arXiv subjects

Joseph Lipman

Publications and source records attributed to Joseph Lipman.

At least 19 recordsLinked to original sources

arXiv2021

Cohomology with supports; idempotent pairs

This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about cohomology with supports--often under "finitary" conditions that obtain, e.g., under noetherian hypotheses--and its local and global interactions with quasi-coherence and with colimits, are reviewed from both the topological and scheme-theoretic perspectives. Some refinements of standard results are needed to accommodate certain features involving unbounded complexes and general systems of supports. An important attribute of such cohomology is "tensor-coreflectiveness," in its avatar--ultimately in the context of closed categories--as "idempotent pair," a notion which plays an important role in the sequel. Some basic facts about linearly topologized noetherian rings and their maps, related to cohomology with supports, and subsumed under properties of idempotent pairs, are brought forth; and similarly for the less-familiar context of formal schemes.

math.AG↗

arXiv2019

Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on category-theoretic considerations. It is not obvious that the two resulting theories are essentially the same. This is a semi-expository account of the connection between these approaches, a nontrivial matter involving some alluring relations, for instance among differential forms, residues and duality. In particular, it emerges that the culminating Ideal Theorem in Hartshorne's "Residues and Duality" holds for arbitrary essentially-finite-type maps of noetherian schemes and bounded-below complexes with quasi-coherent cohomology. What appears in this first part mostly concerns pseudo-coherent finite maps. The rest is being prepared.

math.AG↗

arXiv2017

On the fundamental class of an essentially smooth scheme-map

Let f: X -> Z be a separated essentially-finite-type flat map of noetherian schemes, and δ: X --> X \times_Z X the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor Lδ^*δ_* f^* to the relative dualizing functor f^! A compatibility between this C_f and derived tensor product is shown. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.

math.AG↗

arXiv2014

Relation between two twisted inverse image pseudofunctors in duality theory

Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian schemes a pseudofunctor f^\times right-adjoint to Rf_*, and a pseudofunctor f^! agreeing with f^\times when f is proper, but equal to the usual inverse image f^* when f is etale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by "compactly supported" versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.

math.AG↗

arXiv2014

Bivariance, Grothendieck duality and Hochschild homology, II: the fundamental class of a flat scheme-map

Fix a noetherian scheme S. For any flat map f: X->Y of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where \H(Z) is the (pre-)Hochschild complex of an S-scheme Z and f^! is the twisted inverse image coming from Grothendieck duality theory. When Y=S and f is essentially smooth of relative dimension n, this gives an isomorphism from n-th degree relative differential forms [ =H^{-n}(\H(X)) ] to f^!O_S[-n]. The basic results concern transitivity of c(-) vis-à-vis compositions X->Y->Z, and compatibility of c(-) with flat base change. These properties imply that c(-) orients the flat maps in the bivariant theory of part I, compatibly with essentially étale base change. Furthermore, c(-) leads to a dual oriented bivariant theory, whose homology is the classical Hochschild homology of flat S-schemes. When Y=S, c(-) is used to define a duality map \H(X)->RHom(\H(X),f^!O_S), an isomorphism if f is essentially smooth. These results apply in particular to flat essentially finite type maps of noetherian rings.

math.AG↗

arXiv2013

Adjoint associativity: an invitation to algebra in infinity-categories

There appeared not long ago a Reduction Formula for derived Hochschild cohomology, that has been useful e.g., in the study of Gorenstein maps and of rigidity w.r.t. semidualizing complexes. The formula involves the relative dualizing complex of a ring homomorphism, so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via a number of noncanonical projective and injective resolutions of differential graded objects. Recent efforts aim at more intrinsic approaches, hopefully upgradable to "higher" contexts--like bimodules over algebras in infinity-categories. This would lead to wider applicability, for example to ring spectra; and the methods might be globalizable, revealing some homotopical generalizations of aspects of Grothendieck duality. (The original formula has a geometric version, proved by completely different methods coming from duality theory.) A first step is to extend Hom-Tensor adjunction--adjoint associativity--to the infinity-category setting.

math.CT↗

arXiv2010

Bivariance, Grothendieck duality and Hochschild homology

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a fixed noetherian scheme S. The theory takes values in the category of symmetric graded modules over the graded-commutative ring \oplus_i H^i(S,O_S). In degree i, the cohomology and homology H^0(S,O_S)-modules thereby associated to such an x: X -> S, with Hochschild complex H_x, are Ext^i(H_x, H_x) and Ext^{-i}(H_x, x^!O_S). This lays the foundation for a sequel that will treat orientations in bivariant Hochschild theory through canonical relative fundamental class maps, unifying and generalizing previously known manifestations, via differential forms, of such maps.

math.AG↗

arXiv2010

Reflexivity and rigidity for complexes, II: Schemes

We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings.

math.AG↗

arXiv2009

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite type flat maps f: X->Y of noetherian schemes, with f^!(O_Y) in place of D.

math.AC↗

arXiv2009

Reflexivity and rigidity for complexes. I. Commutative rings

A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, a number of new results concerning derived reflexivity with respect to C are established. Noteworthy is the statement that derived C-reflexivity is a local property; it implies that a finite R-module M has finite G-dimension over R if it is locally of finite G-dimension.

