Papers

We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that ordinary finite-flat descent applies naturally to $A\to C$, whereas the passage $C\to C_{\mathrm{red}}$ may destroy nilpotent information.

Let $G$ be a group and $k$ be a commutative ring. The Gorenstein homological dimension $\text{Ghd}_k G$ and the Gorenstein cohomological dimension $\text{Gcd}_k G$ of $G$ over $k$ are defined as the Gorenstein flat and the Gorenstein projective dimension, respectively, of the trivial $kG$-module $k$. We prove that the Gorenstein cohomological dimension of a countable group $G$ is at most one more than its Gorenstein homological dimension and, more generally, that $\text{Gcd}_k G\le\text{Ghd}_k G+m+1$ for every $m\ge0$ and every group $G$ of cardinality at most $\aleph_m$. When the supremum sfli$k$ of the flat dimensions of the injective $k$-modules is finite, this gives $\text{Ghd}_k G\le\text{Gcd}_k G\le\text{Ghd}_k G+1$ for every countable group $G$, so that the two dimensions are finite simultaneously. These results are Gorenstein analogues of Bieri's inequality for the homological and the cohomological dimension of countable groups. They rest on a module-theoretic theorem of independent interest, which is a Gorenstein version of results of Jensen and Osofsky on the projective dimension of flat modules: over an arbitrary ring, every $\aleph_m$-presented Gorenstein flat module has projectively coresolved Gorenstein flat dimension at most $m+1$. An appendix characterizes the modules of PGF-dimension at most $n$ as the direct summands of the strongly $n$-PGF modules.

2-step ideals are ideals $I$ of the polynomial ring that satisfy $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ where $\mathfrak{m}$ is the maximal ideal generated by variables. This class of ideals was recently introduced in [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] with the aim to obtain new irreducible components of $\mathrm{Hilb}^d(\mathbb{A}^n)$ and in particular to obtain new loci in $\mathrm{Hilb}^d(\mathbb{A}^3)$ of large dimension. In this paper we use the correspondence between Hilbert schemes and varieties of commuting matrices to define loci of commuting matrices that correspond to 2-step ideals. Then we estimate the dimensions of the obtained loci to get new proofs for the estimates of dimensions of loci of 2-step ideals in the case $n=3$. In this case we are also able to omit some assumptions in the dimension estimates in the above mentioned paper.

If $f$ is a tuple of functions satisfying an algebraic ODE and $P\in{\mathbb C}(x)[f]$, it is common in applications to transcendental number theory to consider upper bounds for the order of zero of $P(x,f)$ at a given point in terms of $\operatorname{deg}_x P,\operatorname{deg}_f P$. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field $\mathbb C(x)$ is replaced by a number field $K$ and $\operatorname{deg}_x P$ is replaced by the logarithmic height $\operatorname{h}(P)$. Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.

We propose a local-cohomology obstruction to the existence of small Cohen–Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\equiv4H$ numerically. Put $c(Y,H)=15H^2/8-\chi(\OO_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of $\bigoplus_{n\geq0}H^0(Y,\OO_Y(nH))$ satisfies $\dim_\C H^2_\mm(M)\geq c(Y,H)\rk M$. The argument combines elementary transformations of reflexive extensions, Harder–Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch–Kummer family, $H^2=1080$ and $\chi(\OO_Y)=1926$, so the bound is $99\rk M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen–Macaulay modules.

Let $\mathbf A$ be a totally nonnegative matrix over a real closed non-Archimedean valued field, and suppose that its valuation $A$ is tropically totally nonnegative. Although the valuations of the classical eigenvalues are weakly majorized by the tropical eigenvalues of $A$, equality can fail through cancellation. We give a necessary and sufficient condition for equality. The only obstruction is dominant cancellation in the principal minors that give the essential coefficients of the tropical characteristic polynomial. In other words, equality holds exactly when the classical and tropical Newton polygons agree at the breakpoints determined by the diagonal entries. We describe the exceptional lifts by explicit polynomial equations in the leading coefficients. We also study these equations on each irreducible component of the admissible parameter space. Finally, we give a complete $3\times 3$ criterion and a family that contains both ordinary lifts with spectral equality and exceptional totally nonnegative lifts without it.

