| Summary: | 碩士 === 國立中正大學 === 應用數學研究所 === 87 === In the study of the projective $n$-space over a field $k$, we consider the homogeneous coordinate ring $R=k[X_0,\dots,X_n]/I(V)$ where $V$ is a projective variety and $I(V)$ is its vanishing ideal. The ideal is a homogeneous (or graded) ideal with respect to the usual degree of polynomials. In this case the ring $R$ is endowed with a natural grading which is the degree of polynomials. The ring $R$ can be written as a graded ring $R=\bigoplus_{i=0}^{\infty}R_i$ where $R_i$ is the set of 0 and homogeneous polynomials of degree $i$.
When we study the geometric properties of projective varieties we often instead study $\Proj R$ the set of homogeneous prime ideals of the ring $R$ which do not contain the irrelevant ideal $R_+=\bigoplus_{i=1}^{\infty}R_i$. There is a natural abstraction of the concept of homogeneous coordinate rings to the concept of general graded rings. We are particularly concerned with homogeneous prime ideals.
Roughly speaking, this thesis discuss some graded analogies regarding dimension, grade, depth and projective dimension. The graded rings are sometimes easier to handle than the common rings since they have some additional algebraic structures. We use the Cohen-Macaulay property to demonstrate this point.
To achieve these goals we need some homological tools. Thus this thesis also devoted many pages to prepare for the knowledge of homological algebra.
In Chapter I we introduce two additive functors Ext and Tor, the two most basic and well-known derived functors. To define derived functors we have to start with the basic concepts like projective resolutions, additive functors, homology and so on. Some basic properties are fully discussed. We will examine in great details the well-definedness of Ext and Tor. Since they are defined in terms of derived functors, they will be quite easy to handle. Especially, the snake lemma which deals with long exact sequences will be used again and again.
Chapter II is denoted to background knowledge regarding graded rings and graded modules. Besides defining what graded rings and graded modules are, we also define some other concepts like the {\slocal}ness, homogeneous localization, etc., as opposed to their non-graded counterparts. The symbol $\boldkey*$
denotes that the relevant notation is a graded analogy. And often they themselves are equipped with a graded structure. The graded situations are often extremely similar to their non-graded counterparts. At worst cases, it only requires slight modification for things to work. We give an example here.
Let $R$ be a graded ring. Then to check $R$ is Noetherian it suffices to check every graded ideal of $R$ is finitely generated (see Proposition~2.2.2).
Chapter III is the main chapter of this thesis. Nakayama's Lemma is an extremely important tool in the field of commutative algebra. In \S1 we give two graded variations of this lemma. Another very useful result is Prime Avoidance. We also have the graded analogy. In a graded ring it is possible to find a homogeneous element which ``avoids'' finitely many primes under a slightly stronger condition that the target ideal must be positively graded.
The concept of the Krull dimension is central to commutative algebra. Prime ideals of a ring correspond to subvarieties of affine varieties. Similarly, homogeneous prime ideals of a graded ring correspond to subvarieties of projective varieties. Thus it is meaningful to study the length of chains of homogeneous primes. Thus the concept of {\sdime} comes naturally to us. Let $R$ be a Noetherian graded ring, $M$ a finitely generated graded $R$-module and $p$ a prime ideal of $R$. Then $\sdim M_{(p)}=\dim M_p$ if $p$ is graded while $\sdim M_{(p)}=\dim M_p-1$ if $p$ is not graded.
In \S4 we define another important concept {\sgrade}. Let $R$ be a Noetherian graded ring, $M$ a finitely generated graded $R$-module and $I$ an ideal such that $IM\ne M$. We define the {\sgrade} of $I$ on $M$ to be the length of a maximal homogeneous $M$-sequence in $I$. Since we know that grade can be computed using Ext, we are motivated to study $\sExt$. Under the usual conditions, together with the condition that $I$ is generated by homogeneous elements of positive degree, we also have that {\sgrade} of $I$ on $M$ is the minimum of $i$ of the non-vanishing $\sExt_R^i(R/I,M)$. In this case we do have $\sgra IM=\gra IM$. Let $(R,m)$ be a Noetherian {\slocal} ring and $M$ a finitely generated graded $R$-module, the {\sgrade} of $m$ on $M$ is called the {\sdepth} of $M$.
In \S5 we compare {\sdepth} and depth, and we have the following result. Let $(R,m)$ be a Noetherian {\slocal} ring such that $m$ is generated by homogeneous elements of positive degree. Let $M$ be a finitely generated graded $R$-module. Then the {\sdepth} of $M$ equals the depth of $M_m$. This is not surprising at all since the key lemma of all of these results concering regular sequences is the graded prime avoidance.
In \S6 we study the ${}^{\boldkey*}$Cohen-Macaulay, or simply {\sCM}, property. For a finitely generated graded module $M$ over a Noetherian {\slocal} ring, the {\sCM} property means that the {\sdepth} of the module $M$ equals the {\sdime} of $M$. For global cases, the {\sCM} property must be checked locally, but only at the homogeneous primes. However, for a graded module to be {\sCM} it becomes C-M automatically. Thus to check if a graded module is C-M it suffices to check the homogeneous primes.
In \S7 we state the graded Auslander-Buchsbaum Theorem. The non-graded Auslander-Buchsbaum Theorem is not only of theoretical importance but also a convenient formula for computing the depth of a module. We know that the projective dimension of $M$ over $R$ is the supremum of $i$ of the
non-vanishing $\Tor_i^R(M,R/m)$ if $(R,m)$ is local. But for {\slocal} ring $(R,m)$, we have ascertained the existance of a graded minimal free resolution of any finitely generated graded $R$-module. Thus we can also compute the projective dimension of $M$ over $R$ in the same way. We now have the final
important graded analogy. Let $(R,m)$ be a Noetherian {\slocal} ring where $m$ is generated by homogeneous elements of positive degree. Let $M$ be a non-zero finitely generated graded $R$-module. If $\pd_R M<\infty$, then $\pd M+\sdep M=\sdep R$.
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