A Presentation of the Kähler Differential Module for a Fat Point Scheme in P n 1 × · · · × P nk

Let Y be a fat point scheme in Pn1 × · · · × Pnk over a field K of characteristic zero. In this paper we introduce the multi-graded Kähler differential module for Y and we establish a short exact sequence of this module in terms of the thickening of Y.


Introduction
In [1], G. de Dominicis and M. Kreuzer introduced some methods using algebraic differential forms into the study of finite sets of points in the projective n-space P n over a field K of characteristic zero.Explicitly, given a finite set of points X in P n with homogeneous vanishing ideal I X in R = K[X 0 , . . ., X n ] and homogeneous coordinate ring R X = R/I X , the Kähler differential module for X is the R X -module Ω 1 R X /K = J/J 2 where J is the kernel of the multiplication map µ : R X ⊗ K R X → R X .One of interesting results in [1] is the canonical exact sequence for the Kähler differential module 0 → I (2)  X /I 2 Based on this exact sequence, the structure of this module can be precisely described in several special cases, for instance, if X is the complete intersection of hypersurfaces of degrees d 1 , . . ., d n then it follows that the Hilbert function of Ω 1 R X /K is given by HF Ω 1 R X /K (i) = (n + 1) HF X (i − 1) − n j=1 HF X (i − d j ) for all i ∈ Z. Later, in [2,4], the differential algebra techniques were extended to fat point schemes in P n and in P 1 × P 1 .
In this paper we will consider the natural question of whether these differential algebraic methods can be applied to study fat point schemes of a multiprojective space P n 1 × • • • × P n k and, especially, we look closely at the generalization of the above canonical exact sequence to fat point schemes in . ., X 1n 1 , . . ., X k0 , . . ., X kn k ] and let R Y = S /I Y be the multi-graded coordinate ring of Y.We show that the Kähler differential module Ω 1 R Y /K admits the following exact sequence.
This result shows that one can compute the Hilbert function of the Kähler differential module for Y from the Hilbert functions of Y and V, in particularly, to compute the Hilbert function HF Ω 1 R Y /K (i), we need to compute HF Ω 1 R Y /K (i) for only a finite number of i ∈ Z k .

Basic Facts and Notation
Let K be a field of characteristic zero.Let k ≥ 2 be a positive integer, let i denote the tuple (i 1 , . . ., i k ) ∈ Z k , and let |i| = l i l .We write i j if i l ≤ j l for every l = 1, . . ., k.Also, let {e 1 , . . ., e k }, e i = (0, . . ., 1, . . ., 0), be the canonical basis of Z k .The multi-graded coordinate ring of P n 1 × • • • × P n k is the the polynomial ring S = K[X 10 , . . ., X 1n 1 , . . ., X k0 , . . ., X kn k ] equipped with the Z k -grading defined by deg we let S i be the homogeneous component of degree i of S , i.e., the K-vector space with basis Given an ideal I ⊆ S , we set homogeneous ideal of S then the quotient ring S /I also inherits the structure of a multi-graded ring via (S /I) i := S i /I i for all i ∈ Z k .
A finitely generated S -module M is a Z k -graded S -module if it has a direct sum decomposition with the property that S i M j ⊆ M i+ j for all i, j ∈ Z k .
Definition 2.1.Let M be a finitely generated Z k -graded S -module.The Hilbert function of M is the numerical function HF M : In particular, for a Z k -homogeneous ideal I of S , the Hilbert function of S /I satisfies If M is a finitely generated Z k -graded S -module such that HF M (i) = 0 for i (0, . . ., 0), we write the Hilbert function of M as an infinite matrix, where the initial row and column are indexed by 0.
A point in the space where [a j0 : a j1 : • • • : a jn j ] ∈ P n j .Its vanishing ideal is the bihomogeneous prime ideal of the form where L jl = a jl X j0 − a j0 X jl and deg(L jl ) = e j for l = 1, . . ., n j .
When X = {P 1 , . . ., P s } is a set of s distinct points in The number m j is called the multiplicity of the point P j .
(c) The fat point scheme If Y is a fat point scheme in P n , then there exists a linear form that is a non-zerodivisor for the homogeneous coordinate ring of Y. Using this property, the following lemma for fat point schemes in Remark 2.4.After a suitable change of coordinates, we can assume that L i = X i0 for i = 1, . . ., k.This also implies that X i0 does not vanish at any point of Supp(Y) = X.In this case if x i j denotes the image of X i j in R Y , then x 10 , . . ., x k0 are non-zerodivisors of R Y .
As a consequence of the lemma and [8, Proposition 1.9] we get several basis properties of the Hilbert function of Y.

A Presentation of the Kähler Differential Module
In the following we let Y be a fat point scheme in P n 1 × • • • × P n k supported at X.We also denote the image of X i j in R Y by x i j .According to Remark 2.4, we may always assume that x 10 , . . ., x k0 are non-zerodivisors of R Y .
In the multi-graded algebra we have the Z k -homogeneous ideal J = ker(µ), where µ : ITM Web of Conferences 20, 01007 (2018) https://doi.org/10.1051/itmconf/20182001007ICM 2018 More generally, for any Z k -graded K-algebra T/S we can define in the same way the Kähler differential module Ω 1 T/S , and the universal derivation of T/S (cf.[6, Section 2]).
Some following properties of the module of Kähler differentials follows from [6, Propositions 4.12 and 4.13].
]. Also, we see that and Moreover, we get the exact sequence of Z k -graded R X -modules In this case the Hilbert function of In general, we have a presentation of module of Kähler differential as follows.
Theorem 3.5.Let Y = m 1 P 1 + • • • + m s P s be a fat point scheme in P n 1 × • • • × P n k , and let V be the thickening of Y. Then we have the exact sequence of Z k -graded R Y -modules ITM Web of Conferences 20, 01007 (2018) https://doi.org/10.1051/itmconf/20182001007ICM 2018 To prove Theorem 3.5, we will first require the following lemma.Lemma 3.6.
Since F ∈ I m P , the polynomial F can be written as , and set Clearly, β ∈ Γ ∅.Let (i, j) be the smallest tuple w.r.t.Lex such that there exists β ∈ Γ such that b i j 0. Write Γ = { β 1 , . . ., β s } such that Note that if β, β ∈ Γ, β β and β i j = β i j = , then (β i j+1 , . . ., β kn k ) (β i j+1 , . . ., β kn k ).By Macaulay's Basis Theorem (cf.[5, Theorem 1.5.7]),we have It follows that ω I m P Ω 1 S /K (since in the right-hand side of ( ) the first summand has β i j and the last summand does not contain dX i j ).Therefore we get ω I m P Ω 1 S /K , as we wanted to show.
We are now ready to state and prove the main result of this section.
Proof.Let ϕ : I Y /I V → Ω 1 S /K /I Y Ω 1 S /K be the map given by ϕ(F + I V ) = dF + I Y Ω 1 S /K for all F ∈ I Y .It is easy to check that the map ϕ is well-defined, Z k -homogeneous of degree (0, . . ., 0), and R Y -linear.For any Z k -homogeneous element F ∈ I Y \ I V , we have F ∈ I and let m 1 , . . ., m s are positive integers.The ideal