Prime and Cohen-Macaulay ideals of 2-minors of matrices of variables with linear resolution are classified.

Let $E$ be the exterior algebra on $n$ generators over a field, and let $F$ be a graded free $E$-module with $r$ generators, of degrees $d_1 \le \dots \le d_r$. We determine the convex hull of the set of Hilbert functions of the quotients $F/M$, where $M$ runs over the monomial submodules of Amata and Crupi. The hull is the Minkowski sum of $r$ shifted copies of one and the same simplex: the Kozlov simplex, which Kozlov proved in 1997 to be the convex hull of the $f$-vectors of simplicial complexes on $n$ vertices. For $r=1$ the statement is Kozlov's theorem, so this is a rank-$r$ generalization of it. We give a self-contained proof, including a short proof of Kozlov's theorem from the local Lubell-Yamamoto-Meshalkin inequality, and an explicit facet description of the Kozlov simplex that appears not to be recorded in the literature. Along the way we isolate the class of submodules for which the argument works, the ideal-direct-sum modules, and show by example that the Hilbert function of a graded submodule outside that class need not be a shifted sum of $f$-vectors at all. The combinatorial content is stated separately, in language that uses no algebra, as a result on $r$-vectors of simplicial complexes and their $ff$-vectors. Finally we describe the vertex structure of the Minkowski sum – which sums of vertices of the summands survive as vertices – proving the answer for the all-ones leg vector at every rank and for the Kozlov leg vector at rank two.

Using N. Honda's construction of twistor spaces, bimeromorphic invarince of $\partial\bar\partial$-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the $\partial\bar\partial$-lemma that contains a smooth compact surface for which the lemma fails. This disproves heredity in the smallest possible ambient dimension and yields counterexamples to L. Alessandrini's modification question in every complex dimension at least four. We establish a derived blow-up formula for generalized Bott–Chern complexes with coefficients, compatible with the comparison maps to coefficient cohomology. We also obtain a Dolbeault blow-up formula with bounded derived coefficients. For compact complex manifolds, we introduce a relative $\partial\bar\partial$-property, characterize it using the Frölicher spectral sequence and Hodge filtrations, and prove a blow-up criterion. Finally, counterexamples show the limitations of the classical formulae for coherent coefficients and singular ambient spaces, and the failure of the naive Bott–Chern Künneth formula.

Let $(K^h,v^h)$ be a Henselization of a valued field $(K,v)$, $\mu$ any extension of $v$ to $K[x]$, and $\mu^h$ the canonical extension of $\mu$ and $v^h$ to $K^h[x].$ In this paper, we study two new invariants associated with a Mac Lane-Vaquié chain (MLV) of $\mu$, namely, the defect of each augmentation step and the regularity of an MLV chain of $\mu$. This, in turn, extends the notion of regularity of a complete sequence of abstract key polynomials for $\mu$ to the regularity of $\mu$ itself. We also prove that the regularity of $\mu$ is a sufficient condition for the equality of the depths of $\mu$ and $\mu^h$. Furthermore, it is a necessary and sufficient condition for the equality of the defect and relative gap of each augmentation step of an MLV chain of $\mu.$ Finally, we show that if $\mu$ has finite degree, then the regularity of $\mu$ is equivalent to $ \operatorname{deg}(\mu)=\operatorname{deg}(\mu^h)$.

We introduce the coproduct of ideals and the notion of coprime ideal, extending the Heyting-algebra perspective on ideal lattices to arbitrary commutative rings. The resulting coprime spectrum is a topological space, defines a covariant functor on CF-morphisms of commutative rings, i.e. morphisms that extend ideals and are coprime faithful, classifies fields among integral domains, and in dual rings is homeomorphic to the prime spectrum. This offers an elementary, lattice-theoretic counterpart to the geometric study of nilpotents.

