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# Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities

Yairon Cid-Ruiz  Department of Mathematics, KU Leuven, Celestijnenlaan 200B,
3001 Leuven, Belgium<EMAIL_ADDRESS>
###### Abstract.

We define and study the natural multigraded extension of the relative
multiplicities introduced by Simis, Ulrich and Vasconcelos. We call these new
invariants _relative mixed multiplicities_. We show that they have a stable
value equal to the mixed Buchsbaum–Rim multiplicity of Kleiman and Thorup.
Furthermore, we prove that integral dependence and birationality can be
detected via the vanishing of relative mixed multiplicities.

###### Key words and phrases:

relative mixed multiplicities, mixed Buchsbaum-Rim multiplicities, joint blow-
up, Rees algebra, integral dependence, birationality

###### 2020 Mathematics Subject Classification:

13H15, 14C17, 13D40, 13A30, 13B22

##  1\. Introduction

Let $A\subseteq B$ be an inclusion of standard $\mathbb{N}^{p}$-graded
algebras over a Noetherian local ring
$R=[A]_{(0,\ldots,0)}=[B]_{(0,\ldots,0)}$. Assume that ${\rm
length}_{R}\left([B]_{{\mathbf{e}}_{i}}/[A]_{{\mathbf{e}}_{i}}\right)<\infty$
for all $1\leq i\leq p$. Our main goal is to define and study a set of new
invariants that we call _relative mixed multiplicities_ and that provide the
natural multigraded extension of the _relative multiplicities_ introduced in
seminal work of Simis, Ulrich and Vasconcelos [SUV_MULT]. In the
$\mathbb{N}$-graded setting (i.e., when $p=1$), they introduced a series of
relative multiplicities $e_{t}(A,B)$, one for each $t\geq 1$, associated to
the numerical functions ${\rm
length}_{R}\left([B]_{n}\big{/}[A]_{n-t+1}[B]_{t-1}\right)$. They also showed
that the relative multiplicities $e_{t}(A,B)$ have a stable value
$e_{\infty}(A,B)$, and that the vanishing of $e_{\infty}(A,B)$ detects
integrality and the vanishing of $e_{1}(A,B)$ detects both integrality and
birationality. The relative multiplicity $e_{1}(A,B)$ was considered before by
Kirby and Rees [KIRBY_REES] (including a multigraded setting), whereas
$e_{\infty}(A,B)$ was introduced by Kleiman and Thorup [KLEIMAN_THORUP_GEOM]
under the name of _generalized Buchsbaum–Rim multiplicity_.

In this paper, we extend the above invariants and results to a multigraded
setting. Our first main result is the following theorem that serves as the
stepping stone to define the notion of relative mixed multiplicities.

######  Theorem A (Theorem 5.1).

Fix a tuple $\mathbf{t}=(t_{1},\ldots,t_{p})\in\mathbb{Z}_{+}^{p}$ of positive
integers. Then the following function

$\lambda_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})\;=\;{\rm
length}_{R}\left(\frac{\left[B\right]_{(n_{1},\ldots,n_{p})}}{\left[A\right]_{(n_{1}-t_{1}+1,\ldots,n_{p}-t_{p}+1)}\left[B\right]_{(t_{1}-1,\ldots,t_{p}-1)}}\right)$

coincides with a polynomial $P_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})$ for all
$n_{i}\gg 0$. Moreover, $P_{\mathbf{t}}^{A,B}$ has total degree at most
$\dim\left(\text{MultiProj}(B)\right)$ and its normalized leading coefficients
are nonnegative integers.

Let $X=\text{MultiProj}(B)$ and $r=\dim(X)$. The polynomial
$P_{\mathbf{t}}^{A,B}$ from Theorem A can be written as

$P_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})\;=\;\sum_{\beta\in\mathbb{N}^{p},\,|\beta|=r}e_{\mathbf{t}}\left(\beta;A,B\right)\,\frac{n_{1}^{\beta_{1}}\cdots
n_{p}^{\beta_{p}}}{\beta_{1}!\cdots\beta_{p}!}\;+\;\text{\rm(lower degree
terms)}.$

We say that the nonnegative integers $e_{\mathbf{t}}\left(\beta;A,B\right)\geq
0$ are the _relative mixed multiplicities of $B$ over $A$_. We show that these
new invariants are nonincreasing on the parameter
$\mathbf{t}\in\mathbb{N}^{p}$ and that they have a stable value. Indeed, from
Section missing 5, we obtain the inequality
$e_{\mathbf{t}^{\prime}}(\beta;A,B)\leq e_{\mathbf{t}}(\beta;A,B)$ for any
$\mathbf{t}^{\prime}=(t_{1}^{\prime},\ldots,t_{p}^{\prime})\in\mathbb{Z}_{+}^{p}$
with $\mathbf{t}^{\prime}\geq\mathbf{t}$ and the existence of the following
limit

$e_{\infty}(\beta;A,B)\;=\;\lim_{t_{1}\to\infty,\ldots,t_{p}\to\infty}e_{(t_{1},\ldots,t_{p})}\left(\beta;A,B\right).$

As it turns out, this stable value coincides with the _mixed Buchsbaum–Rim
multiplicity_ of Kleiman and Thorup [KLEIMAN_THORUP_MIXED]:

$e_{\infty}(\beta;A,B)\;=\;\operatorname{{\rm
br}}_{\beta}\left([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B\right)\;=\;\int_{P}c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{1})\right)^{\beta_{1}}\cdots
c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{p})\right)^{\beta_{p}}\,\cap\,\left[P\right]_{r};$

here $P=P_{X}^{Z_{1},\ldots,Z_{p}}$ denotes the Kleiman–Thorup transform of
$X=\text{MultiProj}(B)$ with respect to $Z_{1},\ldots,Z_{p}$, where
$Z_{i}=V_{+}\left([A]_{{\mathbf{e}}_{i}}B\right)\subset X$ is the closed
subscheme determined by $[A]_{{\mathbf{e}}_{i}}$ (see Section missing 2 and
Section missing 2 for further details).

Our proof of Theorem A is based upon using two main ingredients. Firstly, we
consider the stable value $e_{\infty}(\beta;A,B)=\operatorname{{\rm
br}}_{\beta}([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B)$, which
is geometrical by nature. For that, we extend the joint blow-up construction
of Kleiman and Thorup to an arbitrary multigraded setting in Section missing
2, and this allows us to define the mixed Buchsbaum–Rim multiplicity
$\operatorname{{\rm
br}}_{\beta}([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B)$.
Secondly, in Section missing 4, we identify and introduce new joint blow-up
constructions that measure the difference between each relative mixed
multiplicity $e_{\mathbf{t}}(\beta;A,B)$ and the stable value
$e_{\infty}(\beta;A,B)$. Therefore, our approach is to define
$e_{\mathbf{t}}(\beta;A,B)$ as a sum of intersection numbers (multidegrees) in
terms of the joint blow-up construction of Kleiman and Thorup and our new
joint blow-up construction. More precisely, in Section missing 5, we show the
equality

$e_{\mathbf{t}}\left(\beta;A,B\right)\;=\;\operatorname{{\rm
br}}_{\beta}\left([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B\right)\;+\;j_{\beta}^{\\#}\big{(}[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};\mathscr{H}_{\mathbf{t}-\mathbf{1}}\big{)},$

where
$j_{\beta}^{\\#}([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};\mathscr{H}_{\mathbf{t}-\mathbf{1}})$
is a mixed multiplicity of one of our new joint blow-up constructions
considered over the ideal
$\mathscr{H}_{\mathbf{t}-\mathbf{1}}=\big{(}[B]_{(t_{1}-1,\ldots,t_{p}-1)}\big{)}\subset
B$ (see Section missing 4 for further details).

Similarly to the $\mathbb{N}$-graded case, we obtain that the distinguished
relative mixed multiplicities $e(\beta;A,B)=e_{(1,\ldots,1)}(\beta;A,B)$ and
$e_{\infty}(\beta;A,B)$ can detect integral dependence and birationality. Our
second main result is the following theorem.

######  Theorem B (Theorem 6.1).

Consider the associated morphism

$f\;:\;U\;\subseteq\;X=\text{MultiProj}(B)\;\longrightarrow\;Y=\text{MultiProj}(A)$

where $U=X\setminus V_{+}(A_{++}B).$ Then the following statements hold:

  1. (i)

If we obtain a finite morphism $f:X\rightarrow Y$, then
$e_{\infty}\left(\beta;A,B\right)=0$ for all $\beta\in\mathbb{N}^{p}$ with
$|\beta|=r$. The converse holds if $B$ is equidimensional and catenary.

  2. (ii)

If we obtain a finite birational morphism $f:X\rightarrow Y$, then
$e\left(\beta;A,B\right)=0$ for all $\beta\in\mathbb{N}^{p}$ with $|\beta|=r$.
The converse holds if $B$ is equidimensional and catenary.

To the best of our knowledge, even in the $\mathbb{N}$-graded case, the result
of Theorem B seems to be new (see Section missing 6 and Section missing 6):
for the converse statements of Theorem B, it is typically assumed that $B$ is
equidimensional and _universally_ catenary (see [SUV_MULT,
KLEIMAN_THORUP_GEOM, KATZ_RED_MOD]).

Outline. The basic outline of this paper is as follows. In Section missing 2,
we extend the joint blow-up construction of Kleiman and Thorup to an arbitrary
multigraded setting. We present some basic results regarding a multigraded
version of $j$-multiplicities in Section missing 3. Our new joint blow-up
construction is introduced in Section missing 4. We prove Theorem A in Section
missing 5. Finally, Section missing 6 contains the proof of Theorem B and some
applications.

Notation. Let $p\geq 1$ be a positive integer. If
${\mathbf{n}}=(n_{1},\ldots,n_{p}),{\mathbf{m}}=(m_{1},\ldots,m_{p})\in\mathbb{Z}^{p}$
are two multi-indexes, we write ${\mathbf{n}}\geq{\mathbf{m}}$ whenever
$n_{i}\geq m_{i}$ for all $1\leq i\leq p$, and ${\mathbf{n}}>{\mathbf{m}}$
whenever $n_{j}>m_{j}$ for all $1\leq j\leq p$. For each $1\leq i\leq p$, let
${\mathbf{e}}_{i}\in\mathbb{N}^{p}$ be the $i$-th elementary vector
${\mathbf{e}}_{i}:=\left(0,\ldots,1,\ldots,0\right)$. Let
$\mathbf{0}\in\mathbb{N}^{p}$ and $\mathbf{1}\in\mathbb{N}^{p}$ be the vectors
$\mathbf{0}:=(0,\ldots,0)$ and $\mathbf{1}:=(1,\ldots,1)$ of $p$ copies of $0$
and $1$, respectively. For any
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{Z}^{p}$, we define its weight as
$\lvert{\mathbf{n}}\rvert:=n_{1}+\cdots+n_{p}$. We say that an
$\mathbb{N}^{p}$-graded ring $B$ is _standard_ if $B$ is generated as an
algebra over $[B]_{(0,\ldots,0)}$ by finitely many elements of the form
$b\in[B]_{{\mathbf{e}}_{i}}$ for some $1\leq i\leq p$. For a standard
$\mathbb{N}^{p}$-graded ring $B$, we set the following:

  * –

$B_{+}:=\bigoplus_{\mathbf{v}>\mathbf{0}}[B]_{\mathbf{v}}$ is the ideal of
elements of positive total degree.

  * –

$B_{++}:=\bigoplus_{\mathbf{v}\geq\mathbf{1}}[B]_{\mathbf{v}}$ is the
multigraded irrelevant ideal.

  * –

$\text{MultiProj}(B):=\big{\\{}{\mathfrak{p}}\in\text{Spec}(B)\mid\text{${\mathfrak{p}}$
is $\mathbb{N}^{p}$-graded and ${\mathfrak{p}}\not\supseteq B_{++}$}\big{\\}}$
is the corresponding multiprojective scheme.

  * –

$V_{+}(I):=\big{\\{}{\mathfrak{p}}\in\text{MultiProj}(B)\mid{\mathfrak{p}}\supseteq
I\big{\\}}$ for any $\mathbb{N}^{p}$-graded ideal $I\subset B$.

##  2\. An extension of a joint blow-up construction of Kleiman and Thorup

Here, we provide a multigraded extension of a joint blow-up construction of
Kleiman and Thorup [KLEIMAN_THORUP_MIXED], and we closely follow their
approach. As a consequence, we can introduce the notion of mixed Buchsbaum-Rim
multiplicities in an arbitrary multigraded setting. Throughout this section
the following setup is fixed.

######  Setup .

Let $B$ be a standard $\mathbb{N}^{p}$-graded algebra over a Noetherian local
ring $R=\left[B\right]_{(0,\ldots,0)}$. Denote by ${\mathfrak{m}}\in R$ the
maximal ideal of $R$. Let
$X=\text{MultiProj}(B)\subset{\mathbb{P}}_{R}^{m_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{m_{p}}$
be the multiprojective scheme determined by $B$, embedded in a product of
projective spaces over $R$. Let $q\geq 1$ be a positive integer. For each
$1\leq i\leq q$, let $\mathbf{d}_{i}\in\mathbb{N}^{p}$ be a tuple of
nonnegative integers and $H_{i}\subseteq[B]_{\mathbf{d}_{i}}$ be a nonzero
$R$-submodule. Let $Z_{i}\subset X$ be the closed subscheme determined by
$H_{i}$ and $\mathcal{J}_{i}\subset\mathcal{O}_{X}$ be the corresponding ideal
sheaf. Let $Z\subset X$ be the closed subscheme defined by the ideal sheaf
$\mathcal{J}=\mathcal{J}_{1}\cdots\mathcal{J}_{q}\subset\mathcal{O}_{X}$.

