Patricia Klein Research Statement My area of research is commutative algebra. The goal of the research I propose is to gain a better understanding of Koszul homology itself and also, through the use of homological eI (M ) algebra, Hilbert-Samuel multiplicities. Much of my research concerns the ratio `(M/IM ) for a module M over a local ring (R, m) and an m-primary ideal I. In 1960, Lech [Lec60] gave an upper bound for this ratio independent of I when M = R, and a lower bound has been sought since at least 1996 [SV96]. Linquan Ma, Pham Hung Quy, and I show that the ratio is bounded away from 0 and infinity in the greatest possible generality. Independently, I give a simpler proof of Lech’s original result and give explicit bounds on the ratio in question in special cases. I also give a new characterization of the property of being Cohen– Macaulay on the punctured spectrum in terms of Koszul homology. This research statement includes background on all of these topics as well as my previous work and a description of research projects to be undertaken as a postdoc. Background on commutative algebra. Broadly speaking, the goal of commutative algebra is to study solutions to polynomial equations. Commutative algebra is closely related to algebraic geometry, which is the study of the shapes that solutions to polynomial equations produce. For example, one may gain information about y 2 + x2 = 1 either by studying the algebraic properties of the equation itself or by graphing the unit circle, which y 2 + x2 = 1 describes, and studying the geometry of the circle. By taking derivatives, we can show that as x increases between -1 and 0, if y is positive then it is also increasing. We learn the same fact by looking at the portion of the graph of the unit circle in the second quadrant. One of the primary goals of commutative algebra and algebraic geometry is detecting and managing singularities. A singularity is a generalization of the notion of a point of nondifferentiability on a manifold. Understanding singularities is not only of theoretical interest to commutative algebraists and algebraic geometers but also has myriad applications, including phylogenetics [ARS17, ERSS05, AR08], disclosure limitation [SF04, DFR+ 09, HS02], string theory [Asp13], and statistics [Gib10, PRW00, GSS05, DSS08, Wat01], for example. Mathematicians began to explore many of the tools of commutative algebra and algebraic geometry in the late nineteenth and early twentieth centuries with only the goal of understanding pure mathematics in mind. Then, in the same way that a rigorous notion of the limit suddenly found an application when Newton and Leibniz invented calculus and were then able to make vast gains in physics, the already well-developed fields of commutative algebra and algebraic geometry found a wealth of applications in the wake of the advent of computers, when the the reframing of certain biological, physical, and statistical questions in terms of algebra became computationally tractable. The goal of my research is to grow the understanding of the fields of commutative algebra and algebraic geometry so that they will be broad and deep enough to answer the questions of applied mathematicians and statisticians when they come asking. The algebraic objects I study are called rings, which describe geometric varieties. For example, the ring R = C[x, y]/(y − x2 ) refers to all polynomials in two variables over the complex numbers subject to the condition that y = x2 . The variety this ring cuts out is the parabola y = x2 in the Cartesian plane. The tools that I use to study singularities of varieties are the Hilbert-Samuel multiplicity (or simply multiplicity) and Koszul homology, 1

which can be used to compute the multiplicity. We can compute the multiplicity at different points on our variety to tell us whether or not the variety has a singularity at that point. The multiplicity of R is 1 at the origin, which corresponds to the fact that parabolas are smooth. This relationship is true quite broadly. Under mild hypotheses, a variety does not have a singularity at a certain point if and only if the multiplicity of its corresponding ring at the localization at the maximal ideal corresponding to that point tis 1 . Large multiplicities indicate bad singularities. For example, the nodal curve graphed below has a multiplicity of 2, which captures that the graph has a mild-looking singularity at the origin, and the cusp, which is much pointier, has multiplicity 10.