We extend quasi-projective dimension and quasi-projective length from finitely generated modules to homologically finite complexes by using finite filtrations in the derived category. Our definitions recover the original invariants of Gheibi–Jorgensen–Takahashi for modules and behave well under exact functors, which simplifies the proofs of several results. We establish the Auslander–Buchsbaum formula, the derived depth and width formulas, and the dependency formula for complexes of finite quasi-projective dimension. Extending a result of Gheibi–Jorgensen–Takahashi, we show that every homologically finite complex has finite quasi-projective dimension over a suitable complete intersection ring. We also prove a new intersection theorem and a descent theorem for Serre's conditions. Finally, we obtain vanishing results for Tor, Ext, and Tate (co)homology, and also symmetry of eventual Ext vanishing.

Let $(R,\mathfrak m)$ be a Noetherian local ring and let $\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$ be graded $\mathfrak m$-primary families. We introduce weak asymptotic and strong homogeneous joint reductions and prove corresponding extensions of Rees's mixed multiplicity theorem relating joint reductions and mixed multiplicities. Actually, for weak asymptotic joint reductions, mixed multiplicities are realized as normalized limits of Hilbert–Samuel multiplicities, yielding an asymptotic Rees theorem. For multigraded admissible collections, a strong homogeneous joint reduction $\{x_{ij}\}$ of type $\mathbf d=(d_1,\ldots,d_s)$, with $x_{ij}\in I_{a_{ij}}^{(i)}$, satisfies the formula \[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \] We also prove a localized converse under formal equidimensionality, recovering the classical converse of Rees in the adic case. As an application, we derive a Böger-type theorem for graded families, whose adic specialization recovers Böger's classical theorem. In contrast, the converse to the weak asymptotic Rees theorem fails even for an adic family on a one-dimensional regular local ring.

We consider local cohomology sheaves on compact Hermitian symmetric spaces supported in Schubert varieties, which carry natural structures as mixed Hodge modules. Our main result gives an explicit root-theoretic construction, uniform across Lie types, for the D-simple composition factors and weight filtration on these modules. For Lagrangian and orthogonal Grassmannians, we translate these formulas into combinatorial rules involving strict partitions and shifted Dyck patterns, analogous to existing formulas on ordinary Grassmannians. As an application, we expand upon work of Raicu–Weyman to describe the weight filtration on local cohomology supported in symmetric determinantal and Pfaffian varieties. We also give formulas for the Hodge rational homology level and local cohomological defect of every Schubert variety in a compact Hermitian symmetric space. Finally, we tabulate the local cohomology weight filtrations for all Schubert varieties in the two exceptional Hermitian pairs.

In this paper, we study the ${\rm v}$-number of the symbolic-ordinary discrepancy modules $M_t(I) = I^{(t)} / I^t$ over standard graded Noetherian rings. We show that this coincides with the ${\rm v}$-number of $I^t$ at embedded primes. Moreover, under mild assumptions on $I$, we show that the ${\rm v}$-number of $N_t(I) = I^{(t)} / I^{(t+1)}$ is the same as the ${\rm v}$-number of $I^{(t+1)}$. Furthermore, we provide combinatorial formulas for these ${\rm v}$-numbers when $t = 2, 3$ and $I$ is the edge ideal of a graph. Consequently, we classify all graphs for which ${\rm v}(I^2) = {\rm v}(I^{(2)})$.

Let $\pi:S\twoheadrightarrow R$ be a split ring epimorphism with nilpotent kernel $J$, and let $R\hookrightarrow S$ be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if $X$ is a left $S$-module satisfying $\Tor_i^S(R,X)=0$ for every $i>0$, then \[ \pd_S X=\pd_R(R\otimes_S X) \] in $\mathbb N\cup\{0,\infty\}$. In particular, for every flat left $S$-module $F$, \[ \pd_SF=\pd_R(F/JF), \] and $F/JF$ is flat over $R$. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over $S$ and over $R$. Consequently, left $n$-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring $R$ and an arbitrary $R$-module $M$, this applies to the idealization $S=R\ltimes M$ without any projectivity, flatness, finite generation, or finite flat-dimension assumption on $M$. We obtain a list of equivalent characterizations of the condition that $R\ltimes M$ is $n$-perfect, together with \[ \cotD(R\ltimes M)=\cotD(R). \] We further derive exact preservation of the flat-projective-dimension spectrum, strict $n$-perfectness, pointwise localization, arbitrary iterated idealizations, $n$-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.