Let
$\widehat{X}:=\operatorname{\mathchanc{Spec}}_{X}\left(\mathcal{O}_{X}[t]\right)=X\times_{\text{Spec}(R)}\mathbb{A}_{R}^{1}$
be the relative affine line over $X$. We view $X$ as a closed subscheme
embedded in $\widehat{X}$ as the principal divisor $\\{t=0\\}$. Let
$\widehat{\mathcal{B}}:={\mathcal{B}\ell}_{Z_{1},\ldots,Z_{q}}\big{(}\widehat{X}\big{)}$
be the joint blow-up of $\widehat{X}$ along the closed subschemes
$Z_{1},\ldots,Z_{q}$. This scheme can be obtained by successively blowing-up
$\widehat{X}$ along $Z_{1},\ldots,Z_{q}$. It arises from the
$\mathbb{N}^{q}$-graded quasi-coherent $\mathcal{O}_{\widehat{X}}$-algebra

$\mathscr{R}=\mathscr{R}\big{(}(\mathcal{J}_{1},t),\ldots,(\mathcal{J}_{q},t)\big{)}\;:=\;\bigoplus_{(n_{1},\ldots,n_{q})\in\mathbb{N}^{q}}(\mathcal{J}_{1},t)^{n_{1}}\cdots(\mathcal{J}_{q},t)^{n_{q}};$

i.e., the multi-Rees algebra of the ideal sheaves
$(\mathcal{J}_{1},t),\ldots,(\mathcal{J}_{q},t)\subset\mathcal{O}_{\widehat{X}}$.
As we wish to embed $\widehat{\mathcal{B}}$ in a product of projective spaces,
we consider the “twisted” $\mathbb{N}^{q}$-graded algebra

$\mathscr{T}\;:=\;\bigoplus_{(n_{1},\ldots,n_{q})\in\mathbb{N}^{q}}(\mathcal{J}_{1},t)^{n_{1}}\cdots(\mathcal{J}_{q},t)^{n_{q}}\,\otimes_{\mathcal{O}_{\widehat{X}}}\,\mathcal{O}_{\widehat{X}}\left(n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}\right).$

The algebras $\mathscr{R}$ and $\mathscr{T}$ yield the same scheme
$\widehat{\mathcal{B}}$ (see [HARTSHORNE, Lemma II.7.9] for the
$\mathbb{N}$-graded case). Since the sheaf
$(\mathcal{J}_{i},t)(\mathbf{d}_{i})$ is generated by global sections, we have
a surjection
$\mathcal{O}_{\widehat{X}}^{r_{i}+1}\twoheadrightarrow(\mathcal{J}_{i},t)(\mathbf{d}_{i})$
for some $r_{i}\geq 0$. In turn, this naturally gives the surjection

$\text{Sym}_{\mathcal{O}_{\widehat{X}}}\left(\mathcal{O}_{\widehat{X}}^{r_{1}+1}\right)\otimes_{\mathcal{O}_{\widehat{X}}}\cdots\otimes_{\mathcal{O}_{\widehat{X}}}\text{Sym}_{\mathcal{O}_{\widehat{X}}}\left(\mathcal{O}_{\widehat{X}}^{r_{q}+1}\right)\;\twoheadrightarrow\;\mathscr{T}.$

Consequently, we obtain a closed immersion
$\iota:\widehat{\mathcal{B}}\hookrightarrow{\mathbb{P}}_{\widehat{X}}^{r_{1}}\times_{\widehat{X}}\cdots\times_{\widehat{X}}{\mathbb{P}}_{\widehat{X}}^{r_{q}}\cong\widehat{X}\times_{R}{\mathbb{P}}_{R}^{r_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{r_{q}}$.
Then we set

$\mathcal{O}_{\widehat{\mathcal{B}}}({\mathbf{e}}_{i})\,:=\,\iota^{*}\mathcal{O}_{{\mathbb{P}}_{\widehat{X}}^{r_{1}}\times_{\widehat{X}}\cdots\times_{\widehat{X}}{\mathbb{P}}_{\widehat{X}}^{r_{q}}}({\mathbf{e}}_{i})$

for any ${\mathbf{e}}_{i}\in\mathbb{N}^{q}$ with $1\leq i\leq q$. Thus our
choice of $\mathcal{O}_{\widehat{\mathcal{B}}}({\mathbf{e}}_{i})$ equals the
corresponding tautological line bundle determined by $\mathscr{R}$ but twisted
by $\mathcal{O}_{\widehat{X}}(\mathbf{d}_{i})$.

By an abuse of notation, we shall write $\mathcal{O}_{X}({\mathbf{e}}_{i})$
for some ${\mathbf{e}}_{i}\in\mathbb{N}^{p}$ with $1\leq i\leq p$ and
$\mathcal{O}_{\widehat{\mathcal{B}}}({\mathbf{e}}_{j})$ for some
${\mathbf{e}}_{j}\in\mathbb{N}^{q}$ with $1\leq j\leq q$. Thus, depending on
the context, ${\mathbf{e}}_{i}$ could denote an elementary basis vector either
in $\mathbb{N}^{p}$ or in $\mathbb{N}^{q}$.

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $B$-module and
$\mathscr{M}:=\widetilde{M}$ be the corresponding coherent sheaf on $X$. Let
$\widehat{\mathscr{M}}:=\mathscr{M}[t]$ be the
$\mathcal{O}_{\widehat{X}}$-module given as the pullback of $\mathscr{M}$.
Denote by $\widehat{\mathcal{B}}(\mathscr{M})$ the coherent
$\mathcal{O}_{\widehat{\mathcal{B}}}$-module associated to the following
$\mathbb{N}^{q}$-graded $\mathscr{T}$-module

$T(\mathscr{M})\;:=\;\bigoplus_{(n_{1},\ldots,n_{q})\in\mathbb{N}^{q}}(\mathcal{J}_{1},t)^{n_{1}}\cdots(\mathcal{J}_{q},t)^{n_{q}}\widehat{\mathscr{M}}\;\otimes_{\mathcal{O}_{\widehat{X}}}\,\mathcal{O}_{\widehat{X}}\left(n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}\right),$

which is a quasi-coherent $\mathcal{O}_{\widehat{X}}$-module.

The main objects of interest in this section are introduced in the following
definition.

######  Definition .

Let $P=P_{X}^{Z_{1},\ldots,Z_{q}}:=\widehat{\mathcal{B}}\times_{\widehat{X}}Z$
be the restriction of $\widehat{\mathcal{B}}$ to the preimage of $Z$. Let
$\pi:P\rightarrow X$ be the natural projection and
$j:P\hookrightarrow\widehat{\mathcal{B}}$ be the natural closed immersion. We
say that the _Kleiman–Thorup transform_ of $\mathscr{M}$ with respect to
$Z_{1},\ldots,Z_{q}$ is the coherent $\mathcal{O}_{P}$-module given by

$P(\mathscr{M})=P_{X}^{Z_{1},\ldots,Z_{q}}(\mathscr{M})\;:=\;j^{*}\left(\widehat{\mathcal{B}}(\mathscr{M})\right).$

The lemma below provides a basic dimension computation that will be needed
(also, see [KLEIMAN_THORUP_GEOM, §3], [huneke2006integral, Chapter 5]).

######  Lemma .

We have the inequality
$\dim\left(\text{Supp}(P(\mathscr{M}))\right)\leq\dim\left(\text{Supp}(\mathscr{M})\right)$.

###### Proof.

Let $U=\text{Spec}(A)\subset X$ be an affine open subscheme and $N$ be a
finitely generated $A$-module with $\mathscr{M}{\mid_{U}}=\widetilde{N}$. Let
$J_{i}\subset A$ be the ideal corresponding to $U\cap Z_{i}$, and set
$K_{i}=(J_{i},t)A[t]$. Consider the module
$L=\mathscr{R}_{N[t]}\left(K_{1},\ldots,K_{q}\right):=(N[t])\left[K_{1}T_{1},\ldots,K_{q}T_{q}\right]$
over the multi-Rees algebra
$\mathscr{R}_{A[t]}\left(K_{1},\ldots,K_{q}\right):=(A[t])\left[K_{1}T_{1},\ldots,K_{q}T_{q}\right]$
with standard $\mathbb{N}^{q}$-grading induced by setting
$\deg(t)=\mathbf{0}\in\mathbb{N}^{q}$ and
$\deg(T_{i})={\mathbf{e}}_{i}\in\mathbb{N}^{q}$. Notice that
$P(\mathscr{M}){\mid_{\pi^{-1}(U)}}$ is the sheaf associated to the
$\mathbb{Z}^{q}$-graded module $L/(J_{1}\cdots J_{q},t)L$. Since
$\dim\left(L/(J_{1}\cdots J_{q},t)L\right)\leq\dim(L)-1\leq\dim(N)+q$, it
follows that
$\dim\big{(}\text{Supp}(P(\mathscr{M}){\mid_{\pi^{-1}(U)}})\big{)}\leq\dim(N)$
(see, e.g., [HYRY_MULTIGRAD, §1], [SPECIALIZATION_ARON, §3.1]), as required. ∎

We have the following theorem that allows us to define mixed Buchsbaum–Rim
multiplicities. It extends the results of [KLEIMAN_THORUP_MIXED, §8] to an
arbitrary multigraded setting.

######  Remark .

As in the work of Kleiman and Thorup [KLEIMAN_THORUP_GEOM,
KLEIMAN_THORUP_MIXED], the intersection numbers stated below are well-defined
by utilizing the rudiments of intersection theory developed by Thorup in
[THORUP_INT_TH], which are suitable to work over our base Noetherian local
ring $R$. This treatment of Thorup extends the necessary results from Fulton’s
book [FULTON_INTERSECTION_THEORY].

######  Theorem 2.1.

Assume Section missing 2. Let $M$ be a finitely generated
$\mathbb{Z}^{p}$-graded $B$-module and $\mathscr{M}=\widetilde{M}$ be the
corresponding coherent $\mathcal{O}_{X}$-module. Set
$r=\dim(\text{Supp}(\mathscr{M}))$, and suppose that
$Z\cap\text{Supp}(\mathscr{M})$ is contained in the closed fiber
$X\times_{\text{Spec}(R)}\text{Spec}(R/{\mathfrak{m}})$ of $X$. Then the
following function

$\lambda_{M}^{H_{1},\ldots,H_{q}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{q})\;=\;{\rm
length}_{R}\left(\frac{[M]_{(v_{1},\ldots,v_{p})+n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}}}{H_{1}^{n_{1}}\cdots
H_{q}^{n_{q}}\,[M]_{(v_{1},\ldots,v_{p})}}\right)$

coincides with a polynomial
$P_{M}^{H_{1},\ldots,H_{q}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{q})$ for all
$v_{i}\gg 0$ and $n_{i}\gg 0$. Moreover, $P_{M}^{H_{1},\ldots,H_{q}}$ has
total degree at most $r$ and we can write

$P_{M}^{H_{1},\ldots,H_{q}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{q})\;=\;\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}^{p},\beta\in\mathbb{N}^{q}\\\
|\alpha|+|\beta|=r\end{subarray}}e(\alpha,\beta)\,\frac{v_{1}^{\alpha_{1}}\cdots
v_{p}^{\alpha_{p}}n_{1}^{\beta_{1}}\cdots
n_{q}^{\beta_{q}}}{\alpha_{1}!\cdots\alpha_{p}!\beta_{1}!\cdots\beta_{q}!}\;+\;\text{\rm(lower
degree terms)}$

where the normalized coefficient $e(\alpha,\beta)$ equals the intersection
number

$\int_{P}c_{1}\left(\pi^{*}(\mathcal{O}_{X}({\mathbf{e}}_{1}))\right)^{\alpha_{1}}\cdots
c_{1}\left(\pi^{*}(\mathcal{O}_{X}({\mathbf{e}}_{p}))\right)^{\alpha_{p}}\,c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{1})\right)^{\beta_{1}}\cdots
c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{q})\right)^{\beta_{q}}\,\cap\,\left[P(\mathscr{M})\right]_{r}.$

###### Proof.

From our conventions above, we can identify $P$ as a closed subscheme of a
product of projective spaces over $R$ via the following composition of natural
closed immersions

$\displaystyle
f\;:\;P\;\hookrightarrow\;\widehat{\mathcal{B}}\times_{\text{Spec}(R[t])}\text{Spec}\left(R[t]/(t)\right)\;\hookrightarrow\;X\times_{R}$
$\displaystyle{\mathbb{P}}_{R}^{r_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{r_{q}}$
$\displaystyle\;\hookrightarrow\;{\mathbb{P}}:={\mathbb{P}}_{R}^{m_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{m_{p}}\times_{R}{\mathbb{P}}_{R}^{r_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{r_{q}}.$

For any $\mathbf{v}=(v_{1},\ldots,v_{p})\in\mathbb{N}^{p}$ and
$\mathbf{n}=(n_{1},\ldots,n_{q})\in\mathbb{N}^{q}$, the pullback of
$\mathcal{O}_{\mathbb{P}}(\mathbf{v},\mathbf{n})$ to $P$ is given by

$f^{*}\big{(}\mathcal{O}_{{\mathbb{P}}}(\mathbf{v},\mathbf{n})\big{)}\;\cong\;\pi^{*}\big{(}\mathcal{O}_{X}(\mathbf{v})\big{)}\,\otimes_{\mathcal{O}_{P}}\,\mathcal{O}_{P}(\mathbf{n}).$

To simplify notation, let $\mathcal{F}:=P(\mathscr{M})$,
$\mathcal{F}({\mathbf{n}}):=P(\mathscr{M})\otimes_{\mathcal{O}_{P}}\mathcal{O}_{P}({\mathbf{n}})$
and

$\mathcal{F}(\mathbf{v},\mathbf{n})\;:=\;P(\mathscr{M})\otimes_{\mathcal{O}_{P}}f^{*}\big{(}\mathcal{O}_{{\mathbb{P}}}(\mathbf{v},\mathbf{n})\big{)}\;\cong\;P(\mathscr{M})\otimes_{\mathcal{O}_{P}}\pi^{*}\big{(}\mathcal{O}_{X}(\mathbf{v})\big{)}\otimes_{\mathcal{O}_{P}}\mathcal{O}_{P}(\mathbf{n})$

for any $\mathbf{v}\in\mathbb{N}^{p}$ and $\mathbf{n}\in\mathbb{N}^{q}$. From
[KLEIMAN_THORUP_GEOM, Lemma 4.3], it follows that the function

(1)
$\displaystyle\begin{split}\chi\big{(}f_{*}(\mathcal{F})(\mathbf{v},\mathbf{n})\big{)}&=\sum_{i}{(-1)}^{i}{\rm
length}_{R}\big{(}\text{H}^{i}\left({\mathbb{P}},f_{*}(\mathcal{F})(\mathbf{v},\mathbf{n})\right)\big{)}\\\
&=\sum_{i}{(-1)}^{i}{\rm
length}_{R}\big{(}\text{H}^{i}\left(P,\mathcal{F}(\mathbf{v},\mathbf{n})\right)\big{)}\end{split}$

equals a polynomial $P_{\mathcal{F}}(\mathbf{v},{\mathbf{n}})$ of total degree
equal to $\dim(\text{Supp}(\mathcal{F}))\leq r=\dim(\text{Supp}(\mathscr{M}))$
(see Section missing 2). For $\alpha\in\mathbb{N}^{p},\beta\in\mathbb{N}^{q}$
with $|\alpha|+|\beta|=r$, the coefficient of
$\mathbf{v}^{\alpha}\mathbf{n}^{\beta}$ in
$P_{\mathcal{F}}(\mathbf{v},\mathbf{n})$ is equal to
$\frac{1}{\alpha_{1}!\cdots\alpha_{p}!\beta_{1}!\cdots\beta_{q}!}$ times the
intersection number

$\int_{\mathbb{P}}\,c_{1}\left(\mathcal{O}_{\mathbb{P}}({\mathbf{e}}_{1},\mathbf{0})\right)^{\alpha_{1}}\cdots
c_{1}\left(\mathcal{O}_{\mathbb{P}}({\mathbf{e}}_{p},\mathbf{0})\right)^{\alpha_{p}}\,c_{1}\left(\mathcal{O}_{\mathbb{P}}(\mathbf{0},{\mathbf{e}}_{1})\right)^{\beta_{1}}\cdots
c_{1}\left(\mathcal{O}_{\mathbb{P}}(\mathbf{0},{\mathbf{e}}_{q})\right)^{\beta_{q}}\,\cap\,\left[f_{*}(\mathcal{F})\right]_{r};$

where, depending on the context, we have either
$\mathbf{0}=(0,\ldots,0)\in\mathbb{N}^{p}$ or
$\mathbf{0}=(0,\ldots,0)\in\mathbb{N}^{q}$. By the projection formula, this
latter intersection number coincides with $e(\alpha,\beta)$.