nodal curve: y 3 = x3 + x2

cusp: y 31 = x10

More formal background on Koszul homology and multiplicities. All rings in this proposal will be commutative Noetherian local rings with unity and all modules unital and finitely generated. Let f = f1 , . . . , fn be a sequence of elements in the ring R and M an R-module. We now define the Koszul complex of f on M , denoted K• (f ; M ). If n = 1, f1

then K• (f1 ; M ) is the complex 0 → M − → M → 0 with the left M in degree 1 and the right M in degree 0, and if n > 1, we define the Koszul complex to be the total complex of K• (f1 , . . . , fn−1 ; M ) ⊗R K• (fn ; M ). We may think of Koszul homology as encoding the relations among any i of the fj on M modulo relations given by any i-1 of the fj . When the fj form a regular sequence on R, the Koszul complex forms a free resolution of R/(f ). This complex is the best understood of all free resolutions and is particularly valued in cases where a goal is computability (see, for example, [EG12, Vas04]). The Koszul complex is essential in proving foundational results on the rigidity of Tor over regular rings [Aus61, L+ 66]. Furthermore, the Koszul complex, and in particular its homology, denoted Hi (f ; M ) for the ith Koszul homology module, carries a great deal of algebro-geometric information. For example, the depth of (f ) on M is equal to the least i such that Hn−i (f ; M ) 6= 0, and M is Cohen–Macaulay on the punctured spectrum (i.e. after localization at any prime ideal P 6= m) if and only if all of the higher Koszul homology of some (equivalently, every) system of parameters on M is finite length. A system of parameters on M is a sequence of d = dim(M ) elements f1 , . . . , fd such that M/(f1 , . . . , fd )M is finite length. An ideal generated by a system of parameters on R is called a parameter ideal. We define the Hilbert–Samuel multiplicity of an m-primary ideal I of the local ring (R, m) `(M/I t M ) on a d-dimensional R-module M to be eI (M ) := d! lim , where `(−) denotes the t→∞ td length of a module. Multiplicities were first introduced in this generality by Samuel in 1951 [Sam51] and since then have been studied in great depth by commutative algebraists and algebraic geometers and have found applications in areas of math ranging from Banachspace operators to Galois theory (see, for example, [HY02, MER95, CLU08, Esc07a, Kis09, El´ı90, Esc07b, DS99]). A theorem of Serre’s [Ser97] states that the multiplicity of a pad X rameter ideal I on the d-dimensional module M may be computed as (−1)i `(Hi (f ; M )) i=0

for any minimal generating set f = f1 , . . . , fd of I. A result of Lech’s [Lec57] is that 2