Let $U$ be an affine log Calabi–Yau variety. Although special and finitely generated valuations are defined using a log CY-Fano compactification of $U$, we prove that the corresponding skeleta in the dual complex are independent of this choice, and hence invariant under $\mathrm{Aut}(U)$. We also show that a finitely generated valuation is special if and only if it is maximal with respect to a natural partial order induced by regular functions on $U$. We then consider the affine log Calabi–Yau threefold obtained as the complement of the Markov cubic surface in $\mathbb{A}^3$. We describe the action of the three Vieta involutions on the special skeleton and relate it to the $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane. We also show that, for every finite triangulation of the dual complex, the special skeleton fails to be locally closed on some simplex. Via the cone construction, this provides a counterexample to a conjecture of the author and Xu.

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition–deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.

We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span. We show that the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We apply these results, for every integer $q\geq 1$, to a homogeneous $A$-hypergeometric system of lattice rank 2 with $5q$ columns. We show that the canonical formal solution space of this system has dimension $12q^2$ and that the span of the series obtained by intrinsic perturbation has codimension exactly $\lceil 3q^2/4\rceil$ in that space.

We construct commutative rings whose derived categories have non-spatial frames of smashing ideals, thereby answering in the negative questions of Balchin–Stevenson and Aoki.

We determine the multiplicative structure on the Tor algebras associated to the coordinate rings of general points in $\mathbb{P}^3$ in terms of the number of points. We show that in codimension four, the Waring rank of a form determines the Tor algebra class of its apolar Artinian Gorenstein ring for a dense subset of the forms of fixed Waring rank and sufficiently large degree.

In a question recorded in 1978, Vasconcelos asked whether the integral closure of a one-dimensional coherent local domain in its fraction field is Prüfer. We construct such a domain whose integral closure is not Prüfer, giving a negative answer.

Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$. Let $X\to\mathbb{P}^1_K$ be a cyclic cover of degree $n$ prime to the characteristic of $k$. In joint work with Obus and Srinivasan (arXiv:2609.20553), we define an integer called the conductor-discriminant contribution $\text{cdc}(y)$ associated to a multiplicity $2$ point $y$ of the branch divisor of the normalization in $K(X)$ of a regular $\mathcal{O}_K$-model $\mathcal{Y}$ of $\mathbb{P}^1_{K}$; modulo several key results about $\text{cdc}(y)$, we prove a conductor-discriminant inequality for $X$, extending previous work of Ogg, Saito, Liu, Srinivasan, and Obus$-$Srinivasan. In this companion paper, we supply the necessary technical results for $\text{cdc}(y)$. In particular, we show $\text{cdc}(y)$ is non-negative except under highly restrictive conditions on $n$ and the structure of the branch divisor at $y$. Moreover, if $\text{cdc}(y)$ is negative, we show the spectrum of the complete local ring of any point lying over $y$ under the normalization of $\mathcal{Y}$ in $K(X)$ is a rational double point. Along the way, we show the non-negativity of a related quantity, the conductor exponent-discriminant contribution $\text{cedc}(y)$, which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.

We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve $X$ is given by $y^n = f(x)$ with $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.

Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$. We also construct a nine-dimensional projective $F$-rational variety $\mathfrak{X}$ and a closed embedding $\mathfrak{X}\hookrightarrow\mathbb{P}^{29}_k$ under which the same conclusion holds.