We need two observations regarding the natural projection
$\pi:P\subset{\mathbb{P}}_{X}^{r_{1}}\times_{X}\cdots\times_{X}{\mathbb{P}}_{X}^{r_{q}}\rightarrow
X$. We set
$\overline{T}(\mathscr{M}):=T(\mathscr{M})\otimes_{\mathcal{O}_{\widehat{X}}}\mathcal{O}_{Z}$.
For any affine open subscheme $U\subset X$, by utilizing [KLEIMAN_THORUP_GEOM,
Lemma 4.2], we obtain that
$\pi_{*}\left(\mathcal{F}({\mathbf{n}})\right){\mid_{U}}\cong\left[\overline{T}(\mathscr{M})\mid_{U}\right]_{\mathbf{n}}$
and $R^{i}\pi_{*}\left(\mathcal{F}({\mathbf{n}})\right){\mid_{U}}=0$ for all
${\mathbf{n}}\gg\mathbf{0}$ and $i\geq 1$. Hence we globally get

$\pi_{*}\left(\mathcal{F}({\mathbf{n}})\right)\;\cong\;\left[\overline{T}(\mathscr{M})\right]_{\mathbf{n}}\qquad\text{
and }\qquad R^{i}\pi_{*}\left(\mathcal{F}({\mathbf{n}})\right)=0$

for all ${\mathbf{n}}\gg\mathbf{0}$ and $i\geq 1$. So, the Leray spectral
sequence

$E_{2}^{i,j}=\text{H}^{i}\left(X,R^{j}\pi_{*}(\mathcal{F}(\mathbf{v},\mathbf{n}))\right)\;\Rightarrow\;\text{H}^{i+j}\left(P,\mathcal{F}(\mathbf{v},\mathbf{n})\right)$

degenerates and yields the isomorphism

(2)
$\text{H}^{i}\big{(}P,\mathcal{F}(\mathbf{v},{\mathbf{n}})\big{)}\;\cong\;\text{H}^{i}\big{(}X,\left[\overline{T}(\mathscr{M})\right]_{\mathbf{n}}(\mathbf{v})\big{)}$

for all $i\geq 0$, $\mathbf{v}\in\mathbb{N}^{p}$ and
${\mathbf{n}}\gg\mathbf{0}\in\mathbb{N}^{q}$.

We now study the algebraic counterpart of $\overline{T}(\mathscr{M})$. Given a
$\mathbb{Z}^{p}$-graded $B$-module $N$, we define the $\mathbb{Z}^{p}$-graded
$B$-module

$\overline{T}(N)\;:=\;\bigoplus_{(n_{1},\ldots,n_{q})\in\mathbb{N}^{q}}\frac{(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}N[t]}{(H_{1}\cdots
H_{q},t)(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}N[t]}\;\otimes_{B}\,B\left(n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}\right),$

where $\deg(t)=\mathbf{0}\in\mathbb{N}^{p}$. We consider the Noetherian
$\mathbb{Z}^{p}$-graded algebra $S:=\overline{T}(B)$. Notice that $S$ is a
quotient of the multi-Rees algebra
$\mathscr{R}_{B[t]}\left((H_{1},t),\ldots,(H_{q},t)\right):=(B[t])\big{[}(H_{1},t)T_{1},\ldots,(H_{q},t)T_{q}\big{]}$
with $\mathbb{Z}^{p}$-grading induced by the grading of $B$ and by setting
$\deg(T_{i})=-\mathbf{d}_{i}\in\mathbb{Z}^{p}$ and
$\deg(t)=\mathbf{0}\in\mathbb{N}^{p}$. Then $\overline{T}(M)$ is a finitely
generated $\mathbb{Z}^{p}$-graded $S$-module. Notice that
$B_{++}S=S_{++}:=\bigoplus_{\mathbf{v}\geq\mathbf{1}}[S]_{\mathbf{v}}$, where
$\mathbf{1}=(1,\ldots,1)\in\mathbb{N}^{p}$. Therefore, we get the vanishing
$\big{[}\text{H}_{B_{++}}^{i}\big{(}\overline{T}(M)\big{)}\big{]}_{\mathbf{v}}\cong\big{[}\text{H}_{S_{++}}^{i}\big{(}\overline{T}(M)\big{)}\big{]}_{\mathbf{v}}=0$
for all $i\geq 0$ and $\mathbf{v}\gg\mathbf{0}$ (this follows verbatim to the
$\mathbb{Z}$-graded result from [CHARDIN_POWERS, Theorem 2.1]). By utilizing
[HYRY_MULTIGRAD, Corollary 1.5], we obtain

(3)
$\text{H}^{0}\big{(}X,\overline{T}(\mathscr{M})(\mathbf{v})\big{)}\;\cong\left[\overline{T}(M)\right]_{\mathbf{v}}\quad\text{
and }\quad\text{H}^{i}\big{(}X,\overline{T}(\mathscr{M})(\mathbf{v})\big{)}=0$

for all $\mathbf{v}\gg\mathbf{0}$ and $i\geq 1$.

By combining (missing) 1, (missing) 2 and (missing) 3, we get tuples of
integers $\mathbf{v}^{\prime}\in\mathbb{N}^{p}$ and
$\mathbf{n}^{\prime}\in\mathbb{N}^{q}$, such that

(4) $P_{\mathcal{F}}(\mathbf{v},{\mathbf{n}})\;=\;{\rm
length}_{R}\left(E_{\mathbf{v},\mathbf{n}}\right)$

for all $\mathbf{v}\geq\mathbf{v}^{\prime}$ and
$\mathbf{n}\geq\mathbf{n}^{\prime}$, where

$E_{\mathbf{v},\mathbf{n}}\;:=\;\left[\frac{(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}M[t]}{(H_{1}\cdots
H_{q},t)(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}M[t]}\right]_{\mathbf{v}+n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}}.$

We write

$(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}M[t]\;=\;\sum_{0\leq j_{i}\leq
n_{i}}H_{1}^{j_{1}}\cdots
H_{q}^{j_{q}}\,t^{|{\mathbf{n}}|-(j_{1}+\cdots+j_{q})}\,M[t]\;=\;\sum_{k=0}^{|{\mathbf{n}}|}L_{k}t^{|{\mathbf{n}}|-k}M[t],$

where $L_{k}$ is the $R$-module $\sum_{\begin{subarray}{c}0\leq j_{i}\leq
n_{i}\\\ j_{1}+\cdots+j_{q}=k\end{subarray}}H_{1}^{j_{1}}\cdots
H_{q}^{j_{q}}$; by standard convention, we have
$H_{i}^{0}=R=[B]_{\mathbf{0}}$. Since $H_{1}\cdots H_{q}L_{k}M\subset
L_{k+1}M$ for $0\leq k<|{\mathbf{n}}|$, we get the equality

$(H_{1}\cdots
H_{q},t)(H_{1},t)^{n_{1}}\cdots(H_{q},t)^{n_{q}}M[t]\;=\;\Big{(}H_{1}\cdots
H_{q}L_{|{\mathbf{n}}|}+\sum_{k=1}^{|{\mathbf{n}}|}L_{k}t^{|{\mathbf{n}}|+1-k}+t^{|{\mathbf{n}}|+1}\Big{)}M[t].$

Therefore, we obtain the isomorphism of $R$-modules

$E_{\mathbf{v},\mathbf{n}}\;\cong\;\left[\frac{L_{|{\mathbf{n}}|}M}{H_{1}\cdots
H_{q}L_{|{\mathbf{n}}|}M}\,\;\;\bigoplus\;\;\,\bigoplus_{k=0}^{|{\mathbf{n}}|-1}\frac{L_{k}M}{L_{k+1}M}\right]_{\mathbf{v}+n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}}$

and consequently the equality

(5) ${\rm length}_{R}\left(E_{\mathbf{v},\mathbf{n}}\right)={\rm
length}_{R}\left(\frac{[M]_{\mathbf{v}+n_{1}\mathbf{d}_{1}+\cdots+n_{q}\mathbf{d}_{q}}}{H_{1}^{n_{1}+1}\cdots
H_{q}^{n_{q}+1}\,[M]_{\mathbf{v}-\mathbf{d}_{1}-\cdots-\mathbf{d}_{q}}}\right)=\lambda_{M}^{H_{1},\ldots,H_{q}}(\mathbf{v}-\mathbf{d}_{1}-\cdots-\mathbf{d}_{q},\mathbf{n}+\mathbf{1}).$

Finally, (missing) 4 and (missing) 5 yield the equality
$\lambda_{M}^{H_{1},\ldots,H_{q}}(\mathbf{v},\mathbf{n})=P_{\mathcal{F}}(\mathbf{v}+\mathbf{d}_{1}+\cdots+\mathbf{d}_{q},\mathbf{n}-\mathbf{1})$
for all
$\mathbf{v}\geq\mathbf{v}^{\prime}-\mathbf{d}_{1}-\cdots-\mathbf{d}_{q}$ and
$\mathbf{n}\geq\mathbf{n}^{\prime}+\mathbf{1}$. The polynomials
$P_{\mathcal{F}}(\mathbf{v}+\mathbf{d}_{1}+\cdots+\mathbf{d}_{q},\mathbf{n}-\mathbf{1})$
and $P_{\mathcal{F}}(\mathbf{v},\mathbf{n})$ have the same top degree part.
Therefore, by setting
$P_{M}^{H_{1},\ldots,H_{q}}(\mathbf{v},\mathbf{n}):=P_{\mathcal{F}}(\mathbf{v}+\mathbf{d}_{1}+\cdots+\mathbf{d}_{q},\mathbf{n}-\mathbf{1})$,
the proof of the theorem is complete. ∎

Finally, after having the result of Theorem 2.1, we can make the following
definition.

######  Definition .

Under the notation and assumptions of Theorem 2.1, for any
$\alpha\in\mathbb{N}^{p}$ and $\beta\in\mathbb{N}^{q}$ with
$|\alpha|+|\beta|\geq r$, we say that the _mixed Buchsbaum–Rim multiplicity of
$M$ of type $(\alpha,\beta)$ with respect to $H_{1},\ldots,H_{q}$_ is the
following nonnegative integer

$\operatorname{{\rm
br}}_{\alpha,\beta}\left(H_{1},\ldots,H_{q};M\right)\;:=\;\begin{cases}e(\alpha,\beta)&\;\;\text{
if }|\alpha|+|\beta|=r\\\ 0&\;\;\text{ if }|\alpha|+|\beta|>r.\end{cases}$

Notice that $\operatorname{{\rm
br}}_{\alpha,\beta}\left(H_{1},\ldots,H_{q};M\right)$ can be defined either in
geometrical terms as an intersection number associated to the Kleiman–Thorup
transform $P(\mathscr{M})$ or in algebraic fashion by utilizing the Hilbert-
like function $\lambda_{M}^{H_{1},\ldots,H_{q}}$. For any
$\beta\in\mathbb{N}^{p}$ with $|\beta|\geq r$, we also define

$\operatorname{{\rm br}}_{\beta}(H_{1},\ldots,H_{q};M)\;:=\;\operatorname{{\rm
br}}_{\mathbf{0},\beta}\left(H_{1},\ldots,H_{q};M\right)\;=\;\int_{P}c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{1})\right)^{\beta_{1}}\cdots
c_{1}\left(\mathcal{O}_{P}({\mathbf{e}}_{q})\right)^{\beta_{q}}\,\cap\,\left[P(\mathscr{M})\right]_{r}.$

##  3\. Mixed $j$-multiplicities

In this short section, we quickly develop some basic results regarding a
multigraded version of _$j$ -multiplicities_ (see [ACHILLES_MANARESI_J_MULT],
[FLENNER_O_CARROLL_VOGEL, §6.1]). These results will be needed in the next
section. The setup below is used throughout this section.

######  Setup .

Let $(R,{\mathfrak{m}})$ be a Noetherian local ring and $B$ be a standard
$\mathbb{N}^{p}$-graded $R$-algebra. Set $X=\text{MultiProj}(B)$.

