Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7. Regular rings 8. Flatness revisited 9. Derivations 10. I-smoothness 11. Applications of complete local rings Appendices References Index.

Commutative Ring Theory

1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Complexity and Real Computation

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

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Combinatorial Commutative Algebra

Significance: Chronic wounds include, but are not limited, to diabetic foot ulcers, venous leg ulcers, and pressure ulcers. They are a challenge to wound care professionals and consume a great deal of healthcare resources around the globe. This review discusses the pathophysiology of complex chronic wounds and the means and modalities currently available to achieve healing in such patients. Recent Advances: Although often difficult to treat, an understanding of the underlying pathophysiology and specific attention toward managing these perturbations can often lead to successful healing. Critical Issues: Overcoming the factors that contribute to delayed healing are key components of a comprehensive approach to wound care and present the primary challenges to the treatment of chronic wounds. When wounds fail to achieve sufficient healing after 4 weeks of standard care, reassessment of underlying pathology and consideration of the need for advanced therapeutic agents should be undertaken. However, selection ...

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Challenges in the Treatment of Chronic Wounds.

Foot infections are a common and serious problem in persons with diabetes Diabetic foot infections (DFIs) typically begin in a wound, most often a neuropathic ulceration While all wounds are colonized with microorganisms, the presence of infection is defined by ≥2 classic findings of inflammation or purulence Infections are then classified into mild (superficial and limited in size and depth), moderate (deeper or more extensive), or severe (accompanied by systemic signs or metabolic perturbations) This classification system, along with a vascular assessment, helps determine which patients should be hospitalized, which may require special imaging procedures or surgical interventions, and which will require amputation Most DFIs are polymicrobial, with aerobic gram-positive cocci (GPC), and especially staphylococci, the most common causative organisms Aerobic gram-negative bacilli are frequently copathogens in infections that are chronic or follow antibiotic treatment, and obligate anaerobes may be copathogens in ischemic or necrotic wounds Wounds without evidence of soft tissue or bone infection do not require antibiotic therapy For infected wounds, obtain a post-debridement specimen (preferably of tissue) for aerobic and anaerobic culture Empiric antibiotic therapy can be narrowly targeted at GPC in many acutely infected patients, but those at risk for infection with antibiotic-resistant organisms or with chronic, previously treated, or severe infections usually require broader spectrum regimens Imaging is helpful in most DFIs; plain radiographs may be sufficient, but magnetic resonance imaging is far more sensitive and specific Osteomyelitis occurs in many diabetic patients with a foot wound and can be difficult to diagnose (optimally defined by bone culture and histology) and treat (often requiring surgical debridement or resection, and/or prolonged antibiotic therapy) Most DFIs require some surgical intervention, ranging from minor (debridement) to major (resection, amputation) Wounds must also be properly dressed and off-loaded of pressure, and patients need regular follow-up An ischemic foot may require revascularization, and some nonresponding patients may benefit from selected adjunctive measures Employing multidisciplinary foot teams improves outcomes Clinicians and healthcare organizations should attempt to monitor, and thereby improve, their outcomes and processes in caring for DFIs

2012 Infectious Diseases Society of America Clinical Practice Guideline for the Diagnosis and Treatment of Diabetic Foot Infections

Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.

Commutative Algebra: with a View Toward Algebraic Geometry

Let R be a regular local ring, and let A = R/(x), where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A -module becomes periodic of period 1 or 2 after at most dim A steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. Let A be a commutative ring, and let x G A be a nonzero divisor. How does homological algebra over A/(x) = B differ from that over AI In this paper we will study a certain natural endomorphism / of complexes of free A / (x)-modu\es which seems to reflect some of the difference. For example, the (homotopic) triviality of t is an obstruction (closely related to the usual one in Ext2,) to the lifting of a complex of free 5-modules to a complex of free /I-modules. More generally, if x,, . . . , xn is an A -sequence, we study « natural endomorphisms /,,..., tn of complexes of free A/(xx, . . . , x")-modules, and try to use them to explain the way in which free resolutions over A/(xx, . . . , x") differ from free resolutions over A (the construction and elementary properties of these endomorphisms is given in §1). In this paper, we will study the case in which A is a regular local ring and B = A/(xx, . . . , x") is not regular. (It would also be very interesting to understand the case in which both A and A/(x) = B were regular-with, say, A of mixed characteristic and B ramified or of characteristic p.) In this case, the homological algebra over A is dominated, roughly speaking, by the fact that minimal /I-free resolutions are finite; we seek to understand the eventual behavior of minimal 5-free resolutions in terms of the tt. For example, if « = 1, so that B = A/(x), we prove that / is eventually an isomorphism, so that every minimal 5-free resolution becomes periodic of period 2 after at most 1 + dim B steps (§6). We also show that the 5-modules with periodic resolutions are the maximal Cohen-Macaulay modules without free direct summands. Since the periodic part of a periodic resolution over A/(x) (or more generally, over A/(xx, . . . , xn), if x,, . . . , x" is an A -sequence) is easy to describe explicitly (§5), this yields information on maximal Cohen- Macaulay modules.

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Homological algebra on a complete intersection, with an application to group representations

Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

This book gives a new foundation for the theory of links in 3-space modeled on the modern developmentby Jaco, Shalen, Johannson, Thurston et al. of the theory of 3-manifolds. The basic construction is a method of obtaining any link by "splicing" links of the simplest kinds, namely those whose exteriors are Seifert fibered or hyperbolic. This approach to link theory is particularly attractive since most invariants of links are additive under splicing. Specially distinguished from this viewpoint is the class of links, none of whose splice components is hyperbolic. It includes all links constructed by cabling and connected sums, in particular all links of singularities of complex plane curves. One of the main contributions of this monograph is the calculation of invariants of these classes of links, such as the Alexander polynomials, monodromy, and Seifert forms.

David Eisenbud

Papers

Commutative Algebra: with a View Toward Algebraic Geometry

Homological algebra on a complete intersection, with an application to group representations

Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

Three-Dimensional Link Theory and Invariants of Plane Curve Singularities.

Linear Free Resolutions and Minimal Multiplicity