`(M/(f1t1 , . . . , fdtd )M ) mini ti →∞ −−−−−−→ 1 whenever f1 , . . . , fd are parameters on M . It is natue(f t1 ,...,f td ) (M ) 1 d ral in light of these two results to ask whether it is the case that for all parameter ideals `(M/IM ) is close to 1 I = (f1 , . . . , fd ) in sufficiently high powers of the maximal ideal, eI (M ) `(Hi (f ; M )) and whether is close to 0 for all i > 0. My thesis answers this question quite `(M/IM ) broadly in the complete equidimensional case. Theorem 1. If M is an equidimensional module over the complete local ring (R, m, κ) of dimension d ≥ 1 that is either of equal characteristic or in which char(κ) is a parameter, then the following four conditions are equivalent: (1) For every sequence of parameter ideals (fn1 , . . . , fnd ) = In ⊆ mn and every i < k, `(H i (fn1 , . . . , fnd ; M )) lim = 0, n→∞ `(M/In M ) √ (2) sup{`(H i (f1 , . . . , fd ; M )) | f1 , . . . , fd = m, i < k} < ∞, i (M )) < ∞ for all i < k. (3) `(Hm Condition (3) is well-known to be equivalent to the condition depthP MP ≥ height(P )+k −d for all prime ideals P 6= m. As part of the proof that (1) implies (3), I give examples of se`(R/In ) n→∞ quences of parameter ideals In ⊆ mn such that −−−→ 1 for R = k[[x1 , . . . , xd ] `(R/(P + In )) and P = (x1 , . . . , xd−1 ) for any d ≥ 2. This result is surprising because dim(R) = d and dim(R/P ) = 1. I am, furthermore, able to use Theorem 1 to construct examples of rings `(R/In ) (R, m) with a sequence of parameter ideals In ⊆ mn such that 6→ 1 as n → ∞. eIn (R) Example 1. Let R = ( x3k[x,y,z] sk[u, v])[w] localized at the homogeneous maximal ideal, +y 3 +z 3 where s denotes the Segre product.. Because R is normal, it is in particular depth 2, and so its only nonvanishing Koszul homology modules are i = 0, 1. It is not Cohen-Macaulay on the punctured spectrum and so has some sequence of parameter ideals (fn1 , . . . , fnd ) = In `(Hi (fn1 , . . . , fnd ; R)) such that for some 1 ≤ i ≤ 4, 6→ 0 as n → ∞ by Theorem 1. `(R/In ) `(H1 (fn1 , . . . , fnd ; R)) 6→ Therefore, we may pick a sequence of parameter ideals In such that `(R/In ) 0. Recalling that H0 (fn1 , . . . , fnd ; R) ∼ = R/In , it follows that eIn (R) `(H0 (fn1 , . . . , fnd ; R)) − `(H1 (fn1 , . . . , fnd ; R)) = `(R/In ) `(R/In ) `(H1 (fn1 , . . . , fnd ; R)) =1− 6→ 1 as n → ∞. `(R/In ) Project 1. I propose to work on several lines inquiry extending from Theorem 1 as a postdoc. In the setting of Theorem 1, when M is not Cohen-Macaulay on the punctured spectrum, the theorem tells us that there exist sequences of parameter ideals In such that `(H i (fn1 , . . . , fnd ; M )) lim 6= 0, while Theorem 4, below, gives the existence of an upper n→∞ `(M/In M ) 3

bound for the ratio. One might ask, then, whether or not there exists some sequence of parameter ideals giving a limit anywhere between 0 and that upper bound or whether only certain numbers can appear as a limit of this type. Similar questions, especially in light of `(R/In ) [Lec57], are whether there exists a sequence of parameter ideals In such that →r eIn (R) for each r between 1 and the lower bound guaranteed by Theorem 2, below, and which sequences In give a limit of 1. Characterizing sequences of ideals giving a limit of 1 would extend Lech’s 1957 result, which shows that some such sequences exist, and my results, such as Example 1, which show that not all sequences of parameter ideals in high powers of the maximal ideal give a limit of 1. I also expect to be able to remove the requirement that char(κ) be a parameter in R in the mixed-characteristic case. This line of attack will require a careful and detailed study of properties of the duals of local cohomology modules. I extend this understanding of the relationship between Kosul homology, multiplicities, and lengths of quotients by parameter ideals in theorems 2, 3, and 5, which are joint work with Linquan Ma and Pham Hung Quy and in theorems 4, 6, and 7, which are independent ˆ ) = dim(R/P ˆ ) thesis work. We will say that an (R, m)-module M is quasi-unmixed if dim(M ˆ ) where − ˆ denotes m-adic completion. Study of the relationship between for all P ∈ min(M `(M/IM ) and eI (M ) first arose in the context of Buchsbaum modules and modules that are Cohen–Macaulay on the punctured spectrum and, separately, in generalizing Bezout’s theorem [Buc10, ATH00, Vog95, Tru86, conjectured by St¨ uckrad  STC78]. It was originally  `(M/IM ) √ and Vogel in 1996 [SV96] that sup | I = m < ∞ whenever M is quasieI (M ) unmixed, at which point they also showed that to be a necessary condition. In 2000, Allsop and Tuˆan Hoa proved the conjecture when dim(M ) ≤ 3 or M is Cohen–Macaulay on the punctured spectrum [ATH00]. Our result is the general case. Theorem 2. [KMQ] If Mis a quasi-unmixed module over the local ring (R, m), then  `(M/IM ) √ sup | I = m < ∞. eI (M ) Conversely, Lech’s Inequality [Lec60] states that for a local ring (R, m) and an m-primary ideal I, we have eI (R) ≤ d! · em (R) · `(R/I), which Theorem 3 extends to the module case, though at the cost of an explicit formula for the upper bound. Theorem 3. [KMQ] Let M be a module over the local ring (R, m). There exists a constant CM such that for all m-primary ideals I of R, eI (M ) ≤ CM · `(M/IM ). Armed with the information that eI (M ) and `(M/IM ) are commensurate and thinking in terms of Serre’s expansion of multiplicity in terms of Koszul homology, a third natural question is whether the higher Koszul homology of I on M must all be small or whether the lengths of homology modules may be arbitrarily large compared to the length of M/IM , in which case we would observe cancelation in the alternating sum giving the multiplicity. Theorem 4 answers this third question when I is a parameter ideal. Theorem 4. [Kle] Suppose that M is a d-dimensional module over the d-dimensional local   `(Hi (f1 , . . . , fd ; M )) p | (f1 , . . . , fd ) = m, 0 ≤ i ≤ d < ∞. ring (R, m). Then sup `(M/(f1 , . . . , fd )M ) In [NR54], Northcott and Rees show that whenever R has an infinite residue field, every m-primary ideal I has a parameter ideal J with the same integral closure, in which case 4