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $B$-module and
$\mathscr{M}:=\widetilde{M}$ be the corresponding coherent
$\mathcal{O}_{X}$-module. Recall that
$\dim(\mathscr{M})=\dim\big{(}M/\text{H}_{B_{++}}^{0}(M)\big{)}-p$ (see, e.g.,
[HYRY_MULTIGRAD, §1], [SPECIALIZATION_ARON, §3.1]). Notice that
$\text{H}_{{\mathfrak{m}}}^{0}(M)$ is finitely generated over
$B/{\mathfrak{m}}^{k}B$ for $k\gg 0$, and this latter ring is a standard
$\mathbb{N}^{p}$-graded algebra over the Artinian local ring
$R/{\mathfrak{m}}^{k}$. Then we may define the mixed $j$-multiplicities of $M$
as the mixed multiplicities of $\text{H}_{{\mathfrak{m}}}^{0}(M)$. For details
on the notion of _mixed multiplicities_ , the reader is referred to
[MIXED_MULT, HERMANN_MULTIGRAD]. Fix $\beta\in\mathbb{N}^{p}$ with
$|\beta|\geq\dim\big{(}\text{Supp}(\mathscr{M})\big{)}$.

######  Definition .

The _mixed $j$-multiplicity of $M$ of type $\beta$_ is given by
$j_{\beta}(M):=e\big{(}\beta;\,\text{H}_{{\mathfrak{m}}}^{0}(M)\big{)}.$

We have the following expected results (cf., [FLENNER_O_CARROLL_VOGEL, §6.1]).

######  Lemma .

Let $0\rightarrow M^{\prime}\rightarrow M\rightarrow
M^{\prime\prime}\rightarrow 0$ be a short exact sequence of finitely generated
$\mathbb{Z}^{p}$-graded $B$-modules. Then we have the equality
$j_{\beta}(M)\;=\;j_{\beta}(M^{\prime})+j_{\beta}(M^{\prime\prime}).$

###### Proof.

The proof follows similarly to [FLENNER_O_CARROLL_VOGEL, Proposition 6.1.2].
We have an exact sequence
$0\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(M^{\prime})\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(M)\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(M^{\prime\prime})$.
Let
$Q:=\text{Coker}\left(\text{H}_{{\mathfrak{m}}}^{0}(M)\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(M^{\prime\prime})\right)$.
For any minimal prime ${\mathfrak{p}}$ of $M$ that belongs to the closed fiber
$X\times_{\text{Spec}(R)}\text{Spec}(R/{\mathfrak{m}})$, we obtain
$M_{\mathfrak{p}}^{\prime}=\text{H}_{{\mathfrak{m}}}^{0}(M_{\mathfrak{p}}^{\prime})$,
$M_{\mathfrak{p}}=\text{H}_{{\mathfrak{m}}}^{0}(M_{\mathfrak{p}})$ and
$M_{\mathfrak{p}}^{\prime\prime}=\text{H}_{{\mathfrak{m}}}^{0}(M_{\mathfrak{p}}^{\prime\prime})$.
Hence $\dim(\text{Supp}(\widetilde{Q}))<\dim(\text{Supp}(\mathscr{M}))$, and
the result follows from the additivity of mixed multiplicities. ∎

######  Corollary .

We have that $j_{\beta}(M)=j_{\beta}\big{(}M/\text{H}_{B_{++}}^{0}(M)\big{)}$.

###### Proof.

Since $j_{\beta}\big{(}\text{H}_{B_{++}}^{0}(M)\big{)}=0$, the result follows
from Section missing 3. ∎

######  Lemma .

We have the equality

$j_{\beta}(M)\;=\;\sum_{{\mathfrak{p}}}{\rm
length}_{B_{\mathfrak{p}}}(M_{\mathfrak{p}})\cdot e(\beta;B/{\mathfrak{p}})$

where ${\mathfrak{p}}$ runs over the minimal primes of $M$ that belong to the
closed fiber $X\times_{\text{Spec}(R)}\text{Spec}(R/{\mathfrak{m}})$ and
satisfy $\dim(B/{\mathfrak{p}})=|\beta|+p$.

###### Proof.

The proof is similar to [FLENNER_O_CARROLL_VOGEL, Proposition 6.1.3]. However
we now use the additivity result of Section missing 3. ∎

##  4\. New joint blow-up constructions

This section introduces new joint blow-up constructions that will be our
second main ingredient to define relative mixed multiplicities. The setup
below is used throughout this section.

######  Setup .

Let $B$ be a standard $\mathbb{N}^{p}$-graded algebra over a Noetherian local
ring $R=\left[B\right]_{(0,\ldots,0)}$ with maximal ideal
${\mathfrak{m}}\subset R$. For each $1\leq i\leq p$, let
$H_{i}\subseteq[B]_{{\mathbf{e}}_{i}}$ be a nonzero $R$-submodule, and
$I_{i}=\left(H_{i}\right)\subset B$ be the ideal generated by $H_{i}$. Let
$I=I_{1}+\cdots+I_{p}\subset B$ be the ideal given as the sum of the ideals
$I_{i}$. Let $X=\text{MultiProj}(B)$.

We consider the multi-Rees algebra

$\mathscr{R}\;=\;\mathscr{R}\left(I_{1},\ldots,I_{p}\right)\;:=\;\bigoplus_{(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\;T_{1}^{n_{1}}\cdots
T_{p}^{n_{p}}\;\subset\;B[T_{1},\ldots,T_{p}],$

where $T_{1},\ldots,T_{p}$ are new variables. We endow $\mathscr{R}$ with a
standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-grading. For any homogeneous
element $b\in B$, we set
$\deg_{\mathscr{R}}(b)=\deg_{B}(b)\oplus\mathbf{0}\in\mathbb{N}^{p}\oplus\mathbb{N}^{p},$
and for any nonzero element $h\in H_{i}$, we set
$\deg_{\mathscr{R}}(hT_{i})=\mathbf{0}\oplus{\mathbf{e}}_{i}\in\mathbb{N}^{p}\oplus\mathbb{N}^{p}.$
This implies that $\mathscr{R}$ is generated by elements of total degree one,
and thus it is standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded as an
$R$-algebra.

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $B$-module and
$\mathscr{M}:=\widetilde{M}$ be the corresponding coherent
$\mathcal{O}_{X}$-module. Then we obtain the following finitely generated
$(\mathbb{Z}^{p} \oplus\mathbb{N}^{p})$-graded $\mathscr{R}$-module

$\mathscr{R}_{M}\;=\;\mathscr{R}\left(I_{1},\ldots,I_{p};M\right)\;:=\;\bigoplus_{(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M\;T_{1}^{n_{1}}\cdots
T_{p}^{n_{p}}\;\subset\;M[T_{1},\ldots,T_{p}].$

For any $\mathbf{v}=(v_{1},\ldots,v_{p})\in\mathbb{Z}^{p}$ and
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$, the corresponding
$(\mathbf{v},{\mathbf{n}})$-graded part of $\mathscr{R}_{M}$ is given by

$\left[\mathscr{R}_{M}\right]_{\mathbf{v},{\mathbf{n}}}\;=\;\left[I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M\right]_{\mathbf{v}+{\mathbf{n}}}.$

We now introduce our new joint blow-up constructions. We take the Rees algebra
of the ideal $I\mathscr{R}$ over the previously defined multi-Rees algebra
$\mathscr{R}$, that is:

$\mathfrak{C}\;=\;\mathfrak{C}(I_{1},\ldots,I_{p})\;:=\;\mathscr{R}\left(I\mathscr{R}\right)\;=\;\bigoplus_{k=0}^{\infty}\,I^{k}\mathscr{R}\,T^{k}\;\subset\;\mathscr{R}[T].$

We put a standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-grading on
$\mathfrak{C}$ by naturally extending the one of $\mathscr{R}$. We say that
$\mathfrak{C}$ has an “internal grading” induced by $\mathscr{R}$. For any
homogeneous element $w\in\mathscr{R}$, we set
$\deg_{\mathfrak{C}}(w)=\deg_{\mathscr{R}}(w)$, and for any nonzero element
$h\in H_{i}\subset I$, we set $\deg_{\mathfrak{C}}(hT)=\deg_{\mathscr{R}}(h).$
Hence $\mathfrak{C}$ is generated by elements of total degree one, and so it
is also standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded as an
$R$-algebra. The corresponding associated graded ring

$\mathfrak{G}\;=\;\mathfrak{G}(I_{1},\ldots,I_{p})\;:=\;\operatorname{gr}_{I\mathscr{R}}(\mathscr{R})\;=\;\mathfrak{C}/I\mathfrak{C}\;=\;\bigoplus_{k=0}^{\infty}\,\frac{I^{k}\mathscr{R}}{I^{k+1}\mathscr{R}}$

also obtains a standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-grading as an
$R$-algebra. We have the following finitely generated
$(\mathbb{Z}^{p}\oplus\mathbb{N}^{p})$-graded modules

$\mathfrak{C}_{M}\;=\;\mathfrak{C}(I_{1},\ldots,I_{p};M)\;:=\;\mathscr{R}\left(I\mathscr{R};\mathscr{R}_{M}\right)\;=\;\bigoplus_{k=0}^{\infty}\,I^{k}\mathscr{R}_{M}T^{k}\;\subset\;\mathscr{R}_{M}[T]$

and

$\mathfrak{G}_{M}\;=\;\mathfrak{G}(I_{1},\ldots,I_{p};M)\;:=\;\mathfrak{C}_{M}/I\mathfrak{C}_{M}\;=\;\bigoplus_{k=0}^{\infty}\,\frac{I^{k}\mathscr{R}_{M}}{I^{k+1}\mathscr{R}_{M}}$

over $\mathfrak{C}$ and $\mathfrak{G}$, respectively.

Let $\mathfrak{X}=\text{MultiProj}(\mathfrak{G})$ be the multiprojective
scheme determined by $\mathfrak{G}$ (which can naturally be embedded in a
product of the form
${\mathbb{P}}_{R}^{m_{1}}\times_{R}\cdots\times_{R}{\mathbb{P}}_{R}^{m_{2p}}$),
and $\widetilde{\mathfrak{G}_{M}}$ be the corresponding coherent
$\mathcal{O}_{\mathfrak{X}}$-module.

######  Lemma .

We have the inequality
$\dim(\text{Supp}(\widetilde{\mathfrak{G}_{M}}))\leq\dim(\text{Supp}(\mathscr{M}))$.

###### Proof.

Let $\overline{M}=M/\text{H}_{B_{++}}^{0}(M)$. Notice that the
$(\mathbf{v},\mathbf{n})$-graded part of $\mathfrak{G}_{M}$ is given by

$\displaystyle\big{[}\mathfrak{G}_{M}\big{]}_{\mathbf{v},{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{\infty}\left[\frac{I^{k}\mathscr{R}_{M}}{I^{k+1}\mathscr{R}_{M}}\right]_{\mathbf{v},{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{\infty}\left[\frac{I^{k}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M}{I^{k+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M}\right]_{\mathbf{v}+{\mathbf{n}}}\;=\;\bigoplus_{k=0}^{|\mathbf{v}|-{\rm
indeg}(M)}\left[\frac{I^{k}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M}{I^{k+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}M}\right]_{\mathbf{v}+{\mathbf{n}}}$

where ${\rm indeg}(M):=\inf\big{\\{}v\in\mathbb{Z}\mid[M]_{\mathbf{v}}\neq
0\text{ for some $\mathbf{v}\in\mathbb{Z}^{p}$ with
$|\mathbf{v}|=v$}\big{\\}}$. Since
$[M]_{\mathbf{v}}\cong\left[\overline{M}\right]_{\mathbf{v}}$ for all
$\mathbf{v}\gg\mathbf{0}$, it follows that
$\big{[}\mathfrak{G}_{M}\big{]}_{\mathbf{v},{\mathbf{n}}}\cong\big{[}\mathfrak{G}_{\overline{M}}\big{]}_{\mathbf{v},{\mathbf{n}}}$
for all $\mathbf{v}\gg\mathbf{0}$ and $\mathbf{n}\gg\mathbf{0}$. Therefore
$\widetilde{\mathfrak{G}_{M}}\cong\widetilde{\mathfrak{G}_{\overline{M}}}$. We
have $\dim(\mathfrak{G}_{\overline{M}})\leq\dim(\overline{M})+p$ (see, e.g,
[huneke2006integral, Chapter 5]). Hence we obtain

$\dim(\text{Supp}(\widetilde{\mathfrak{G}_{M}}))\;\leq\;\dim(\mathfrak{G}_{\overline{M}})-2p\;\leq\;\dim(\overline{M})-p\;=\;\dim(\text{Supp}(\mathscr{M})),$

as required. ∎

As a consequence of the above dimension computation, we can introduce the
following invariants. Similar invariants were introduced in the
$\mathbb{N}$-graded case by Ulrich and Validashti
[ULRICH_VALIDASHTI_INT_DEP_MOD].

######  Definition .

For any $\alpha,\beta\in\mathbb{N}^{p}$ with
$|\alpha|+|\beta|\geq\dim(\text{Supp}(\mathscr{M}))$, we consider the
following mixed $j$-multiplicities

$j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}\;:=\;j_{\alpha,\beta}(\mathfrak{G}_{M})\quad\text{
and }\quad
j_{\alpha,\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};M\big{)}\;:=\;j_{\alpha,\beta}(B_{+}\mathfrak{G}_{M});$

see Section missing 3.

We also set
$j_{\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}:=j_{\mathbf{0},\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}$
and
$j_{\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};M\big{)}:=j_{\mathbf{0},\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};M\big{)}$
for any $\beta\in\mathbb{N}^{p}$ with
$|\beta|\geq\dim(\text{Supp}(\mathscr{M}))$.

The next remark tells us that under the conditions needed to define relative
mixed multiplicities, we have that the mixed $j$-multiplicities of
$B_{+}\mathfrak{G}_{M}$ are just mixed multiplicities.

######  Remark .

Assume ${\rm length}_{R}\left([B]_{{\mathbf{e}}_{i}}/H_{i}\right)<\infty$ for
all $1\leq i\leq p$. Then
$\overline{\mathfrak{G}}:=\mathfrak{G}/(0:_{\mathfrak{G}}B_{+}\mathfrak{G})$
is a standard $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded algebra over the
Artinian local ring
$\Big{[}\overline{\mathfrak{G}}\Big{]}_{\mathbf{0},\mathbf{0}}=R/\mathfrak{a}$
with
$\mathfrak{a}=\bigcap_{i=1}^{p}\text{Ann}_{R}\left([B]_{{\mathbf{e}}_{i}}/H_{i}\right)$.
Notice that $B_{+}\mathfrak{G}_{M}$ is always a finitely generated
$(\mathbb{Z}^{p}\oplus\mathbb{N}^{p})$-graded module over
$\overline{\mathfrak{G}}$. Therefore, we obtain that
$j_{\alpha,\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};M\big{)}=e(\alpha,\beta;B_{+}\mathfrak{G}_{M})$.