We investigate an algebraic approach to the Syndrome Decoding Problem, based on a reformulation of the Hamming weight constraint and its integration with the Information Set Decoding paradigm. We begin with a systematic analysis of the Hamming variety, deriving its defining equations in terms of elementary symmetric functions. Since these equations may have high degree, we exploit convolution identities for elementary symmetric functions, together with factorizations based on Lucas' identity, to derive an equivalent formulation with auxiliary variables and equations of bounded degree. Building on this modeling, we generalize the ISD paradigm through an ISD-like decoding strategy, implemented by the GBDecode algorithm, in which only a subset of an information set is fixed. This approach reduces the size of the combinatorial search space at the cost of solving the associated multivariate nonlinear systems. To handle this algebraic component, we employ the MultiSolve algorithm, which replaces a single Grobner basis computation with a collection of computations on simpler systems, obtained by exhaustively assigning a varying number of indeterminates over the finite field. This provides a tunable balance between combinatorial search and algebraic solving. We evaluate the resulting approach experimentally on instances of the Syndrome Decoding Problem for random binary linear codes, using parameters corresponding to the NIST Security Category 1 parameter set of the Classic McEliece cryptosystem. The experiments assess the feasibility of this combinatorial-algebraic approach and provide insights into the practical behavior of Grobner basis techniques within an ISD-like decoding framework.

Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let $\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $\Phi \colon\underline{CM}(A) \rightarrow \underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then $dim A = dim B$. (2) If $M$ is a maximal \CM \ $A$-module then $$\text{curv}_A(M) = \text{curv}_B (\Phi(M)),$$ here $\text{curv}_A(M) = \limsup_n \sqrt[n]{\ell(\text{Tor}^A_n(M,k))}$. (3) $A$ satisfies Serre's condition $R_i$ if and only if $B$ satisfies $R_i$. (4) $A$ is a complete intersection on the punctured spectrum of $A$ if and only if $B$ is a complete intersection on the punctured spectrum of $B$. We also show that $\Phi$ imposes constraints of residue field of $B$ in terms of residue field of $A$ and vice-versa. Finally if $I$ is an ideal in $A$ such that the extended Rees algebra $\mathcal{R}(I) = A[It, t^{-1}]$ is Gorenstein then we construct a triangulated functor $\Psi \colon \underline{CM}^\mathbb{Z}(\mathcal{R}(I)) \rightarrow \underline{CM}(A)$ where $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\mathcal{R}(I)$-modules. We show that $\Psi$ induces an equivalence $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))/\ker \Psi \rightarrow \underline{CM}(A)$. We give some applications of this map.

Problem L of Fröberg–Lundqvist–Oneto–Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for \(n\) general points of \(\PP^{n-1}\). Besides a closed formula for \[ \HS(S/I^m)-\HS(S/I^{(m)}), \] we determine all minimal monomial generators of \(I^{(m)}\), and describe the symbolic Rees algebra. We also show that containment \(I^{(m)}\subseteq I^r\) is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant \(\walpha\), the resurgence \(\Res\), and the asymptotic resurgence \(\aRes\): \[ \walpha(I)=\frac{n}{n-1}, \qquad \Res(I)=\aRes(I)=\frac{2(n-1)}{n}. \] We also take the first step beyond \(n\) points: for \(n+1\) general points of \(\PP^{n-1}\) — again a rigid, non-monomial configuration — we identify the defining quadrics, resolve the case \(n=3\) completely (a complete intersection, with \(J^{(m)}=J^m\) for all \(m\) and resurgence \(1\)), and propose an exact Waldschmidt-constant formula \(\walpha=\frac{n+1}{n-1}\) for all \(n\), verified computationally in every case we could check.

We study density functions associated with filtrations of graded ideals in standard graded domains. Extending earlier work, we develop a general theory for arbitrary filtrations and establish their fundamental properties, including existence, continuity, log-concavity, and integral representations. We show that every density function is locally uniformly approximable by piecewise polynomial density functions arising from truncated Noetherian filtrations. We prove differentiability properties of saturation density functions and give a numerical characterization of when the symbolic powers of a graded radical ideal in a polynomial ring coincide with the integral closures of its ordinary powers. Finally, we establish several inequalities for the mixed multiplicities of equigenerated ideals.

We construct a commutative ring whose big finitistic projective dimension is $1$ and whose big finitistic injective dimension is $2$. The example contradicts the comparison suggested by Bass for commutative rings.