math.AC↗

arXiv2007

Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor

For a map f: X -> Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, that is, the right adjoint f^\times of the derived functor Rf_* respects small direct sums. This is equivalent to the existence of a functorial isomorphism f^\times O_{Y} \otimes^L Lf^*(-) \to f^\times (-); to quasi-properness (preservation by Rf_* of pseudo-coherence, or just properness in the noetherian case) plus boundedness of Lf^* (finite tor-dimensionality), or of the functor f^\times; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of the local definition of pseudo-coherence of complexes, as well as a refinement of the known fact that the derived category of complexes with quasi-coherent homology is generated by a single perfect complex.

math.AG↗

arXiv2007

A vanishing theorem for finitely supported ideals in regular local rings

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It follows that the adjoint of a finitely supported ideal I of order a has order sup(a+1-d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, the adjoint of I is also finitely supported. Also: if this I is a normal ideal, then its reduction number is the least integer > d(1-1/a).

math.AC↗

arXiv2005

Duality and flat base change on formal schemes

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.

alg-geom↗

arXiv2005

A numerical criterion for simultaneous normalization

We investigate conditions for "simultaneous normalizability" of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective complex varieties X_y with parameter y ranging over a normal space--algebraic or analytic--admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure of the structure sheaf O_{X_y} is locally independent of y. When the X_y are curves projectivity is not needed, and the statement reduces to the well known δ-constant criterion of Teissier. Proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta, etc.) to more abstract algebraic ones.

math.AG↗

arXiv2004

Pseudofunctorial behavior of Cousin complexes on formal schemes

On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X functorially represent derived-category objects F by the local cohomologies H_x^{codim(x)}(F) (x \in X) together with "residue maps" from the cohomology at x to that at each immediate specialization of x; this representation is faithful when restricted to F which are Cohen-Macaulay (CM), i.e., H_x^i(F)=0 unless i=codim(x). Formal schemes provide a framework for treating local and global duality as aspects of a single theory. One motivation has been to gain a better understanding of the close relation between local properties of residues and global variance properties of dualizing complexes (which are CM). Our construction, depending heavily on local phenomena, is inspired by, but generalizes and makes more concrete, that of the classical pseudofunctor (-)^Δtaking values in residual complexes, on which the proof of Grothendieck's (global) Duality Theorem in Hartshorne's "Residues and Duality" is based. Indeed, it is shown in a subsequent paper by Sastry that (-)^# is a good concrete approximation to the fundamental duality pseudofunctor (-)^!. The pseudofunctor (-)^# takes residual complexes to residual complexes, so contains a canonical representative of (-)^Δ; and it generalizes as well several other functorial (but not pseudofunctorial) constructions of residual complexes which appeared in the 1990s.

math.AG↗

arXiv2003

Residues and duality for Cousin complexes

We construct a canonical pseudofunctor ^# on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor ^! to the full subcategory of the derived category having Cohen-Macaulay complexes as objects (a subcategory equivalent to the category of Cousin complexes, once a codimension function has been fixed). Specifically, for Cousin complexes M and any scheme map f:X -> Y as above, there is a functorial derived-category map γ: f^# M -> f^! M inducing a functorial isomorphism in the category of Cousin complexes f^# M \iso E(f^! M) (where E is the Cousin functor). γitself is an isomorphism if the complex f^! M is Cohen-Macaulay--which will be so whenever the map f is flat or whenever the complex M is injective. Also, f^# takes residual (resp. injective) complexes on Y to residual (resp. injective) complexes on X; and so the pseudofunctor ^# generalises--and makes canonical--the "variance theory" of residual complexes developed in Chapter VI of Hartshorne's "Residues and Duality." Moreover, we generalise the Residue Theorem of loc.cit., p.369 by defining a functorial Trace map of graded modules Tr_f(M): f_*f^# M -> M (a sum of local residues) such that whenever f is proper, Tr_f(M) is a map of complexes and the pair (f^# M, Tr_f(M)) represents the functor Hom(f_*C, M) of Cousin complexes C.

alg-geom↗

arXiv2003

Integrally closed ideals in two-dimensional regular local rings are multiplier ideals

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete ones primary for the maximal ideal are those of order one.

math.AC↗

arXiv1998

Equisingularity and simultaneous resolution of singularities

The purpose, mainly expository and speculative, of this paper---an outgrowth of a survey lecture at the September 1997 Obergurgl working week---is to indicate some (not all) of the efforts that have been made to interpret equisingularity, and connections among them; and to suggest directions for further exploration. Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by ``dimensionality type," an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratification by simultaneous resolvability of singularities. The two approaches are known to be equivalent for families of plane curve singularities. In higher dimension we ask whether constancy of dimensionality type along a smooth subvariety W of V implies the existence of a simultaneous resolution of the singularities of V along W. (The converse is false.) The underlying idea is to follow the classical inductive strategy of Jung--begin by desingularizing the discriminant of a generic projection--to reduce to asking if there is a canonical resolution process which when applied to quasi-ordinary singularities depends only on their characteristic monomials. This appears to be so in dimension 2. In higher dimensions the question is quite open.

math.AC↗