The following theorem shows the existence of a polynomial that will be
essential to our approach to defining relative mixed multiplicities. This
existence result is a byproduct of considering the above joint blow-up
constructions. The leading coefficients of this polynomial will coincide with
the mixed multiplicities of $B_{+}\mathfrak{G}_{\mathscr{H}}$ for a certain
ideal $\mathscr{H}\subset B$.

######  Theorem 4.1.

Assume Section missing 4 and that ${\rm
length}_{R}\left([B]_{{\mathbf{e}}_{i}}/H_{i}\right)<\infty$ for all $1\leq
i\leq p$. Set $r=\dim(X)$. Fix a tuple
$\mathbf{t}=(t_{1},\ldots,t_{p})\in\mathbb{N}^{p}$ of nonnegative integers,
and let $\mathscr{H}_{\mathbf{t}}\subset B$ be the ideal generated by the
elements in the graded part $[B]_{\mathbf{t}}$. Then the following function

$\lambda_{\mathbf{t}}^{H_{1},\ldots,H_{p}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{p})={\rm
length}_{R}\left(\frac{{H_{1}}^{n_{1}}\cdots{H_{p}}^{n_{p}}\,[B]_{(v_{1},\ldots,v_{p})}}{H_{1}^{v_{1}+n_{1}-t_{1}}\cdots
H_{p}^{v_{p}+n_{p}-t_{p}}\,[B]_{(t_{1},\ldots,t_{p})}}\right)$

coincides with a polynomial
$P_{\mathbf{t}}^{H_{1},\ldots,H_{p}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{p})$
for all $v_{i}\gg 0$ and $n_{i}\gg 0$. Moreover,
$P_{\mathbf{t}}^{H_{1},\ldots,H_{p}}$ has total degree at most $r$ and we can
write

$P_{\mathbf{t}}^{H_{1},\ldots,H_{p}}(v_{1},\ldots,v_{p},n_{1},\ldots,n_{p})\;=\;\sum_{\begin{subarray}{c}\alpha\in\mathbb{N}^{p},\beta\in\mathbb{N}^{p}\\\
|\alpha|+|\beta|=r\end{subarray}}e\left(\alpha,\beta\right)\,\frac{v_{1}^{\alpha_{1}}\cdots
v_{p}^{\alpha_{p}}n_{1}^{\beta_{1}}\cdots
n_{p}^{\beta_{p}}}{\alpha_{1}!\cdots\alpha_{p}!\beta_{1}!\cdots\beta_{p}!}\;+\;\text{\rm(lower
degree terms)}$

where
$e(\alpha,\beta):=j_{\alpha,\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};M\big{)}=e\left(\alpha,\beta;B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\right)$
is the mixed multiplicity of $B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}$ of
type $(\alpha,\beta)$.

###### Proof.

We may assume that $\mathbf{v}>\mathbf{t}$. Let $c=|\mathbf{v}|-|\mathbf{t}|$.
We study the finitely generated $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded
$\mathfrak{G}$-module $B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}$. From the
above conventions, the $(\mathbf{v},{\mathbf{n}})$-graded part is given by

$\displaystyle\big{[}B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\big{]}_{\mathbf{v},{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{\infty}\left[B_{+}\cdot\frac{I^{k}\mathscr{R}_{\mathscr{H}_{\mathbf{t}}}}{I^{k+1}\mathscr{R}_{\mathscr{H}_{\mathbf{t}}}}\right]_{\mathbf{v},{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{\infty}\left[B_{+}\cdot\frac{I^{k}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}{I^{k+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}\right]_{\mathbf{v}+{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{c-1}\left[B_{+}\cdot\frac{I^{k}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}{I^{k+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}\right]_{\mathbf{v}+{\mathbf{n}}}$
$\displaystyle\;=\;\bigoplus_{k=0}^{c-1}\left[\frac{I^{k}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}{I^{k+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}}\right]_{\mathbf{v}+{\mathbf{n}}}.$

Thus, we obtain the equality

${\rm
length}_{R}\left(\big{[}B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\big{]}_{\mathbf{v},{\mathbf{n}}}\right)\;=\;{\rm
length}_{R}\left(\frac{\left[I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}\right]_{\mathbf{v}+{\mathbf{n}}}}{\left[I^{c}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}\right]_{\mathbf{v}+{\mathbf{n}}}}\right).$

It is clear that

$\left[I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}\right]_{\mathbf{v}+{\mathbf{n}}}\;=\;H_{1}^{n_{1}}\cdots
H_{p}^{n_{p}}[B]_{\mathbf{v}}.$

On the other hand, since

$I^{c}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\;=\;\sum_{j_{1}+\cdots+j_{p}=c}I_{1}^{n_{1}+j_{1}}\cdots
I_{p}^{n_{p}+j_{p}}$

and $\big{[}I_{1}^{n_{1}+j_{1}}\cdots
I_{p}^{n_{p}+j_{1}}\mathscr{H}_{\mathbf{t}}\big{]}_{\mathbf{v}+{\mathbf{n}}}=0$
when any $j_{i}>v_{i}-t_{i}$, by the pigeonhole principle it follows that

$\left[I^{c}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\mathscr{H}_{\mathbf{t}}\right]_{\mathbf{v}+{\mathbf{n}}}\;=\;H_{1}^{v_{1}+n_{1}-t_{1}}\cdots
H_{p}^{v_{p}+n_{p}-t_{p}}[B]_{\mathbf{t}}.$

Then the result follows from the existence of (multigraded) Hilbert
polynomials (see, e.g., [KLEIMAN_THORUP_GEOM, Lemma 4.3], [HERMANN_MULTIGRAD,
Theorem 4.1], [MIXED_MULT, Theorem 3.4]). Indeed, there is a polynomial
$P_{\mathbf{t}}^{H_{1},\ldots,H_{q}}(\mathbf{v},\mathbf{n}):=P_{B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}}(\mathbf{v},\mathbf{n})$
of total degree equal to
$\dim(\text{Supp}(\widetilde{B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}}))$
such that

$P_{\mathbf{t}}^{H_{1},\ldots,H_{p}}(\mathbf{v},\mathbf{n})\;=\;P_{B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}}(\mathbf{v},\mathbf{n})\;=\;{\rm
length}_{R}\big{(}\big{[}B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\big{]}_{\mathbf{v},{\mathbf{n}}}\big{)}\;=\;\lambda_{\mathbf{t}}^{H_{1},\ldots,H_{p}}(\mathbf{v},\mathbf{n})$

for all $\mathbf{v}\gg\mathbf{0}$ and $\mathbf{n}\gg\mathbf{0}$. Finally, we
have the upper bound
$\dim(\text{Supp}(\widetilde{B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}}))\leq
r=\dim(X)$ (see Section missing 4). ∎

We now show an additivity result for the mixed $j$-multiplicities
$j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}$. Our proof is inspired by
the arguments used in the proofs of [FLENNER_O_CARROLL_VOGEL, Theorem 1.2.6]
and [FLENNER_O_CARROLL_VOGEL, Proposition 6.1.7]. We also benefited from the
exposition in [ULRICH_VALIDASHTI_INT_DEP_MOD].

######  Proposition .

Let $0\rightarrow M^{\prime}\rightarrow M\rightarrow
M^{\prime\prime}\rightarrow 0$ be a short exact sequence of finitely generated
$\mathbb{Z}^{p}$-graded $B$-modules. Set $r=\dim(\text{Supp}(\mathscr{M}))$.
Then, for all $\alpha,\beta\in\mathbb{N}^{p}$ with $|\alpha|+|\beta|\geq r$,
we have the equality

$j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}\;=\;j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M^{\prime}\big{)}+j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M^{\prime\prime}\big{)}.$

###### Proof.

We consider the extended Rees algebra

$\mathscr{R}^{+}(I\mathscr{R})\;:=\;\mathscr{R}\left[I\mathscr{R}T,T^{-1}\right]\;=\;\bigoplus_{k\in\mathbb{Z}}I^{k}\mathscr{R}T^{k}\;\subset\;\mathscr{R}[T,T^{-1}].$

By following our grading conventions, we obtain that
$\mathscr{R}^{+}(I\mathscr{R})$ is a standard
$(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded algebra over $R[T^{-1}]$. We
take the localization
$\mathfrak{C}^{+}:=\mathscr{R}^{+}(I\mathscr{R})\otimes_{R[T^{-1}]}R^{\prime}$
with $R^{\prime}:=R[T^{-1}]_{({\mathfrak{m}},T^{-1})R[T^{-1}]}$. Therefore, we
have that $\mathfrak{C}^{+}$ is a standard
$(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded algebra over the local ring
$R^{\prime}$, and so we are free to compute mixed $j$-multiplicities over
$\mathfrak{C}^{+}$ (see Section missing 3). We also consider the extended
Rees-module

$\mathscr{R}^{+}(I\mathscr{R};\mathscr{R}_{M})\;:=\;\bigoplus_{k\in\mathbb{Z}}I^{k}\mathscr{R}_{M}T^{k}\;\subset\;\mathscr{R}_{M}[T,T^{-1}]$

and take the localization
$\mathfrak{C}_{M}^{+}:=\mathscr{R}^{+}(I\mathscr{R};\mathscr{R}_{M})\otimes_{R[T^{-1}]}R^{\prime}$.

Let $\mathscr{Y}=\text{MultiProj}(\mathfrak{C}^{+})$ be the multiprojective
scheme associated to $\mathfrak{C}^{+}$. Let
$N:=\text{Ker}\left(\mathfrak{C}_{M}^{+}\rightarrow\mathfrak{C}_{M^{\prime\prime}}^{+}\right)$
and $L:=N/\mathfrak{C}_{M^{\prime}}^{+}$. We have the following commutative
diagram with exact rows and columns of finitely generated
$(\mathbb{Z}^{p}\oplus\mathbb{N}^{p})$-graded $\mathfrak{C}^{+}$-modules

${0}$${0}$${0}$${0}$${N}$${\mathfrak{C}_{M}^{+}}$${\mathfrak{C}_{M^{\prime\prime}}^{+}}$${0}$${0}$${N}$${\mathfrak{C}_{M}^{+}}$${\mathfrak{C}_{M^{\prime\prime}}^{+}}$${0.}$$T^{-1}$$T^{-1}$$T^{-1}$

The snake lemma yields the short exact sequence

(6) $0\longrightarrow
N/T^{-1}N\longrightarrow\mathfrak{G}_{M}\longrightarrow\mathfrak{G}_{M^{\prime\prime}}\longrightarrow
0.$

Let $U$ and $V$ be the kernel and cokernel of the multiplication map
$L\xrightarrow{T^{-1}}L$, respectively. Hence we consider the exact sequence

(7) $0\longrightarrow U\longrightarrow L\xrightarrow{\quad
T^{-1}\quad}L\longrightarrow V\longrightarrow 0.$

We have the following commutative diagram with exact rows and columns

${0}$${0}$${0}$${U}$${0}$${\mathfrak{C}_{M^{\prime}}^{+}}$${N}$${L}$${0}$${0}$${\mathfrak{C}_{M^{\prime}}^{+}}$${N}$${L}$${0}$${V}$${0.}$$T^{-1}$$T^{-1}$$T^{-1}$

Again, by utilizing the snake lemma, we get the exact sequence

(8) $0\longrightarrow U\longrightarrow\mathfrak{G}_{M^{\prime}}\longrightarrow
N/T^{-1}N\longrightarrow V\longrightarrow 0.$

Since we have the following explicit description

$\frac{\text{Ker}\big{(}\mathscr{R}^{+}(I\mathscr{R};\mathscr{R}_{M})\rightarrow\mathscr{R}^{+}(I\mathscr{R};\mathscr{R}_{M^{\prime\prime}})\big{)}}{\mathscr{R}^{+}(I\mathscr{R};\mathscr{R}_{M^{\prime}})}\;\cong\;\bigoplus_{k\in\mathbb{Z},\,n_{1},\ldots,n_{p}\geq
0}\frac{I^{k}I_{1}^{n_{1}}\cdots I_{p}^{n_{p}}M\cap
M^{\prime}}{I^{k}I_{1}^{n_{1}}\cdots I_{p}^{n_{p}}M^{\prime}},$

the multi-ideal version of the Artin-Rees lemma [huneke2006integral, Theorem
17.1.6] implies that $(T^{-1}I_{1}\cdots I_{p})^{c}\cdot L=0$ for some $c\geq
0$. Consider the coherent $\mathcal{O}_{\mathscr{Y}}$-modules $\widetilde{L}$
and $\mathchanc{Q}=\widetilde{\mathfrak{C}_{M}^{+}/T^{-1}I_{1}\cdots
I_{p}\mathfrak{C}_{M}^{+}}$. It then follows that

$\dim(\text{Supp}(\widetilde{L}))\;\leq\;\dim(\text{Supp}(\mathchanc{Q}))\;\leq\;r=\dim(\text{Supp}(\mathscr{M}));$

we can proceed as in Section missing 4.

Finally, by utilizing the additivity result of Section missing 3 and combining
(missing) 6, (missing) 7 and (missing) 8, we obtain the required equality
$j_{\alpha,\beta}(\mathfrak{G}_{M})=j_{\alpha,\beta}(\mathfrak{G}_{M^{\prime}})+j_{\alpha,\beta}(\mathfrak{G}_{M^{\prime\prime}})$.
∎

######  Corollary .

The following statements holds:

  1. (i)

$j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};M\big{)}=j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};\overline{M}\big{)}$
for all $\alpha,\beta\in\mathbb{N}^{p}$ with
$|\alpha|+|\beta|\geq\dim(\text{Supp}(\mathscr{M}))$, where
$\overline{M}=M/\text{H}_{B_{++}}^{0}(M)$.

