Let $(A,\mathfrak{m} )$ be a hypersurface local ring of dimension $d \geq 1$ and let I be an $\mathfrak{m} $-primary ideal. We show that there is a integer rI$\geq\;-1$ (depending only on I) such that if M is any non-free maximal Cohen–Macaulay (= MCM) A-module the function $n \rightarrow \ell(\operatorname{Tor}^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree rI. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly, a key ingredient is the classification of thick subcategories of the stable category of MCM A-modules (obtained by Takahashi, see [11, 6.6]).

In this paper, we show existence of bountiful examples of Gorenstein local rings A and B such that there is a triangle equivalence between the stable categories CM(A), CM(B).

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$-split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$-adic filtration with the help of the numerical function $e^T_A$. In this article, we explore the relation between Auslander–Reiten (AR)-sequences and $T$-split sequences. For a Gorenstein ring $(A,\mathfrak{m})$, we define a Hom-finite Krull–Remak–Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\underline{\mathrm{CM}}(A)$. This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$.This article has two objectives: first objective is to extend the notion of $T$-split sequences, and second objective is to explore the function $e^T_A$ and $T$-split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is an $\mathfrak{m}$-primary ideal, then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer–Thrall-II for a class of such rings.

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, I an $\mathfrak{m}$-primary ideal. Let N be a nonzero finitely generated A-module. Consider the functions

\begin{equation*}t^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Ext}_A^i(N, A/I^n))\end{equation*}

of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$. We prove that $t^I(N) = e^I(N) = \max\{\dim N, d -1 \}$. A crucial ingredient in the proof is that $D^b(A)_f$, the bounded derived category of A with finite length cohomology, has no proper thick subcategories.

Let R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Let K be a field of characteristic zero and let R = K[X1, . . .,Xn], with standard grading. Let ${\mathfrak m}$ = (X1, . . ., Xn) and let E be the *injective hull of R/${\mathfrak m}$. Let An(K) be the nth Weyl algebra over K. Let I, J be homogeneous ideals in R. Fix i, j ≥ 0 and set M = HiI(R) and N = HjJ(R) considered as left An(K)-modules. We show the following two results for which no analogous result is known in charactersitc p > 0.

1. (i) $H^l_{\mathfrak m}$(TorRν(M, N)) ≅ E(n)al,ν for some al,ν ≥ 0.

2. (ii) For all ν ≥ 0; the finite dimensional vector space TorAn(K)ν(M♯, N) is concentrated in degree -n (here M♯ is the standard right An(K)-module associated to M).

There was a gap in the proof of Theorem 4.1 of [1]. In this corrigendum, we correct the error.

Let (A, ${\mathfrak{m}$ ) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$ -primary ideals with I a reduction of J. For i = 0,. . .,d, let e i J (A) (e i I (A)) be the ith Hilbert coefficient of J (I), respectively. We call the number c i (I, J) = e i J (A) − e i I (A) the ith relative Hilbert coefficient of J with respect to I. If G I (A) is Cohen–Macaulay, then c i (I, J) satisfy various constraints. We also show that vanishing of some c i (I, J) has strong implications on depth G J n (A) for n ≫ 0.

Let A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that

$$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$

$$\begin{linenomath}\begin{subeqnarray*}{\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\{\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right).\end{subeqnarray*}\end{linenomath}$$

Let K be a field and let R be a regular domain containing K. Let G be a finite subgroup of the group of automorphisms of R. We assume that |G| is invertible in K. Let RG be the ring of invariants of G. Let I be an ideal in RG. Fix i ⩾ 0. If RG is Gorenstein then:

1. (i) injdimRGHiI(RG) ⩽ dim Supp HiI(RG);

2. (ii) $H^j_{\mathfrak{m}}$(HiI(RG)) is injective, where $\mathfrak{m}$ is any maximal ideal of RG;

3. (iii) μj(P, HiI(RG)) = μj(P′, HiIR(R)) where P′ is any prime in R lying above P.

Let K be a field of characteristic zero, and let R = K[X1,… ,Xn ]. Let An(K) = K⟨X1,… ,Xn,∂1,… ,∂n⟩ be the nth Weyl algebra over K. We consider the case when R and An (K) are graded by giving deg Xi = ωi and deg ∂i = –ωi for i = 1,…,n (here ωi are positive integers). Set . Let I be a graded ideal in R. By a result due to Lyubeznik the local cohomology modules are holonomic (An (K))-modules for each i≥0. In this article we prove that the de Rham cohomology modules are concentrated in degree —ω; that is, for j ≠ –ω. As an application when A = R/(f) is an isolated singularity, we relate to Hn-1 (∂(f);A), the (n – 1)th Koszul cohomology of A with respect to ∂1 (f),…, ∂n (f).

Let $(A,\mathfrak{m})$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I)={{\oplus }_{n\ge 0}}{{I}^{n}}/\mathfrak{m}{{I}^{n}}$ be the fiber cone of $I$ . We prove certain relations among the Hilbert coefficients ${{f}_{0\,}}(I),\,{{f}_{1}}(I),\,{{f}_{2}}(I)$ of $F(I)$ when the $a$ -invariant of the associated graded ring $G(I)$ is negative.

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras and are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$ -primary ideal of a $d$ -dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$ , the multiplicity $e\left( I \right)$ , and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$ -primary ideals with a $d$ -generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$ .