(a-1)+(b-1).

dim(A) = ab+cd-(b-a+1)-(d-c+1)

and A is Cohen-Macaulay.

Now we specialize generically the x_{ij}'s and the y_{ij}'s generically to linear forms in variables z_1,..,z_n. The ring you want to udenrstand gets identified with A/L where L is generated by ab+cd-n general linear forms in the x_{ij} and y_{ij}.

Now if n\leq (b-a+1)+(d-c+1) then ab+cd-n\geq ab+cd-(b-a+1)-(d-c+1) and ab+cd-(b-a+1)-(d-c+1) of the general linear forms generating L form a maximal regular sequence in A. Let U be the ideal generated by ab+cd-(b-a+1)-(d-c+1) of the general linear forms generating L.

Then reg(A/U)=reg(A)=(a-1)+(b-1) and dim(A/U)=0. The regularity for a 0-dimensional module M is the largest index i such that M_i\neq 0. It follows that the (A/U)_i=0 for i>(a-1)+(b-1). And the same is true for A/L (becasue it is a quotinet of A/U). Hence we have (A/L)_i=0 for i>(a-1)+(b-1). On the size of the z's it says that the ideal (z_1,..,z_n) to power (a-1)+(b-1)+1 is contained in ideal of definiton, which is exactly what you wanted to prove.