eI (R) = eJ (R). We call such an ideal J a minimal reduction of I. After passing to the case of an infinite residue field, the Northcott-Rees result and Theorem 4 combine with `(Hi (f1 , . . . , fd ; M )) is always bounded above independent of I Theorem 5 to show that `(M/IM ) and its minimal reduction J = (f1 , . . . , fd ). These results give an explanation of Lech’s Inequality for modules in terms of the Koszul homology of a minimal reduction of I. Theorem 5. [KMQ] Let M be   a quasi-unimxed module over the local ring (R, m). Then `(M/IM ) ¯ sup ¯ ) | I parameter < ∞ where I denotes the integral closure of I. `(M/IM Another section of my thesis is directed at giving explicit upper and lower bounds for the `(Hi (f ; M )) eI (M ) and upper bounds on that constitute an explanation for those ratio `(M/IM ) `(M/IM ) results. Theorems 6 and 7 are motivated by and partially achieve these goals. Theorem 6. [Kle] Suppose that M is a d-dimensional module over (R, m) of dimension d. Then eI (M ) ≤ d! · `(M/IM ) · em (R) for all m-primary ideals I of R provided that M = m or that M = A ⊆ R is generated by a regular sequence or that R is unramified regular. If R is a domain of characteristic p > 0, fix an algebraic closure F of the fraction field of R, and define R1/p := {s ∈ F | sp ∈ R} as a subring of F. Theorem 7. [Kle] If (R, m) is a local domain of dimension d and characteristic p > 0, then ( ) q `(H i (f ; R)) CN sup | f = (f1 , . . . , fd ), (f ) = m ≤ d , `(R/(f )) p (1 − 1/p) d

where CN is a particular invariant of the cokernel N of an embedding Rp ,→ R1/p . I also give a simpler proof of Lech’s Inequality in the case of rings, first by passing to the associated graded ring, which is equal characteristic, and then by using the technique of reduction to characteristic p > 0. These results bring me to the second research project I propose. Project 2. I propose to continue the search for explicit upper and lower bounds for `(Hi (f ; M )) eI (M ) and upper bounds on that depend only on features of the ring, `(M/IM ) `(M/(f )M )) such as its multiplicity, and of the module M , such as its dimension, rank, or minimal number of generators. My intention is to extend the techniques of my thesis in pursuit of this goal. For example, if the invariant CN of Theorem 7 can be shown to depend only on the dimension of N , then an inductive argument would yield a bound that is an explicit formula. If, furthermore, the constant CN depends on the characteristic only up to scaling by pd , then the bound will be of the right form to employ reduction to characteristic p > 0. In any cases where the bound is less than 1, that result would also yield a lower bound on eI (R) . `(R/I) 5

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