A cancellation theorem for ideals
Craig Huneke
证明了Gorenstein局部环中特殊理想的消去定理。这些定理的形式是：如果KI包含在JI中，则K包含在J中。
We prove cancellation theorems for special ideals in Gorenstein local rings. These theorems take the form that if KI is contained in JI, then K is contained in J.
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Cancellation properties in ideal systems: an $\boldsymbol{e.a.b.}$ not $\boldsymbol{a.b.}$ star operation
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Let $M, N$ be finite modules over a Noetherian local ring $R$, and let $G$ be the associated graded ring of $R$. We show that the bigraded Hilbert series of $gr(Tor^R(M,N))$ is obtained from that of $Tor^G(gr(M),gr(N))$ by negative consecutive cancellations, thus extending a theorem of Rossi and Sharifan.
The Module Cancellation Problem solicits hypotheses that, when imposed on modules $K$, $L$, and $M$ over a ring $S$, afford the implication $K\oplus L\cong K\oplus M\Longrightarrow L\cong M$. In a well-known paper on basic element theory from 1973, Eisenbud and Evans lament the "great scarcity of strong results" in module cancellation research, expressing the wish that, "under some general hypothesis" on finitely generated modules over a commutative Noetherian ring, cancellation could be demonstrated. Singling out cancellation theorems by Bass and Dress that feature "large" projective modules,…
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Rees's theorem for arbitrary ideals
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Graded Version of Some Basic Theorems on Local Cohomology to a Pair of Ideals
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Structure theorems for certain Gorenstein ideals
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Syzygy Theorems via Comparison of Order Ideals on a Hypersurface
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Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals
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A Briancon-Skoda type theorem for graded systems of ideals
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When does the subadditivity theorem for multiplier ideals hold?
Demailly，Ein和Lazarsfeld \cite{DEL}证明了乘子理想的次可加性定理，即理想的理想的积的乘子理想包含在非奇异簇上的单个乘子理想的乘积中。证明了在二维情形下，次可加性定理在对数终端奇点上成立。然而，在高维的情况下，我们有几个反例。考虑了环面环上单项理想的次可加性定理，并在三维环面环上构造了一个反例。
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A Simple Proof of some Generalized Principal Ideal Theorems
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Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
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The colorful Helly theorem and colorful resolutions of ideals
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Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)
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