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Math-2509 · mathematics · Algebra

Lex-plus-powers Betti bounds for Artinian quotients over C

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let $S=C[x_{1},\dots ,x_{n}]$, $n\geq 1$, and let I contain a homogeneous regular sequence of degrees $2\leq a_{1}\leq \cdots \leq a_{n}$. Is there a $lex-plus-(x_{1}^{a_{1}},\dots ,x_{n}^{a_{n}})$ ideal L with the same Hilbert function as I and $\beta ^S_{i,j}(S/I)\leq \beta ^S_{i,j}(S/L)$ for every i,j?

Short English statement

Let $S=C[x_{1},\dots ,x_{n}]$, $n\geq 1$, and let I contain a homogeneous regular sequence of degrees $2\leq a_{1}\leq \cdots \leq a_{n}$. Is there a $lex-plus-(x_{1}^{a_{1}},\dots ,x_{n}^{a_{n}})$ ideal L with the same Hilbert function as I and $\beta ^S_{i,j}(S/I)\leq \beta ^S_{i,j}(S/L)$ for every i,j?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2509-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2509

Over $\mathbb C$, the lex-plus-powers ideal in the theorem has the specified Hilbert function and bounds every Betti number of $S/I$ in its homological and internal bidegree. Thus the same single ideal gives all inequalities requested in the target.

Lex-plus-powers Betti bounds for Artinian quotients over C

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For characteristic-zero regular sequences, the proposed lex-plus-powers ideal simultaneously preserves the Hilbert function and bounds every graded Betti number above.

Mathematical statement of the claimed result

Result 1.
For a characteristic-zero field $F$, a homogeneous ideal $I\subseteq S_F=F[x_1,\ldots,x_n]$ containing a regular sequence of degrees $2\le a_1\le\cdots\le a_c$, $1\le c\le n$, put $P_F=(x_1^{a_1},\ldots,x_c^{a_c})$. In degree $d$, define $J_d$ by adjoining to $(P_F)_d$ the lexicographically greatest $q_d=\dim I_d-\dim(P_F)_d$ monomials outside $P_F$. The source claims $J=\bigoplus_dJ_d$ is an ideal with \[ \dim_F(S_F/I)_d=\dim_F(S_F/J)_d\quad(d\ge0),\qquad \beta^{S_F}_{u,j}(S_F/I)\le\beta^{S_F}_{u,j}(S_F/J) \quad(u\ge0,\ j\in\mathbb Z). \] Here $\beta_{u,j}$ are graded Betti numbers. The additional generators of $I$ have no degree or number restriction. The main complex-field theorem is the full-length Artinian case $c=n$.

Outline of the argument

The division-coefficient commuting forms retain enough filtration information to compare minimal free resolutions, beyond merely counting graded dimensions. The Artinian Betti comparison is then transported through the established regular-sequence reduction and descent to arbitrary characteristic-zero coefficient fields.

Formal verification

Family 200 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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