Dtjytyk

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Hello Math.Stackexchange.com-Community.

Sorry for asking two questions at the same time, however they are part of one single step in a proof and hence closely related.
My questions arise from the following context:

Let $Xto Y$ be a proper morphism between locally noetherian Schemes and let $mathcal{F}$ be a coherent $mathcal{O}_X$-module on $X$ which is flat over $mathcal{O}_Y$.
Then for all $pgeq 0$ the function $h^p(-,mathcal{F})colon Ytomathbb{N}$ which is defined by
$h^p(y,mathcal{F}):=dim_{kappa(y)}H^p(X_y,mathcal{F}|_{X_y})$
is upper semi-continuous in the sense that for all $cinmathbb{R}$ the set ${yin Ymid h^p(y,mathcal{F})leq c}$ is open.

In order to prove this, the following lemma was invoked:

Lemma:
Let $A$ be a noetherian ring.
Let $deltacolon Mto F$ be a morphism between finitely-generated $A$-modules, of which $F$ is free.
Then, the function $gcolon Spec(A)to mathbb{N}$ defined by
$g(mathfrak{p}):=dim_{kappa(mathfrak{p})}(ker(Motimes_A kappa(mathfrak{p})overset{delta_{mathfrak{p}}:=deltaotimes_A kappa(mathfrak{p})}{to} Fotimes_A kappa(mathfrak{p}))$ is upper semi-continuous.

My question arises in the proof of this lemma, which goes as follows:

Proof
Let $mu_1,cdots,mu_sin M$ be elements, such that the images $overline{mu_1},cdots,overline{mu_s}in Motimes_Akappa(mathfrak{p})$ form a basis of that vector space, and such that $delta_{mathfrak{p}}(overline{mu_i})$ vanish for $i=1,cdots,r=g(mathfrak{p})$ and form a basis of $im(delta_{mathfrak{p}})$ for $i=r+1,cdots,s$.

Here comes the part that I don't understand, and also my first question:

Since the assertion is local near any $mathfrak{p}$ we may replace $A$ by $A_f$ for $fnot inmathfrak{p}$ and assume that $M$ is generated by the $mu_1,cdotsmu_s$ as an $A$ module.

The last assertion is clear to me. This is a Nakayama-Style argument.
It is however not clear to me, what "local near $mathfrak{p}$" means, and why this implies, that we may replace $A$ by $A_f$. Aside from this argument, it is also not clear to me, how to prove by hand (i.e. without using the "near $mathfrak{p}$"-Argument) why we may replace $A$ by $A_f$.

I tried and failed as follows:
Assume that we may find a covering by $D(f_j)$'s of $Spec(A)$ such that the morphisms $g_jcolon Spec(A_{f_j})tomathbb{N}, g_j(mathfrak{p}_j)=dim_{kappa(mathfrak{p}_j)}(ker(M_{f_j}otimes_{A_{f_j}}kappa(mathfrak{p}_{f_j})to F_{f_j}otimes_{A_{f_j}}kappa(mathfrak{p}_{f_j}))$ is EDIT: locally constant upper semi-continuous, where by $mathfrak{p}_j$ I mean the prime ideal in $Spec(A_{f_j})$ corresponding to $mathfrak{p}in D(f_j)$.

Then it would suffice if one could show that this implies, that the restrictions of $g$ to the $D(f_j)$ are upper semi-continuous.

In order to show this one would have to show, that for every $cinmathbb{R}$ we have $dim_{kappa(mathfrak{p}_j)}(ker(M_{f_j}otimes_{A_{f_j}}kappa(mathfrak{p}_{f_j})to F_{f_j}otimes_{A_{f_j}}kappa(mathfrak{p}_{f_j}))leq c$ for all $j$ iff $dim_{kappa(mathfrak{p})}(ker(Motimes_A kappa(mathfrak{p})overset{deltaotimes_A kappa(mathfrak{p})}{to} Fotimes_A kappa(mathfrak{p}))leq c$ for all prime ideals $mathfrak{p}in D(f_j)$.

This is the point where I don't see how to continue.

My second question is why we may construct a basis for $M_{f_j}otimes_{A_{f_j}}kappa(mathfrak{p_{f_j}})$ out of the basis for $Motimes_A kappa(mathfrak{p})$ such that it still has the properties we set up in the beginning of the proof, so that we may carry on with the proof after replacing $A$ by $A_f$.

Thank You for your participation!

Let me answer your first question. I will use the following setup:
$A$ a noetherian ring, $M$ a finitely generated $A$-module, and $p$ a prime ideal of $A$. We show that if $M_p$ is generated by the images of $x_1,dots, x_n in M$, then there exists $f in A$ such that $M_f$ is generated by the images of the same set of elements.

Let $N$ be the $A$-submodule of $M$ generated by $x_1,dots, x_n$, and let $K$ be the cokernel of the inclusion map $N to M$.
Our goal is to find an element $f in A$ such that $K_f = 0$.
If $K$ is $0$, then there is nothing to do. Assume that $K neq 0$.
Consider the support of $K$. Since $K$ is finitely generated $Supp(K)$ is determined by an $A$-ideal $I$, and since $p$ is not in it, $I$ is a proper ideal. Hence we may choose any nonzero element in $I$ for $f$.