  2. (ii)

$j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};\mathscr{H}_{\mathbf{t}}\big{)}=j_{\alpha,\beta}\big{(}H_{1},\ldots,H_{p};B\big{)}$
for all $\mathbf{t},\alpha,\beta\in\mathbb{N}^{p}$ with
$|\alpha|+|\beta|\geq\dim(X)$.

###### Proof.

Both statements follow from Section missing 4 and Section missing 4. ∎

Finally, we have the following nonincreasing behavior for the mixed
$j$-multiplicities $j_{\alpha,\beta}^{\\#}$.

######  Corollary .

Let $\mathbf{t},\mathbf{t}^{\prime}\in\mathbb{N}^{p}$ such that
$\mathbf{t}^{\prime}\geq\mathbf{t}$. Then, for all
$\alpha,\beta\in\mathbb{N}^{p}$ with $|\alpha|+|\beta|\geq\dim(X)$, we have

$j_{\alpha,\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};\mathscr{H}_{\mathbf{t}^{\prime}}\big{)}\;\leq\;j_{\alpha,\beta}^{\\#}\big{(}H_{1},\ldots,H_{p};\mathscr{H}_{\mathbf{t}}\big{)}.$

###### Proof.

By Section missing 4 and Section missing 3, we may assume that
$(0:_{B}B_{++})=0$ (i.e., $\operatorname{grade}(B_{++})>0$) and it suffices to
show that

$j_{\alpha,\beta}\big{(}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}}\big{)}\;\geq\;j_{\alpha,\beta}\big{(}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\big{)}.$

Notice that
$\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\cong\mathscr{R}(I,I_{1},\ldots,I_{p};\mathscr{H}_{\mathbf{t}})/B_{+}\mathscr{R}(I,I_{1},\ldots,I_{p};\mathscr{H}_{\mathbf{t}})$
is a (relative) fiber cone and that we have the following explicit description

$\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\;=\;\bigoplus_{\mathbf{v},\mathbf{n}\in\mathbb{N}^{p}}\big{[}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\big{]}_{\mathbf{v},\mathbf{n}}\;=\;\bigoplus_{\mathbf{v},\mathbf{n}\in\mathbb{N}^{p},\mathbf{v}\geq\mathbf{t}}H_{1}^{v_{1}+n_{1}-t_{1}}\cdots
H_{p}^{v_{p}+n_{p}-t_{p}}[B]_{\mathbf{t}};$

see the computations in the proof of Theorem 4.1. Consider the standard
$(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded $R$-algebra
$S:=\mathfrak{G}/B_{+}\mathfrak{G}$. By using the faithfully flat extension
$R\rightarrow R^{\prime}=R[z]_{{\mathfrak{m}}R[z]}$ and making the base change
$B\rightarrow B\otimes_{R}R^{\prime}$, we may assume that the residue field of
$R$ is infinite. Then we can find an element
$x\in[B]_{\mathbf{t}^{\prime}-\mathbf{t}}$ which is a nonzerodivisor (see,
e.g., [BRUNS_HERZOG, Proposition 1.5.12]), and so multiplication by $x$ yields
the following inclusion

${\Big{(}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}\Big{)}(\mathbf{t}-\mathbf{t}^{\prime},\mathbf{0})}$${\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}}}$$x$

of finitely generated $(\mathbb{N}^{p}\oplus\mathbb{N}^{p})$-graded
$S$-modules. Finally, Section missing 3 yields the required inequality
$j_{\alpha,\beta}(\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}}})\leq
j_{\alpha,\beta}(\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}}/B_{+}\mathfrak{G}_{\mathscr{H}_{\mathbf{t}^{\prime}}})$.
∎

##  5\. Relative mixed multiplicities

In this section, we introduce the notion of relative mixed multiplicities. Our
existence proof follows by combining Theorem 2.1 and Theorem 4.1. The setup
below is used during this section.

######  Setup .

Let $A\subseteq B$ be an inclusion of standard $\mathbb{N}^{p}$-graded
algebras over a Noetherian local ring $R=[A]_{(0,\ldots 0)}=[B]_{(0,\ldots
0)}$. Assume that ${\rm
length}_{R}\left([B]_{{\mathbf{e}}_{i}}/[A]_{{\mathbf{e}}_{i}}\right)<\infty$
for all $1\leq i\leq p$. Let $X=\text{MultiProj}(B)$ and $r=\dim(X)$.

The theorem below is the main result of this section.

######  Theorem 5.1.

Assume Section missing 5. Fix a tuple
$\mathbf{t}=(t_{1},\ldots,t_{p})\in\mathbb{Z}_{+}^{p}$ of positive integers.
Then the following function

$\lambda_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})\;=\;{\rm
length}_{R}\left(\frac{\left[B\right]_{(n_{1},\ldots,n_{p})}}{\left[A\right]_{(n_{1}-t_{1}+1,\ldots,n_{p}-t_{p}+1)}\left[B\right]_{(t_{1}-1,\ldots,t_{p}-1)}}\right)$

coincides with a polynomial $P_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})$ for all
$n_{i}\gg 0$. Moreover, $P_{\mathbf{t}}^{A,B}$ has total degree at most $r$
and its normalized leading coefficients are nonnegative integers.

###### Proof.

The proof follows by using Theorem 2.1, Theorem 4.1 and the following short
exact sequence

$0\;\longrightarrow\;\frac{[A]_{\mathbf{n}}[B]_{\mathbf{v}}}{[A]_{\mathbf{v}+\mathbf{n}-\mathbf{t}+\mathbf{1}}[B]_{\mathbf{t}-\mathbf{1}}}\;\longrightarrow\;\frac{[B]_{\mathbf{v}+\mathbf{n}}}{[A]_{\mathbf{v}+\mathbf{n}-\mathbf{t}+\mathbf{1}}[B]_{\mathbf{t}-\mathbf{1}}}\;\longrightarrow\;\frac{[B]_{\mathbf{v}+\mathbf{n}}}{[A]_{\mathbf{n}}[B]_{\mathbf{v}}}\;\longrightarrow\;0.$

From Theorem 4.1, we obtain a polynomial
$P_{\mathbf{t}-\mathbf{1}}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})$
of total degree at most $r$ that satisfies the equality

$P_{\mathbf{t}-\mathbf{1}}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})\;=\;{\rm
length}_{R}\left(\frac{[A]_{\mathbf{n}}[B]_{\mathbf{v}}}{[A]_{\mathbf{v}+\mathbf{n}-\mathbf{t}+\mathbf{1}}[B]_{\mathbf{t}-\mathbf{1}}}\right)$

for all $\mathbf{v}\gg\mathbf{0}$ and $\mathbf{n}\gg\mathbf{0}$. Similarly,
Theorem 2.1 gives a polynomial
$P_{B}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})$
of total degree at most $r$ such that

$P_{B}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})\;=\;{\rm
length}_{R}\left(\frac{[B]_{\mathbf{v}+\mathbf{n}}}{[A]_{\mathbf{n}}[B]_{\mathbf{v}}}\right)$

for all $\mathbf{v}\gg\mathbf{0}$ and $\mathbf{n}\gg\mathbf{0}$. Consider the
polynomial
$P(\mathbf{v},\mathbf{n}):=P_{\mathbf{t}-\mathbf{1}}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})+P_{B}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{v},\mathbf{n})$.
Then there exist vectors $\mathbf{v}^{\prime}\in\mathbb{N}^{p}$ and
$\mathbf{n}^{\prime}\in\mathbb{N}^{p}$ such that

$P(\mathbf{v},\mathbf{n})\;=\;{\rm
length}_{R}\left(\frac{[B]_{\mathbf{v}+\mathbf{n}}}{[A]_{\mathbf{v}+\mathbf{n}-\mathbf{t}+\mathbf{1}}[B]_{\mathbf{t}-\mathbf{1}}}\right)$

for all $\mathbf{v}\geq\mathbf{v}^{\prime}$ and
$\mathbf{n}\geq\mathbf{n}^{\prime}$. By taking
$P_{\mathbf{t}}^{A,B}({\mathbf{n}}):=P(\mathbf{v}^{\prime},\mathbf{n}-\mathbf{v}^{\prime})$,
the result of the theorem follows. ∎

With Theorem 5.1 in hand we are ready to introduce our new invariants:
_relative mixed multiplicities_.

######  Definition .

Assume Section missing 5. Fix a tuple
$\mathbf{t}=(t_{1},\ldots,t_{p})\in\mathbb{Z}_{+}^{p}$ of positive integers,
and let $P_{\mathbf{t}}^{A,B}$ as in Theorem 5.1. Write

$P_{\mathbf{t}}^{A,B}(n_{1},\ldots,n_{p})\;=\;\sum_{\beta\in\mathbb{N}^{p},\,|\beta|=r}e_{\mathbf{t}}\left(\beta;A,B\right)\,\frac{n_{1}^{\beta_{1}}\cdots
n_{p}^{\beta_{p}}}{\beta_{1}!\cdots\beta_{p}!}\;+\;\text{\rm(lower degree
terms)}.$

We say that the nonnegative integer $e_{\mathbf{t}}\left(\beta;A,B\right)\geq
0$ is the _relative mixed multiplicity of $B$ over $A$ of type
$(\mathbf{t},\beta)$_. For any $\beta\in\mathbb{N}^{p}$ with $|\beta|>r$, we
set $e_{\mathbf{t}}(\beta;A,B):=0$.

The next lemma shows that relative mixed multiplicities are nonincreasing on
the parameter $\mathbf{t}\in\mathbb{Z}^{p}$ and that they eventually stabilize
with value equal to a mixed Buchsbaum-Rim multiplicity.

######  Lemma .

Let $\beta\in\mathbb{N}^{p}$ with $|\beta|\geq r$. The following statements
hold:

  1. (i)

$e_{\mathbf{t}}\left(\beta;A,B\right)=\operatorname{{\rm
br}}_{\beta}\left([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B\right)+j_{\beta}^{\\#}\big{(}[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};\mathscr{H}_{\mathbf{t}-\mathbf{1}}\big{)}$
for all $\mathbf{t}\in\mathbb{Z}_{+}^{p}$.

  2. (ii)

(Nonincreasing): For any $\mathbf{t}^{\prime}\in\mathbb{Z}_{+}^{p}$ with
$\mathbf{t}^{\prime}\geq\mathbf{t}$, we have
$e_{\mathbf{t}^{\prime}}\left(\beta;A,B\right)\leq
e_{\mathbf{t}}\left(\beta;A,B\right)$.

  3. (iii)

(Stabilization):

$\lim_{t_{1}\to\infty,\ldots,t_{p}\to\infty}e_{(t_{1},\ldots,t_{p})}\left(\beta;A,B\right)\;=\;\operatorname{{\rm
br}}_{\beta}\left([A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}};B\right).$

###### Proof.

(i) This part is clear from the proof of Theorem 5.1.

(ii) The result follows from part (i) and Section missing 4.

(iii) By Theorem 2.1 and Theorem 5.1, there is some
$\mathbf{v}^{\prime}\in\mathbb{N}^{p}$ such that, for all
$\mathbf{t}>\mathbf{v}^{\prime}$ and ${\mathbf{n}}\gg\mathbf{0}$, we have the
equalities

$P_{B}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{t}-\mathbf{1},{\mathbf{n}})\;=\;{\rm
length}_{R}\left(\frac{[B]_{\mathbf{n}+\mathbf{t}-\mathbf{1}}}{[A]_{\mathbf{n}}[B]_{\mathbf{t}-\mathbf{1}}}\right)=P_{\mathbf{t}}^{A,B}({\mathbf{n}}+\mathbf{t}-\mathbf{1}).$

So, for all $\mathbf{t}\geq\mathbf{v}^{\prime}$, by comparing the leading
coefficients of the polynomials
$P_{B}^{[A]_{{\mathbf{e}}_{1}},\ldots,[A]_{{\mathbf{e}}_{p}}}(\mathbf{t}-\mathbf{1},{\mathbf{n}})$
and $P_{\mathbf{t}}^{A,B}({\mathbf{n}}+\mathbf{t}-\mathbf{1})$ in
${\mathbf{n}}$, the claim follows. ∎

In our applications in the next section, the relative mixed multiplicities
$e_{(1,\ldots,1)}(\beta;A,B)$ and the stable values will play an important
role. Hence we introduce the following notation.

######  Notation .

For all $\beta\in\mathbb{N}^{p}$ with $|\beta|\geq r$, set

$e(\beta;A,B):=e_{(1,\ldots,1)}(\beta;A,B)\qquad\text{ and }\qquad
e_{\infty}\left(\beta;A,B\right):=\lim_{t_{1}\to\infty,\ldots,t_{p}\to\infty}e_{(t_{1},\ldots,t_{p})}\left(\beta;A,B\right).$

##  6\. Applications

In this section, we use relative mixed multiplicities to provide integral
dependence and birationality criteria in a multigraded setting. These results
show that relative mixed multiplicities extend the fact that relative
multiplicities can detect integral dependence and birationality in an
$\mathbb{N}$-graded setting (see [SUV_MULT]). Throughout this section we use
the following setup.

######  Setup .

Let $(R,{\mathfrak{m}},\kappa)$ be a Noetherian local ring, and $A\subseteq B$
be an inclusion of standard $\mathbb{N}^{p}$-graded algebras over
$R=[A]_{(0,\ldots 0)}=[B]_{(0,\ldots 0)}$. Assume that ${\rm
length}_{R}\left([B]_{{\mathbf{e}}_{i}}/[A]_{{\mathbf{e}}_{i}}\right)<\infty$
for all $1\leq i\leq p$.

Our goal is to prove the following theorem.

######  Theorem 6.1.

Assume Section missing 6. Consider the associated morphism

$f\;:\;U\;\subseteq\;X=\text{MultiProj}(B)\;\longrightarrow\;Y=\text{MultiProj}(A)$

where $U=X\setminus V_{+}(A_{++}B).$ Set $r=\dim(X)$. Then the following
statements hold:

  1. (i)

If we obtain a finite morphism $f:X\rightarrow Y$, then
$e_{\infty}\left(\beta;A,B\right)=0$ for all $\beta\in\mathbb{N}^{p}$ with
$|\beta|=r$. The converse holds if $B$ is equidimensional and catenary.

