I came across these notes from a talk by Hoschter which talks about superheight of an ideal and it mentions Krull's ideal height theorem on P2-P3 in terms of superheight. Here are the notes: http://www.math.lsa.umich.edu/~hochster/swb2.pdf. Does the Peskine–Szpiro intersection theorem imply Krull's ideal height theorem? I couldn't find any reference to this and wonder if anyone can either explain or point out some references about how to prove Krull's ideal height theorem from the Peskine–Szpiro intersection theorem.
Peskine–Szpiro Intersection Theorem and Krull’s Ideal Height Theorem – Commutative Algebra
ac.commutative-algebra
Best Answer
-The Superheight Theorem implies Krull's Height Theorem: See page 6 of Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley .
The Superheight Theorem implies the Intersection Theorem: See page 6 of Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley .
The Superheight Theorem follows from the New Intersection Theorem: See the proof of Theorem 9.4.4 of [Bruns, Winfried; Herzog, Jürgen, Cohen-Macaulay rings. ZBL0909.13005.].
The Superheight Theorem follows from the Intersection Theorem, at least in the case of equicharacteristic zero: I do not know a reference for this, however I provide a proof for the equicharacteristic zero case:
Let $M$ be as in the statement of the Superheight Theorem. To prove the statement, as in the proof of Theorem 9.4.4 of [Bruns, Winfried; Herzog, Jürgen, Cohen-Macaulay rings. ZBL0909.13005.], one can assume that $R\rightarrow S$ is a local homomorphism of local rings and $S$ is an $R$-algebra such that $(ann M)S$ is primary to the maximal ideal of $S$. Then passing to the completion, we can assume that $R$ and $S$ are both complete local rings (faithfully flat extension preserves the height and annihilator of a flat base change of $M$ is the extension of the annihilator of $M$ because $M$ is finite). Since $R$ and $S$ are both complete, so they admit coefficient fields and since $R$ (and $S$) has equicharacteristic zero so the coefficient field, $C_R$, of $R$ maps into the coefficient field, $C_S$ of $S$ by the map $R\rightarrow S$. Thus we can factor $R\rightarrow S$ through $R\widehat{\otimes}_{C_R}C_S\rightarrow S$. Since $C_R$ is the coefficient field of $R$, so $R\widehat{\otimes}_{C_R}C_S$ is Noetherian (and complete local). Since $R\widehat{\otimes}_{C_R}C_S$ and $S$ both have the same coefficient field $C_S$, so $R\widehat{\otimes}_{C_R}C_S\rightarrow S$ is a finite ring homomorphism (i.e. $S$ is module-finite over $R\widehat{\otimes}_{C_R}C_S$). Thus, without loss of generality, we can assume that the ring homomorphsim $R\rightarrow S$ is a finite ring homomorphism ($M\otimes_R(R\widehat{\otimes}_{C_R}C_S)$ has the same projective dimension as of $M$ by flatness of the local homomorphism $R\rightarrow R\widehat{\otimes}_{C_R}C_S$). Then since $(0:_RM)S$ is primary to the maximal ideal of $S$, so $M\otimes_RS$ has finite length, and thus the Peskine-Szpiro Intersection Theorem implies that $\text{height}\big((0:_RM)S\big)=\dim(S)\le \text{projdim}(M)$, as was to be proved, because $S$ is module-finite.
For a reference for the details of the facts used about the complete tensor product and finiteness of $S$ over $R\widehat{\otimes}_{C_R}C_S$, the factorization, the reason for the characteristic restriction of the statement (or some arguments around this restriction), or flatness of the complete tensor product $R\widehat{\otimes}_{C_R}C_S$ over $R$ please see some facts/results of arXiv:1911.11290, or/and Remark 5.2+Lemma 5.1 of arXiv:1609.00095.