Does the Peskine–Szpiro intersection theorem imply Krull's ideal height theorem? 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.
 A: *

*Superheight Theorem: Let $M$ be a non-zero finitely generated $R$-module over a Noetherian ring $R$. Then superheight(ann $M$)$\le$ projdim $M$.

-The Superheight Theorem   implies  Krull's Height Theorem:  See page 6 of Class Notes for Math 918: Homological Conjectures, Instructor
Tom Marley .

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*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.