  2. (ii)

If we obtain a finite birational morphism $f:X\rightarrow Y$, then
$e\left(\beta;A,B\right)=0$ for all $\beta\in\mathbb{N}^{p}$ with $|\beta|=r$.
The converse holds if $B$ is equidimensional and catenary.

######  Remark .

We compare Theorem 6.1 to known results in the $\mathbb{N}$-graded case. To
prove the converse statements of Theorem 6.1, one typically assumes that $B$
is equidimensional and _universally_ catenary. For instance, the approach in
[SUV_MULT] uses the associated graded ring $G=\operatorname{gr}_{A_{+}B}(B)$
as the main tool, which yields an elegant and unified treatment. However, if
one considers $G$, one needs $G$ to be equidimensional and catenary. Then, the
standard assumption for $G$ to be equidimensional and catenary is that $B$ is
equidimensional and universally catenary (see [RATLIFF_II, Theorem 3.8],
[SUV_MULT, Lemma 2.2]).

To circumvent any assumption on $G$, we work directly over the algebra $B$,
and our primary tool is the technical result of Section missing 6.

The next corollary slightly improves some criteria from [SUV_MULT] by
weakening an assumption of universally catenary to just catenary. In the
$\mathbb{N}$-graded case (i.e., $p=1$), to recover the original definition of
Simis, Ulrich and Vasconcelos [SUV_MULT], we set
$e_{t}(A,B):=e_{t}\left(\dim(B)-1;A,B\right)$ (notice that one can have the
strict inequality $\dim(X)<\dim(B)-1$).

######  Corollary  (cf., [SUV_MULT, Corollary 2.7, Theorem 3.3]).

Adopt the notation of Theorem 6.1 in an $\mathbb{N}$-graded setting (i.e.,
$p=1$). Let $G=\operatorname{gr}_{A_{+}B}(B)$ and $t\geq 1$. Consider the
following conditions:

  1. (a)

$B$ is integral over $A$ and
$B_{\mathfrak{p}}=\sum_{i=0}^{t-1}[B]_{i}A_{\mathfrak{p}}$ for every minimal
prime ${\mathfrak{p}}$ of $A$.

  2. (b)

$\dim\left([B]_{t}G\right)<\dim(G)=\dim(B)$.

  3. (c)

$e_{t}(A,B)=0$.

In general, the implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) hold. If
$B$ is equidimensional and catenary, the three conditions are equivalent.

###### Proof.

The implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) follow from
[SUV_MULT, Corollary 2.4, Proposition 3.2]. Thus we only need to show (c)
$\Rightarrow$ (a) under the assumption that $B$ is equidimensional and
catenary. Let $\overline{B}=B/(0:_{B}B_{+}^{\infty})$ and
$\overline{A}=A/A\cap(0:_{B}B_{+}^{\infty})$. Notice that $\overline{B}$ is
equidimensional and catenary, and that $A\hookrightarrow B$ is integral if and
only if $\overline{A}\hookrightarrow\overline{B}$ is. As $B$ is
equidimensional, we have either $\dim(\overline{B})=\dim(B)$ or
$\overline{B}=0$. Therefore, we may assume that
$r=\dim(X)=\dim(\overline{B})-1=\dim(B)-1$. Since
$e_{\infty}(r;A,B)=e_{\infty}(A,B)\leq e_{t}(A,B)=0$, Theorem 6.1(i) implies
that $A\hookrightarrow B$ is integral.

Next, we consider the finitely generated graded $A$-modules
$M=\sum_{i=0}^{t-1}[B]_{i}A\subseteq B$ and $B$. The associativity formula of
Section missing 3 yields the equalities

$j_{r}(M)=\sum_{{\mathfrak{p}}}{\rm
length}_{A_{\mathfrak{p}}}\left(M_{\mathfrak{p}}\right)e(A/{\mathfrak{p}})\quad\text{
and }\quad j_{r}(B)=\sum_{{\mathfrak{p}}}{\rm
length}_{A_{\mathfrak{p}}}\left(B_{\mathfrak{p}}\right)e(A/{\mathfrak{p}}),$

where, in both summations, ${\mathfrak{p}}\in\text{Spec}(A)$ runs through the
minimal primes of $A$ such that $\dim(A/{\mathfrak{p}})=r+1$ and
${\mathfrak{p}}\cap R={\mathfrak{m}}$. Notice that $A$ is also
equidimensional. By additivity, we have
$j_{r}(B)-j_{r}(M)=j_{r}(B/M)=e_{t}(A,B)=0$ (see Section missing 3). Since
${\rm length}_{R}([B]_{1}/[A]_{1})<\infty$, we obtain
$M_{\mathfrak{p}}=B_{\mathfrak{p}}$ for all ${\mathfrak{p}}\in\text{Spec}(A)$
with ${\mathfrak{p}}\cap R\subsetneq{\mathfrak{m}}$. Therefore, it follows
that $M_{\mathfrak{p}}=B_{\mathfrak{p}}$ for every minimal prime
${\mathfrak{p}}$ of $A$. ∎

Before proving the above theorem, we need a couple of technical lemmas. The
first one is inspired by Rees’ original proof of his celebrated integral
dependence criterion [REES_CRITERION, REES_AMAO_THM].

######  Lemma .

Let $T$ be a standard graded $R$-algebra, and
$\pi:X=\text{Proj}(T)\rightarrow\text{Spec}(R)$ be the natural projective
morphism. Let $x\in X$ be a closed point and $P\subset T$ be the corresponding
relevant homogeneous prime ideal. Then the following statements hold:

  1. (i)

$\pi$ maps $x$ to the closed point of $\text{Spec}(R)$, and thus
$P\supset{\mathfrak{m}}T$.

  2. (ii)

${\rm
length}_{R}\big{(}\text{H}^{0}\big{(}X,\mathcal{O}_{X}/\,\widetilde{P}\big{)}\big{)}=[\kappa(x):\kappa]$,
where $\kappa(x)=\mathcal{O}_{x}/{\mathfrak{m}}_{x}$ is the residue field at
$x$.

  3. (iii)

$\lambda_{x}(n):={\rm
length}_{R}\big{(}\text{H}^{0}\big{(}X,\mathcal{O}_{X}/\,\widetilde{P^{n}}\big{)}\big{)}$
eventually coincides with a polynomial $P_{x}(n)$ of degree
$d_{x}:=\dim(\mathcal{O}_{x})$ and whose leading coefficient is
$[\kappa(x):\kappa]\cdot e(\mathcal{O}_{x})/d_{x}!$.

  4. (iv)

Let $H=[P]_{d}$ such that $(H)=[P]_{\geq d}=\bigoplus_{v\geq d}[P]_{v}$ (i.e.,
$d$ is larger than or equal to the degrees of a set of homogeneous generators
of $P$). Then the function $L_{d}(v,n):={\rm
length}_{R}\left([T]_{v+nd}\big{/}H^{n}[T]_{v}\right)$ eventually becomes a
polynomial $P_{d}(v,n)$. Furthermore, we actually have $P_{d}(v,n)=P_{x}(n)$.

###### Proof.

(i) The morphism $\pi$ is closed, hence it should send the closed point $x\in
X$ to the unique closed point in $\text{Spec}(R)$. It clearly follows that
$P\supset{\mathfrak{m}}T$.

(ii) Since
$\text{H}^{0}\big{(}X,\mathcal{O}_{X}/\,\widetilde{P}\big{)}\cong\kappa(x)$,
the equality is clear.

(iii) For all $n\geq 1$, we have
$\text{H}^{0}\big{(}X,\mathcal{O}_{X}/\,\widetilde{P^{n}}\big{)}\cong\mathcal{O}_{x}/{\mathfrak{m}}_{x}^{n}$,
and so we obtain

$\lambda_{x}(n)\;=\;[\kappa(x):\kappa]\cdot{\rm
length}_{\mathcal{O}_{x}}(\mathcal{O}_{x}/{\mathfrak{m}}_{x}^{n}).$

This settles the claim, since ${\rm
length}_{\mathcal{O}_{x}}(\mathcal{O}_{x}/{\mathfrak{m}}_{x}^{n})$ is
eventually the Hilbert-Samuel polynomial of the local ring
$(\mathcal{O}_{x},{\mathfrak{m}}_{x})$.

(iv) The fact that $L_{d}(v,n)$ eventually coincides with a polynomial
$P_{d}(v,n)$ follows from the existence of mixed Buchsbaum-Rim multiplicities
(see Theorem 2.1 or [KLEIMAN_THORUP_MIXED]). We fix $n_{0}$ big enough so that
$L_{d}(v,n_{0})$ eventually becomes the polynomial $P_{d}(v,n_{0})$ in $v$.
However, for $v\gg 0$, we have

$L_{d}(v,n_{0})={\rm
length}_{R}\left({\left[T/P^{n_{0}}\right]}_{v+n_{0}d}\right)={\rm
length}_{R}\left(\text{H}^{0}\left(X,\mathcal{O}_{X}/\,\widetilde{P^{n_{0}}}\,(v+n_{0}d)\right)\right)=\lambda_{x}(n_{0}),$

and the last equality follows because $x\in X$ is a closed point. This shows
that $P_{d}(v,n)=P_{x}(n)$. ∎

######  Lemma .

With the same notation of Theorem 6.1, assume that $B$ is equidimensional and
catenary. Then, for any closed point $x\in X$, we have
$d_{x}=\dim(\mathcal{O}_{x})=r$.

###### Proof.

We may assume $(0:_{B}B_{++})=0$, and so $\dim(B)=r+p$. Notice that $B$ is
naturally a *local ring with *maximal ideal given by
${\mathfrak{M}}={\mathfrak{m}}+B_{+}$ and that
$r+p=\dim(B)=\operatorname{ht}({\mathfrak{M}})$; see [BRUNS_HERZOG, §1.5].
Since $B$ is equidimensional and catenary, any maximal chain of homogeneous
prime ideals has length $r+p$. Let $x\in X$ be a closed point with associated
prime $P\subset B$. As $x\in X$ is a closed point,
$\dim(B/P)=\dim\big{(}\overline{\\{x\\}}\big{)}+p=p$. Therefore, we obtain
$d_{x}=\dim(\mathcal{O}_{x})=\dim(B_{P})=\dim(B)-\dim(B/P)=(r+p)-p=r$. ∎

We are now ready to prove the main result of this section.

###### Proof of Theorem 6.1.

We may substitute $A$ and $B$ by $\overline{A}=A/A\cap(0:_{B}B_{++}^{\infty})$
and $\overline{B}=B/(0:_{B}B_{++}^{\infty})$, respectively. By utilizing the
Segre embedding, we obtain the isomorphisms $X\cong\text{Proj}(T)$ and
$Y\cong\text{Proj}(S)$, where $S:=\bigoplus_{n\geq 0}[A]_{(n,\ldots,n)}$ and
$T:=\bigoplus_{n\geq 0}[B]_{(n,\ldots,n)}$. Then we see $f:U\subseteq
X=\text{Proj}(T)\rightarrow Y=\text{Proj}(S)$ as the morphism associated to
the inclusion $S\subseteq T$ of standard graded $R$-algebras.

By considering the substituted polynomial $P_{\mathbf{1}}^{A,B}(n,\ldots,n)$
(see Theorem 5.1), we can compute the equality

$e(S,T)\;=\;\bigoplus_{\beta\in\mathbb{N}^{p},\,|\beta|=r}\frac{r!}{\beta_{1}!\cdots\beta_{p}!}\,e(\beta;A,B).$

Similarly, as a consequence of Section missing 5(iii), by choosing $t>0$ large
enough and considering the substituted polynomial
$P_{(t,\ldots,t)}^{A,B}(n,\ldots,n)$, we can compute the equality

$e_{\infty}(S,T)\;=\;\bigoplus_{\beta\in\mathbb{N}^{p},\,|\beta|=r}\frac{r!}{\beta_{1}!\cdots\beta_{p}!}\,e_{\infty}(\beta;A,B).$

From these reductions, we may instead prove the theorem in terms of the
vanishing of $e(S,T)$ and $e_{\infty}(S,T)$.

(i) Suppose that $S\subseteq T$ is an integral extension. Then we may choose
$t>0$ large enough such that $[T]_{n}=[S]_{n-t+1}[T]_{t-1}$ for all $n\geq 0$.
This shows that $e_{\infty}(S,T)=0$.

Conversely, suppose that $S\subseteq T$ is not an integral extension and that
$B$ is equidimensional and catenary. Then $S_{+}T$ is a relevant ideal of $T$
(i.e., $\sqrt{S_{+}T}\not\supseteq T_{+}$). Let $x\in X$ be a closed point
with associated prime $P\subset T$ such that $P\supseteq S_{+}T$. From Section
missing 6, we get the equality $d_{x}=\dim(\mathcal{O}_{x})=r$. By Section
missing 6, for large enough integers $d>0$ and $v>0$, we obtain

${\rm length}_{R}\big{(}[T]_{v+nd}/[S]_{nd}[T]_{v}\big{)}\;\geq\;{\rm
length}_{R}\big{(}[T]_{v+nd}/H^{n}[T]_{v}\big{)}\;=\;P_{x}(n)\quad\text{ for
}\quad n\gg 0,$

where $H=[P]_{d}$ and $P_{x}(n)$ is a polynomial of degree $r$ with leading
coefficient $[\kappa(x):\kappa]\cdot e(\mathcal{O}_{x})/r!$. It then follows
that

$e_{\infty}(S,T)\;\;=\;\;\frac{r!}{d^{r}}\lim_{n\to\infty}\frac{{\rm
length}_{R}\big{(}[T]_{v+nd}/[S]_{nd}[T]_{v}\big{)}}{n^{r}}\;\;\geq\;\;\frac{r!}{d^{r}}\lim_{n\to\infty}\frac{P_{x}(n)}{n^{r}}\;\;=\;\;\frac{[\kappa(x):\kappa]\cdot
e(\mathcal{O}_{x})}{d^{r}}\;\;>\;\;0.$

Therefore, $e_{\infty}(S,T)\geq 1$ is a positive integer, as required.

