Search

We show that the number of non-isotopic commutative semifields of odd order $p^{n}$ is exponential in $n$ when $n = 4t$ and $t$ is not a power of $2$. We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in $n$ and given by Zhou and Pott [Adv. Math. 234 (2013)]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003)] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004)]., 27 pages. Incorporates reviewer comments. To appear in Transactions of the American Mathematical Society

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain., to appear in Transactions of the AMS

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone., Comment: 13 pages. Substantial improvements in writing, existing theorems strengthened, several results added. arXiv admin note: text overlap with arXiv:1910.11498

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ on the $n$-dimensional affine spaces over finite extensions of $\mathbf F_p$. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree $4$ for alternating groups of degree $p^7-1$ for any odd prime $p$., Comment: New Proposition 7.2; improved bound in Theorem 5.1; minor changes and corrections, main results unchanged. Version accepted for publication

Let $I$ be an equidimensional ideal of a ring polynomial $R$ over $\mathbb{C}$ and let $J$ be its generic linkage. We prove that there is a uniform bound of the difference between the F-pure thresholds of $I_p$ and $J_p$ via the generalized Frobenius powers of ideals. This provides evidence that the F-pure threshold of an equidimensional ideal $I$ is less than that of its generic linkage. As a corollary we recover a result on log canonical thresholds of generic linkage by Niu., Minor revisions

Given a non-positive DG-ring $A$, associated to it are the reduction and coreduction functors $F(-) = \mathrm{H}^0(A)\otimes^{\mathrm{L}}_A -$ and $G(-) = \mathrm{R}\operatorname{Hom}_A(\mathrm{H}^0(A),-)$, considered as functors $\operatorname{\mathsf{D}}(A) \to \operatorname{\mathsf{D}}(\mathrm{H}^0(A))$, as well as the forgetful functor $S:\operatorname{\mathsf{D}}(\mathrm{H}^0(A)) \to \operatorname{\mathsf{D}}(A)$. In this paper we carry a systematic study of the categorical properties of these functors. As an application, a new descent result for vanishing of $\operatorname{Ext}$ and $\operatorname{Tor}$ over ordinary commutative noetherian rings is deduced., Comment: 13 pages, final version, to appear in Proceedings of the AMS

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Musta\c{t}\u{a}, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case., Comment: v2: minor fixes after referee report; 58 pages; comments appreciated

Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a family of quotients of the Rees algebra $R[It]$ has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When $R$ is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either $I$ is $\mathfrak{m}$-primary or $R$ is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored., Comment: To appear in the Proceedings of the American Mathematical Society

We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension., 8 pages; comments are welcome

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M fits into an exact triangle $T \to M \to F \to$ where T is a finite length complex of torsion modules and F is a finite length complex of "free" modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras ${\rm Sym}({\bf C}^{\infty} \oplus \bigwedge^2{\bf C}^{\infty})$ and ${\rm Sym}({\bf C}^{\infty} \oplus {\rm Sym}^2{\bf C}^{\infty})$ are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian., Comment: 29 pages

The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities., Comment: Final version, 50 pages, to appear in J. Algebraic Geom., appendix A will not appear in the journal version, comments are very welcome!

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements

We prove that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP., Comment: 16 pages; minor revision; to appear in the Proc. of the AMS

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”.

Ritt's theorem of zeroes and Seidenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs., 13 pages

Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity $3$. We classify graphs whose parity binomial edge ideals have pure resolution., 10 pages, Suggestions and comments are welcome. arXiv admin note: text overlap with arXiv:1911.10388

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed., Comment: To appear in Trans. Am. Math. Soc

Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete answer to it.

Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro when $A$ is a field of positive characteristic; our result applies, for example, when $A$ is the ring of integers., Comment: 7 pages

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n., Comment: 10 pages, to appear in PAMS

This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb{K}$. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on $\lfloor \frac{n}{2} \rfloor + 1$ sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for $n \leq 4$ we explicitly give minimal separating sets (with respect to inclusion) for all $m$ in case $\text{char}(\mathbb{K}) = 0$ or $\text{char}(\mathbb{K}) > n$., 12 pages, accepted for publication in Proc AMS

Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers., Comment: v2: added Subsections 6.1 (some remarks about compatibility with char. 0) 6.5 (other characterizations of Bernstein-Sato roots) and 6.6 (examples). Relaxed assumptions on R. 33 pages

In this paper we show that determinantal ideals of generic matrices are Knutson ideals. This fact leads to a useful result about Gröbner bases of certain sums of determinantal ideals. More specifically, given I = I 1 + … + I k I=I_1+\ldots +I_k a sum of ideals of minors on adjacent columns or rows, we prove that the union of the Gröbner bases of the I j I_j ’s is a Gröbner basis of I I .

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gr\"obner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety., Comment: Final version

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{X}$ an irreducible affine algebraic variety over $k$. Consider the linear difference equation $$ \sigma(Y)=AY $$ where $A\in \mathrm{GL}_n(k(\mathbb{X})(x))$ and $\sigma$ is the shift operator $\sigma(x)=x+1$. Assume that the Galois group $G$ of the above equation over $\overline{k(\mathbb{X})}(x)$ is defined over $k(\mathbb{X})$ i.e. the vanishing ideal of $G$ is generated by a finite set $S\subset k(\mathbb{X})[X,1/\det(X)]$. For a ${\bf c}\in \mathbb{X}$, denote by $v_{\bf c}$ the map from $k[\mathbb{X}]$ to $k$ given by $v_{\bf c}(f)=f({\bf c})$ for any $f\in k[\mathbb{X}]$. We prove that the set of ${\bf c}\in \mathbb{X}$ satisfying that $v_{\bf c}(A)$ and $v_{\bf c}(S)$ are well-defined and the affine variety in $\mathrm{GL}_n(k)$ defined by $v_{\bf c}(S)$ is the Galois group of $\sigma(Y)=v_{\bf c}(A)Y$ over $k(x)$ is Zariski dense in $\mathbb{X}$. We apply our result to van der Put-Singer's conjecture which asserts that an algebraic subgroup $G$ of $\mathrm{GL}_n(k)$ is the Galois group of a linear difference equation over $k(x)$ if and only if the quotient $G/G^\circ$ by the identity component is cyclic. We show that if van der Put-Singer's conjecture is true for $k=\mathbb{C}$ then it will be true for any algebraically closed field $k$ of characteristic zero., Comment: 35 pages

We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B ( r , s ) B(r,s) of the natural number parameters r r and s s so that for any system of algebraic ordinary differential-difference equations in the variables x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q and y = y 1 , … , y r \mathbfit {y} = y_1, \ldots , y_r , each of which has order and degree in y \mathbfit {y} bounded by s s over a differential-difference field, there is a nontrivial consequence of this system involving just the x \mathbfit {x} variables if and only if such a consequence may be constructed algebraically by applying no more than B ( r , s ) B(r,s) iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over C \mathbb {C} is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

We give examples of surfaces which are Ulrich-wild, i.e. that support families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$., Comment: 15 pages. Several typos have been fixed. The proof of Theorem 1.2 has been simplified. The final version will appear in Proceedings of the A.M.S

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\"omer., Comment: 20 pages

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of J{\o}rgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology., Comment: 41 pages, final version, to appear in Transactions of the AMS

Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $\delta$ of $B[Y_d]=B[y_1,\ldots,y_d]$ is elementary if $\delta(B)=0$ and $\delta(y_i)\in B$, $i=1,\ldots,d$. Then the elements $u_{ij}=\delta(y_i)y_j-\delta(y_j)y_i$, $1\leq i0$, $i=1,\ldots,d$, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009 if $\delta(y_i)=f_i(x_i)$, for any nonconstant polynomials $f_i(x_i)$, $i=1,\ldots,d$, then $B[Y_d]^{\delta}=K[X_d,Y_d]^{\delta}$ is generated by $X_d$ and $U_d=\{u_{ij}=f_i(x_i)y_j-y_if_j(x_j)\mid 1\leq i, Comment: 7 pages LATEX