(ii) After proving part (i), the result of Section missing 6 is already valid.
In particular, the statement about birationality follows. ∎

It should be mentioned that in a multigraded setting the inclusion $A\subseteq
B$ may not be integral and yet we can still obtain a finite morphism:

######  Example .

Let $\mathbb{k}$ be a field and consider the following inclusion of standard
$\mathbb{N}^{2}$-graded $\mathbb{k}$-algebras

$A=\mathbb{k}[x_{1},x_{2},y_{1},y_{2}]\;\;\hookrightarrow\;\;B=\frac{\mathbb{k}[x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}]}{\big{(}x_{3}(y_{1},y_{2},y_{3}),\,y_{3}(x_{1},x_{2},x_{3})\big{)}}$

with $\deg(x_{i})={\mathbf{e}}_{1}\in\mathbb{N}^{2}$ and
$\deg(y_{i})={\mathbf{e}}_{2}\in\mathbb{N}^{2}$. Then
$f:{\text{BiProj}}(B)\subset{\mathbb{P}}_{\mathbb{k}}^{2}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{2}\rightarrow{\text{BiProj}}(A)={\mathbb{P}}_{\mathbb{k}}^{1}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{1}$
is a finite morphism (in fact, it is an isomorphism since
$B/(0:_{B}B_{++})\cong B/(x_{3},y_{3})$). However, the extension
$A\hookrightarrow B$ is not integral.

Finally, we briefly discuss the particular case of graphs of rational maps.

######  Remark .

Let $\mathbb{k}$ be a field and
$\mathfrak{d}_{1}\subseteq\mathfrak{d_{2}}\subseteq\text{H}^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}(d)\big{)}$
be two linear systems. Let
$\mathcal{F}_{1}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s_{1}}$
and
$\mathcal{F}_{2}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s_{2}}$
be the respective rational maps. Let $\Gamma_{i}$ and $E_{i}$ be the graph
(blow-up of the base locus) and exceptional divisor of $\mathcal{F}_{i}$,
respectively. Then the following conditions are equivalent:

  1. (a)

The induced rational map
$f:\Gamma_{2}\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s_{2}}\dashrightarrow\Gamma_{1}\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s_{1}}$
between graphs is a finite birational morphism.

  2. (b)

The projective degrees of $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ coincide
(see [HARRIS, Example 19.4], [DOLGACHEV, §7.1.3]).

  3. (c)

The multidegrees of $E_{1}$ and $E_{2}$ coincide.

The equivalence between (a) and (b) follows from Theorem 6.1. Since
$\mathcal{O}_{\Gamma_{i}}(-E_{i})\cong\mathcal{O}_{\Gamma_{i}}(-d,1)$, we have
the short exact sequence
$0\rightarrow\mathcal{O}_{\Gamma_{i}}(-d,1)\rightarrow\mathcal{O}_{\Gamma_{i}}\rightarrow\mathcal{O}_{E_{i}}\rightarrow
0$, and so we can compute the following equality on multidegrees (see, e.g.,
[POSITIVITY]):

$\deg^{(n_{1},n_{2})}(E_{i})\;=\;d\cdot\deg^{(n_{1}+1,n_{2})}(\Gamma_{i})-\deg^{(n_{1},n_{2}+1)}(\Gamma_{i})\quad\text{
for all }\quad n_{1}+n_{2}=r-1.$

Therefore, since $\deg^{(r,0)}(\Gamma_{i})=1$, it follows that parts (b) and
(c) are equivalent.

Alternatively, the equivalence between parts (a) and (c) could be deduced as a
very particular case of the main result of [PTUV_MULT]. Indeed, important
recent work of Polini, Trung, Ulrich and Validashti [PTUV_MULT] shows that,
over an equidimensional and universally catenary Noetherian local ring, two
ideals $I\subset J$ have the same integral closure if and only if their
multiplicity sequences coincide.

## Acknowledgments

The author was supported by an FWO Postdoctoral Fellowship (1220122N).

## References


mydate[]

Volume 1, Issue 1, February 1, 2016

TC-CCPS Newsletter

Technical Articles

Multi-Modal Attack Detection for Cyber-Physical Additive Manufacturing

Shih-Yuan Yu, Arnav Vaibhav Malawade, Mohammad Abdullah Al Faruque

Department of Electrical Engineering and Computer Science, University of
California, Irvine

##  1 Introduction

Cyber-Physical Additive Manufacturing (AM) constructs a physical 3D object
layer-by-layer according to its digital representation and has been vastly
applied to fast prototyping and the manufacturing of functional end-products
across fields. The computerization of traditional production processes propels
these technological advancements; however, this also introduces new
vulnerabilities, necessitating the study of cyberattacks on these systems
[12]. The AM Sabotage Attack is one kind of kinetic cyberattack that
originates from the cyber domain and can eventually lead to physical damage,
injury, or even death [2]. By introducing inconspicuous yet damaging
alterations in any specific process of the AM digital process chain, the
attackers can compromise the structural integrity of a manufactured component
in a manner that is invisible to a human observer. If the manufactured objects
are critical for their system, those attacks can even compromise the whole
system’s structural integrity and pose a severe safety risk to its users. For
example, an inconspicuous void (less than 1 mm in dimension) placed in the 3D
design of a tensile test specimen can reduce its yield load by 14% [11].
However, security studies primarily focus on securing digital assets [10],
overlooking the fact that AM systems are CPSs.

The AM system, or the printer, is comprised of a set of connected hardware
components, and thus can unintentionally produce analog emissions during the
operation of printing through different physical side-channels such as
acoustics, electromagnetic radiation, vibration, and power. In AM systems, the
information flow in the cyber domain has at least one corresponding control
signal sent to the physical domain. This signal flow, in turn, actuates the
physical processes accordingly, resulting in side-channel emissions that have
a high degree of mutual information with the digital control signals. This
property allows our group to a series of research on utilizing the correlation
between the two domains and validating that physical domain signals match
their cyber domain counterparts [4, 9, 1, 6, 7, 3, 8, 5]. Sabotaging the
structural integrity of 3D objects requires the attacker to make subtle
variations to one or more of the sub-processes in the AM process chain,
resulting in a change in the printer’s control parameters and a corresponding
change to its analog emissions. In this case, monitoring the operation of the
targeted AM systems can be the most direct defense [4]. To detect such
modifications, our group proposes an attack detection system that continuously
monitors and analyzes the different side-channel information leaked during the
operation of AM systems, allowing us to identify unusual analog emissions
resulting from potential sabotage attacks.

##  2 The AM Process Chain

The manufacturing of 3D objects in AM requires a chain of cyber-physical
processes [12]. We primarily describe the process chain for the Fused
Deposition Model (FDM) based AM. As shown in Figure 1, the process chain
begins in the cyber domain with an idea for a 3D object. The first item on the
workflow is to substantiate the design specifications using Computer-Aided
Design (CAD) tools. Next, the generated CAD model gets converted into the
STereoLithography (STL) format that uses a series of triangles to model the
surface geometry. In the Computer-Aided Manufacturing (CAM) process, the
slicing algorithms convert the STL files into a layer-by-layer description
file which instructs the printer on how to create the 3D object. The
description file uses a Numerical Control Programming Language called
G/M-code. The G-codes describe the motion and flow settings of the printer,
determining the speed of the nozzle along each axis and the amount of material
to deposit at each printing step. As an example, G1 F2100 X5 Y6 Z1.2 E2.1
represents a single line of G-code for controlling the movement of the nozzle,
where G1 means coordinated linear motion, F defines travel feed rate (speed)
which is measured in mm/min, and distance and extrusion are measured in mm. On
the other hand, the M-codes control the machine settings such as temperature,
coolant. Lastly, the printer’s firmware interprets each received G/M-code
instruction into the corresponding control signals to actuate the printer
hardware and, in turn, print the object.

Figure 1: The AM process chain.

In the physical domain, the FDM-based printer uses thermoplastic filament
materials to form 3D physical objects. The printer uses motors and belts to
control the heated nozzle to melt the materials, depositing the extrusion onto
the surface of the build plate at a precisely controlled rate. An array of
stepper motors enables a fine-grained degree of control, each controlling
movement along a single axis. These components act as a significant source of
information leakage because of their analog emissions during printer
operation. The analog emissions come from various side-channels of the printer
(vibration, acoustic, magnetic, and power side-channels), which are streams of
information outside of the primary data path containing information which can
be correlated with the data in the primary path. In this case, the primary
data path includes the control signals sent to the printer. The stepper motors
are a major source of vibration and acoustic emissions due to the fluctuating
radial force and torque ripples working on the stator core of the stepper
motor. The varying electric field in the stepper motors also leaks data to the
magnetic side channel. The power side-channel reveals the primary consumers of
power in a 3D printer, such as the heating elements, stepper motors, fans, bed
heater, and the internal control circuitry. In short, through these side-
channels, the unintentional information leakage about the primary data path
enables the inference of control signal values from side-channel information.

##  3 Sabotage Attack Detection for AM Systems

As for the threat model, the attackers can exploit multiple attack surfaces on
the AM process chain (Figure 1) to sabotage manufactured parts. From the cyber
domain, the adversaries can modify the intermediate forms of the 3D object by
altering the integrity of the tools, firmware, or computers in the process
chain, eventually resulting in the modification of the printer’s control
signals and the sabotage of the printed object. Cyber-security techniques can
help mitigate risks further upstream in the digital process chain; however, AM
systems often have network connectivity or physical access ports that can
enable an adversary to exploit firmware vulnerabilities and compromise the
system. To this end, we focus our on detecting adversarial attacks to the
printer firmware that modify the control parameters directly. With this in
mind, we assume that the G-codes sent to the AM system have not been
previously tampered with and are representative of the correct 3D object
model. In this case, the movement of the AM machine’s internal components
during a sabotage attack will not match the movements described by the
instructions given in the G-code file, and thus, will result in different
side-channel emissions. By utilizing these side-channel emissions with the
G-codes sent to the printer, we can infer the values of the control signals
and then determine if the printer firmware has been hacked.

Figure 2: System Architecture.

The architecture of our proposed Sabotage Attack Detection System is described
in Figure 2. To infer the control signals, the System State Estimation Model
learns the relationship between the various side-channel emissions and the
various control signals using supervised machine learning approaches. The
dataset for training consists of the analog emissions from the Benchmark
Printer along with their corresponding control signals, which are parsed from
the G-codes. Once the mapping between analog emissions and control signals has
been learned, our system first infers each control signal by continuously
observing the analog emissions from the Monitoring Printer. The Attack
Detection Algorithm then compares the unmodified control signals to the
inferred signals to determine if a sabotage attack has occurred.

##  4 Results

Figure 3: Experimental setup

Our experimental setup is shown in Figure 3, consisting of an Ultimaker 3
3D-printer, four microphones, three vibration sensors (accelerometers), three
magnetic sensors (magnetometers), and one current sensor. We use timestamps to
match each G-code to its corresponding side-channel emissions recorded from
the sensors. To evaluate the performance of our proposed detection system, we
combined data from several different 3D-prints to produce a 20-gigabyte
dataset containing 60,959 rows with 18,276 features per row. In the following
sections, the selected results for state estimation and attack detection are
shown (see [13] for more results).

###  4.1 AM System State Estimation

The ability of our system to detect sabotage attacks depends on how precisely
our system models the relationship between side-channel information and the
corresponding machine control parameters. To improve our system’s performance,
we use multiple modalities of side-channel data for control signal estimation.
To evaluate our multi-modal approach in comparison to single side-channel
(unimodal) methods, we assessed the movement-axis prediction ($A_{x}$ and
$A_{y}$) performance of our classifiers using data from each modality. The
results of the highest accuracy classifiers in each modality in comparison to
our multi-modal approach and the single acoustic sensor approach presented in
[4] are shown in Figure 4. As shown in the figure, our multi-modal technique
outperforms the uni-modal methods as well as the technique from [4]. Notably,
the acoustic and vibration modalities show the highest uni-modal accuracy and
reach within 5% of the accuracy of the multi-modal technique. Overall, these
results demonstrate that our multi-modal technique results in performance
improvement over uni-modal methods.

Figure 4: The comparison of multimodal approach and unimodal approaches in
axis movement prediction.

###  4.2 AM Sabotage Attack Detection

To evaluate our attack detection performance, we generate synthetic attacks by
adding adversarial modifications to G-Codes. Then, we pass the unmodified
sensor data and the modified G-code files as inputs to our attack detection
system to evaluate its ability to detect mismatches. Our overall row-level
accuracy is 99.17% for movement-axis prediction. For axis-velocity prediction
our average accuracy is 96.64%. Our overall attack detection accuracy for all
control parameters is 98.15%. Although these attacks are synthetic, they
provide empirical benchmarks to demonstrate the attack detection system’s
capabilities.

(a) Original and modified wrench.

(b) Original and modified gear.

(c) Adversarial modifications in details.

Figure 6: The adversarially modified objects used for real-world testing.

To validate these results in a real scenario, we tested two real-world
sabotage attacks: a gear and a wrench. Since gears and wrenches are generally
placed in high-stress scenarios, it is plausible that a kinetic cyberattack
could compromise their structural integrity and significantly increase the
chance of equipment failure. Moreover, these parts serve as analogues to other
load-bearing 3D printed parts to demonstrate that our detection system can
identify kinetic cyberattacks on these types of components. The original and
attacked wrenches are shown in Figure 6(a), and the adversarial modifications
to the gear are shown in Figure 6(b). Although these specific modifications
are visible to the human eye in the final print, this was done primarily for
demonstrative purposes. These modifications can easily be concealed in
internal layers of the print without modifying the outward appearance of the
object, enabling it to pass visual quality inspections done by both machines
and employees. For both test cases, the detection system was able to detect
the attack successfully. These tests demonstrate that our methodology can be
applied to real-world, practical AM tasks.

##  5 Conclusion

AM systems are core to the future of manufacturing and Industry 4.0. Since
these systems are advancing in terms of complexity and connectivity as well as
producing parts for increasingly critical and taxing applications, the
security of these systems is a significant concern. Since AM systems are CPSs,
they present a broad and diverse attack surface, making attack detection and
prevention a daunting task. Our proposed sabotage attack detection system has
demonstrated the ability to detect various attacks by correlating multiple
forms of physical-domain emissions of the AM system with cyber-domain
information. We have achieved an overall detection accuracy of 98.15% on
synthetic benchmarks and demonstrated that our system could detect two real-
world scenarios of sabotage attacks. Although our solution addresses a
significant security risk in AM systems, more work is needed to address the
broad scope of potential AM vulnerabilities.

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