This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call cores of $S$ includes the semigroup rings $k[H]$ of numerical semigroups $H$, but much larger than the class of numerical semigroup rings. For $R=k[H]$ and $M \in \operatorname{Max}R$, our result eventually shows that $\mu_{R}(M) \in \{1,2,\mu(H)\}$ where $\mu_{R}(M)$ (resp. $\mu(H)$) stands for the minimal number of generators of $M$ (resp. $H$), which covers in the specific case the classical result of O. Forster-R. G. Swan., Comment: 10 pages

In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two., 11 pages

We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of P^1 over a discretely valued field to be regular., Final version, to appear in the Journal of Algebraic Geometry. 31 pages

We determine the equations of the blowup of $\mathbb{P}^n$ along a $d$-fold rational normal scroll $S$, and we prove that the Rees ring and special fiber ring of $S \subseteq \mathbb{P}^n$ are Koszul algebras., To appear on Transactions of the American Mathematical Society

International audience; We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_\Delta$ denote the diagonal subalgebra of $S$ corresponding to $\Delta = \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_\Delta$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_\Delta$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul, and Cohen-Macaulay property of the diagonal subalgebras of their Rees algebras., Comment: Typos and errors have been fixed in the current version. Scheduled to appear in Proc. Amer. Math. Soc

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$ where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity. In the first case, we obtain the value $1/|G|$, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value $0$, providing an example of a ring where differential symmetric signature and F-signature are different., Final version. To appear in Proceedings of the American Mathematical Society

The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R), Comment: 10 pages, comments welcome

We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay., 17 pages

We show that the associated form, or equivalently a Macaulay inverse system, of an Artinian complete intersection of type $(d,\dots, d)$ is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities., Comment: 20 pages, to appear in Journal of Algebraic Geometry. v5: agrees with accepted version, except for formatting

In 2007 Huneke and Wiegand announced in an erratum that one of the conclusions of their depth formula theorem is flawed due to an incorrect convention for the depth of the zero module. Since then, the deleted claim has remained unresolved. In this paper we give examples to prove that the deleted claim is false, in general. Moreover, we point out several places in the literature which relied upon this deleted claim or the initial argument from 2007., 5 pages

We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for ${\rm CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the ${\rm CM}_t$ property is a topological invariant of a simplicial complex., 9 pages, 1 figure

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp., Comment: 25 pages Some some corrections are made in this final version

We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and we give a new combinatorial proof of a fundamental result of Miller on the relationship between Artininizations and Alexander duality of monomial ideals., Comment: The published version of this paper, Trans. Amer. Math. Soc. 371:3995--4027 (2019), claimed incorrectly that Theorem 7.1 is a new result when in fact it is a known consequence of a result of Ezra Miller. In this version we fix this and provide the appropriate references

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then sleuth for analogues where the ring $R$ is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal $Q$ inside of tensor products of domains of finite type over an algebraically closed field $\mathbb{F}$, we present binomial- and multinomial expansion criteria for containments of type $Q^{(E r)} \subseteq Q^r$, or even better, of type $Q^{(E (r-1)+1)} \subseteq Q^r$ for all $r>0$. The final section consolidates remarks on how often we can utilize these criteria, presenting an example., 10 pages, a follow-up to (arXiv:1608.02320). Rewritten both to address referee feedback and to reflect more recent developments in this research direction, as posted on arXiv in 2017. Abstract and Bibliography updated

Let $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $ \operatorname{Supp}_R(D(\operatorname{H}^i_{I}(R)))=\operatorname{Spec}(R)$., Comment: 7 pages. Conjecture 1 is modified